Reliability of Large Systems [1 ed.] 0080444296, 9780080444291, 9780080540498

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RELIABILITY OF LARGE SYSTEMS

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RELIABILITY OF LARGE SYSTEMS

Krzysztof Kolowrocki Gdynia Maritime University, Gdynia, Poland

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CONTENTS

P R E F A C E ............................................................................................................ NOTATIONS

ix

..................................................................................................... xvii

LIST OF F I G U R E S

............................................................................................ xxv

LIST OF T A B L E S .............................................................................................. xxix 1. B A S I C N O T I O N S .................................................................................................... 1 2. T W O - S T A T E S Y S T E M S ....................................................................................... 9 3. M U L T I - S T A T E S Y S T E M S

.................................................................................. 23

4. R E L I A B I L I T Y 4.1. Reliability 4.2. Reliability 4.3. Reliability 4.4. Reliability 4.5. Reliability

OF L A R G E T W O - S T A T E S Y S T E M S ........................................ evaluation o f two-state series systems ........................................... evaluation o f two-state parallel systems ........................................ evaluation o f two-state "m out of n" systems ............................... evaluation o f two-state series-parallel systems .............................. evaluation o f two-state parallel-series systems .............................

5. R E L I A B I L I T Y 5.1. Reliability 5.2. Reliability 5.3. Reliability 5.4. Reliability 5.5. Reliability

OF L A R G E M U L T I - S T A T E S Y S T E M S .................................... 87 evaluation of multi-state series systems ........................................ 88 evaluation o f multi-state parallel systems .................................... 106 evaluation o f multi-state "m out o f n" systems ............................ 117 evaluation o f multi-state series-parallel systems ......................... 125 evaluation of multi-state parallel-series systems ......................... 144

6. R E L I A B I L I T Y E V A L U A T I O N OF P O R T AND S H I P Y A R D T R A N S P O R T A T I O N S Y S T E M S ...................................................................... 6.1. Auxiliary results ............................................................................................ 6.2. Reliability o f a port grain transportation system .......................................... 6.3. Reliability o f a port oil transportation system ............................................... 6.4. Reliability o f a port bulk transportation system ............................................. 6.5. Reliability o f a shipyard rope transportation system ....................................

39 39 47 54 60 77

153

153 156 168 178 194

viii

Contents

7. RELIABILITY OF L A R G E MULTI-STATE E X P O N E N T I A L SYSTEMS ...... 7.1. Auxiliary theorems ........................................................................................ 7.2. Algorithms for reliability evaluation of multi-state exponential systems ..... 7.3. Algorithms application to reliability evaluation of exponential systems ......

211 211 219 232

8. R E L A T E D AND OPEN PROBLEMS ............................................................ 8.1. Domains of attraction for system limit reliability functions .......................... 8.2. Speed of convergence of system reliability function sequences ................... 8.3. Reliability of large series-"m out of n" systems ............................................ 8.4. Reliability of large "m out of n"-series systems ............................................ 8.5. Reliability of large hierarchical systems ....................................................... 8.6. Asymptotic approach to systems reliability improvement ............................ 8.7. Reliability of large systems in their operation processes ..............................

243 243 245 252 257 262 275 288

S U M M A R Y ...........................................................................................................

311

B I B L I O G R A P H Y ................................................................................................. 315 I N D E X .................................................................................................................

323

PREFACE

The book is concerned with the application of limit reliability functions to the reliability evaluation of large systems. Two-state and multi-state large systems composed of independent components are considered. The main emphasis is on multi-state systems with degrading (ageing) components because of the importance of such an approach in safety analysis, assessment and prediction, and analysing the effectiveness of operation processes of real technical systems. Many technical systems belong to the class of complex systems as a result of the large number of components they are built of and their complicated operating processes. This complexity very often causes evaluation of system reliability and safety to become difficult. As a rule these are series systems composed of large number of components. Sometimes the series systems have either components or subsystems reserved and then they become parallel-series or series-parallel reliability structures. We meet large series systems, for instance, in piping transportation of water, gas, oil and various chemical substances. Large systems of these kinds are also used in electrical energy distribution. A city bus transportation system composed of a number of communication lines each serviced by one bus may be a model series system, if we treat it as not failed, when all its lines are able to transport passengers. If the communication lines have at their disposal several buses we may consider it as either a parallel-series system or an "m out of n" system. The simplest example of a parallel system or an "m out of n" system may be an electrical cable composed of a number of wires, which are its basic components, whereas the transmitting electrical network may be either a parallel-series system or an "m out of n"-series system. Large systems of these types are also used in telecommunication, in rope transportation and in transport using belt conveyers and elevators. Rope transportation systems like port elevators and ship-rope elevators used in shipyards during ship docking and undocking are model examples of series-parallel and parallel-series systems. Taking into account the importance of the safety and operating process effectiveness of such systems it seems reasonable to expand the two-state approach to multi-state approach in their reliability analysis. The assumption that the systems are composed of multi-state components with reliability states degrading in time without repair gives the possibility for more precise analysis of their reliability, safety and operational processes' effectiveness. This assumption allows us to distinguish a system reliability critical state to exceed which is either dangerous for the environment or does not assure the necessary level of its operational process effectiveness. Then, an important system reliability characteristic is the time to the moment of exceeding the system reliability critical state and its distribution, which is called the system risk function. This

x

Preface

distribution is strictly related to the system multi-state reliability function that is a basic characteristic of the multi-state system. In the case of large systems, the determination of the exact reliability functions of the systems and the system risk functions leads us to very complicated formulae that are often useless for reliability practitioners. One of the important techniques in this situation is the asymptotic approach to system reliability evaluation. In this approach, instead of the preliminary complex formula for the system reliability function, after assuming that the number of system components tends to infinity and finding the limit reliability of the system, we obtain its simplified form. The mathematical methods used in the asymptotic approach to the system reliability analysis of large systems are based on limit theorems on order statistics distributions considered in very wide literature ([3], [9]-[11], [13], [20]-[21], [26]-[29], [33]-[36], [39]-[40], [84], [100], [106], [108], [112], [114]). These theorems have generated the investigation concerned with limit reliability functions of the systems composed of twostate components ([5], [7], [23]-[27], [41]-[43], [50]-[65], [71]-[72], [79]-[84], [94][95], [105], [109]-[111], [115]). The main and fundamental results on this subject that determine the three-element classes of limit reliability functions for homogeneous series systems and for homogeneous parallel systems have been established by Gniedenko in [36]. These results are also presented, sometimes with different proofs, for instance in subsequent works [7], [13], [21], [28], [56] and [71]. The generalisations of these results for homogeneous "m out of n" systems have been formulated and proved by Smirnow in [108], where the seven-element class of possible limit reliability functions for these systems has been fixed. Some partial results obtained by Smimow may be found in [71 ] and additionally with the solution of the speed of convergence problem in [29]. As it has been done for homogeneous series and parallel systems classes of limit reliability functions have been fixed by Chernoff and Teicher in [21] for homogeneous seriesparallel and parallel-series systems. Their results were concerned with so-called "quadratic" systems only. They have fixed limit reliability functions for the homogeneous series-parallel systems with the number of series subsystems equal to the number of components in these subsystems, and for the homogeneous parallel-series systems with the number of parallel subsystems equal to the number of components in these subsystems. These results may also be found for instance in later works [7] and [56]. All the results so far described have been obtained under the linear normalisation of the system lifetimes. Of course, there is a possibility to look for limit reliability functions of large systems under other than linear standardisation of their lifetimes. In this context, the results obtained by Pantcheva ([ 100]) and Cichocki ([25]) are exemplary. Pantcheva in [I00] has fixed the seven-element classes of limit reliability functions of homogeneous series and parallel systems under power standardisation for their lifetimes. Cichocki in [25] has generalised Pantcheva's results to hierarchical seriesparallel and parallel-series systems of any order. The book contains the results described above and their newest generalisations for large two-state systems and their developments for multi-state systems' asymptotic reliability analysis under the linear standardisation of the system lifetimes and the system sojourn times in the state subsets, respectively. Generalisations presented here of the results on limit reliability functions of two-state homogeneous series, and parallel systems for these systems in case they are nonhomogeneous, are mostly taken from [71] and [74]. A more general problem is

Preface

xi

concerned with fixing the classes of possible limit reliability functions for so-called "rectangular" series-parallel and parallel-series systems. This problem for homogeneous series-parallel and parallel-series systems of any shapes, with different number of subsystems and numbers of components in these subsystems, has been progressively solved in [53]-[56], [59] and [61]. The main and new result of these works was the determination of seven new limit reliability functions for homogeneous series-parallel systems as well as for parallel-series systems. This way, new ten-element classes of all possible limit reliability functions for these systems have been fixed. Moreover, in these works it has been pointed out that the type of the system limit reliability function strongly depends on the system shape. These results allow us to evaluate reliability characteristics of homogeneous series-parallel and parallel-series systems with regular reliability structures, i.e. systems composed of subsystems having the same numbers of components. The extensions of these results for non-homogeneous series-parallel and parallel-series systems have been formulated and proved successively in [56], [60]-[63] and [74]. These generalisations additionally allow us to evaluate reliability characteristics of the series-parallel and parallel-series systems with non-regular structures, i.e. systems with subsystems having different numbers of components. In some of the cited works, as well as the theoretical considerations and solutions, numerous practical applications of the asymptotic approach to real technical system reliability evaluation may also be found ([27], [41]-[43], [52], [57], [62], [64], [71][72], [109], [111], [115]). More general and practically important complex systems composed of multi-state and degrading in time components are considered among others in [1]-[2], [4]-[6], [8], [12], [14]-[19], [30]-[32], [38], [45]-[49], [65]-[71], [73]-[79], [83], [85]-[91], [93], [96][99], [101], [104], [107] and [116]-[119]. An especially important role they play in the evaluation of technical systems reliability and safety and their operating process effectiveness is defined in the book for large multi-state systems with degrading components. The most important results regarding generalisations of the results on limit reliability functions of two-state systems dependent on transferring them to series, parallel, "m out of n", series-parallel and parallel-series multi-state systems with degrading components are given in [65]-[77]. Some of these publications also contain practical applications of the asymptotic approach to the reliability evaluation of various technical systems ([65]-[71], [73]-[74], [76]-[77]). The results concerned with the asymptotic approach to system reliability analysis have become the basis for the investigation concerned with domains of attraction for the limit reliability functions of the considered systems ([23], [71]-[72], [81]-[82]). In a natural way they have led to investigation of the speed of convergence of the system reliability function sequences to their limit reliability functions ([71]). These results have also initiated the investigation of limit reliability functions of "m out of n"-series, series-"m out of n" systems ([23], [94]-[95]), and systems with hierarchical reliability structures ([23]-[25]), as well as investigations on the problems of the system reliability improvement and optimisation ([82]-[83]). The aim of the book is to deliver a complete elaboration of the state of art on the method of asymptotic approach to reliability evaluation for as wide as possible a range of large systems. Pointing out the possibility of this method's extensive practical application in the operating processes of these systems is also an important reason for this book. The book contains complete current theoretical results of the asymptotic approach to reliability evaluation of large two-state and multi-state series, parallel, "m out of n",

xii

Preface

series-parallel, and parallel-series systems together with their practical applications to the reliability evaluation of a wide range of technical systems. Additionally some recent partial results on the asymptotic approach to reliability evaluation of"m out of n"-series, series-"m out of n" and hierarchical systems, and their application to large systems reliability improvement and to large systems reliability analysis in their operation processes are presented in the book. The following construction of the book has been assumed. In chapters concerned with two-state systems the results and theorems are presented without the proofs but with exact reference to the literature where their proofs may be found. Moreover, the procedures of the results' practical applications are described and applied to the model two-state systems reliability evaluation. In chapters concerned with multi-state systems the recent theorems about their multi-state limit reliability functions are formulated and briefly justified. Next, the procedures of the result applications are presented and applied to real technical systems reliability and risk evaluation. Moreover, the possibility of the computer aided reliability evaluation of these systems is suggested and its use is presented. The book contains complete actual solutions of the formulated problems for the considered large systems reliability evaluation in the case of any reliability functions of the system components. The book consists of this Preface, eight chapters, Summary and Bibliography. In Chapter 1, some basic notions necessary for further considerations are introduced. The asymptotic approach to the system reliability investigation and the system limit reliability function are defined. In Chapter 2 two-state homogeneous and non-homogeneous series, parallel, "m out of n", series-parallel and parallel-series systems are def'med. Their exact reliability functions are also determined. Basic notions of the system multi-state reliability analysis are introduced in Chapter 3. Further the multi-state homogeneous and non-homogeneous series, parallel, "m out of n", series-parallel and parallel-series systems with degrading components are defined and their exact reliability functions are determined. Moreover, the notions of the multistate limit reliability function of the system, its risk function and other multi-state system reliability characteristics are introduced. Chapter 4 is concerned with limit reliability functions of two-state systems. Threeelement classes of limit reliability functions for homogeneous and non-homogeneous series systems are fixed. Some auxiliary theorems that allow us to justify facts on the methods of those systems' reliability evaluation are formulated and proved. The chapter also contains the application of one of the proven facts to the reliability evaluation of a non-homogeneous gas pipeline that is composed of components with Weibull reliability functions. The accuracy of this evaluation is also illustrated. Three-element classes of possible limit reliability functions for homogeneous and non-homogeneous parallel systems are fixed as well. Some auxiliary theorems that allow us to justify facts on the methods of these systems' reliability evaluation are formulated and proved. The chapter also contains the application of one proved fact to the reliability evaluation of a homogeneous energetic cable used in the overhead electrical energy distribution that is composed of components with Weibull reliability functions. The accuracy of this evaluation is illustrated in a table and a figure. The class of limit reliability functions for a homogeneous "m out of n" system is fixed and the "16 out of 35" lighting reliability is evaluated in this chapter. Chapter 4 contains also the results of investigations on limit

Preface

xiii

reliability functions of two-state homogeneous and non-homogeneous series-parallel systems. Apart from formulated and proved auxiliary theorems that allow us to justify facts on the methods of these systems' reliability evaluation, their ten-element classes of possible limit reliability functions are fixed. In this chapter, in the part concerned with applications there are two formulated and proved facts that determine limit reliability functions of series-parallel systems in the cases where they are composed of components having the same and different Weibull reliability functions. On the basis of those facts the reliability characteristics of a homogeneous gas pipeline composed of two lines of pipe segments and a non-homogeneous water supply system composed of three lines of pipe segments are evaluated. The results of investigations on limit reliability functions of two-state homogeneous and non-homogeneous parallel-series systems are given in this chapter as well. Theorems, which determine ten-element classes of possible limit reliability functions for these systems in the cases where they are composed of identical and different components, are formulated and justified. Moreover, some auxiliary theorems that are necessary in practical reliability evaluation of real technical systems are formulated and proved. In the part concerned with applications one fact is formulated and proved and then applied to evaluation of the reliability of a model homogeneous parallel-series system. Generalisations of the results of Chapter 4 on limit reliability functions of two-state systems consisting in their transferring to multi-state series, parallel, "m out of n", series-parallel and parallel-series systems are done in Chapter 5. The classes of all possible limit, reliability functions for these systems in the cases when they are composed of identical and different (in the reliability sense) components are fixed. The newest theorems that allow us to evaluate the reliability of large technical systems of those kinds are formulated and proved in Chapter 5 as well. Apart from the main theorems fixing the classes of multi-state limit reliability functions of the considered system, some auxiliary theorems and corollaries allowing their direct applications to reliability evaluation of real technical objects are also formulated and proved. Moreover, in this chapter there are wide applications depending on the results applying to the evaluation of reliability characteristics and risk functions of different multi-state transportation systems. The results concerned with multi-state series systems are applied to the reliability evaluation and risk function determination of homogeneous and nonhomogeneous pipeline transportation systems, a homogeneous model telecommunication network and a homogeneous bus transportation system. The results concerned with multi-state parallel systems are applied to reliability evaluation and risk function determination of an energetic cable used in an overhead electrical energy distribution network and to reliability and durability evaluation of a three-level steel rope used in rope transport. Results on limit reliability functions of a homogeneous multi-state "m out of n" system are applied to durability evaluation of a steel rope. A model homogeneous series-parallel system and homogeneous and non-homogeneous series-parallel pipeline systems composed of several lines of pipe segments are estimated as well. Moreover, the reliability evaluation of the model homogeneous parallel-series electrical energy distribution system is performed. Chapter 6 is devoted to the multi-state asymptotic reliability analysis of port and shipyard transportation systems. Theoretical results of this chapter and Chapter 5 are applied to the reliability evaluation and the risk functions determination of some selected port transportation systems. The results of the asymptotic approach to reliability evaluation of non-homogeneous multi-state series-parallel systems are

xiv

Preface

applied to the transportation system used in the Baltic Grain Terminal of the Port of Gdynia for transporting grain from its elevator to the rail carriages. The results of the asymptotic approach to the reliability evaluation of the non-homogeneous multi-state series-parallel systems are applied to the piping transportation system used in the Oil Terminal in Debogorze. This transportation system is set up to take the oil from the tankers that deliver it to the unloading pier located at the breakwater of the Port of Gdynia. The results of the asymptotic approach to reliability evaluation of nonhomogeneous multi-state series-parallel and series systems are applied to the transportation system used in the Baltic Bulk Terminal of the Port of Gdynia for loading bulk cargo on the ships. The results of this chapter and Chapter 5 are also applied to reliability evaluation and risk function determination of the shipyard transportation system. Namely, the results of the asymptotic approach to reliability evaluation of homogeneous multi-state parallel-series systems are applied to the ship-rope transportation system used in the Naval Shipyard of Gdynia for docking ships coming for repair. The reliability analysis performed on the considered systems in this chapter is based on the data concerned with the operation processes and reliability of their components coming from experts, from component technical norms and from their producer's certificates. In Chapter 7 the classes of possible limit reliability functions are fixed for the considered systems in the case where their components have exponential reliability functions. Theoretical results are represented in the form of a very useful guide containing algorithms placed in tables and giving sequential steps for proceeding in the reliability evaluation in each of the possible cases of the considered system shapes. The application of these algorithms for reliability evaluation of the multi-state nonhomogeneous series transportation system, the multi-state model homogeneous seriesparallel, the multi-state non-homogeneous series-parallel pipeline transportation system and the multi-state non-homogeneous parallel-series bus transportation system is illustrated. The evaluation of reliability functions, risk functions, mean values of sojourn times in subsets of states and mean values of sojourn times in particular states for these systems is carried out. The calculations are performed using a computer program based on the algorithms, so allowing automatic evaluation of the reliability of large real technical systems. In Chapter 8 the open problems related to the topics considered in the book are presented. The domains of attraction for previously fixed limit reliability functions of the series, parallel, "m out of n", series-parallel and parallel-series systems are introduced. More exactly, there are formulated theorems giving conditions which reliability functions of the components of the system have to satisfy in order that the system limit reliability function is one of the functions from the system class of all limit reliability functions. Some examples of the result application for series systems are also illustrated. The practically very important problem of the speed of convergence of system reliability function sequences to their limit reliability functions is investigated as well. An exemplary theorem is presented, which allows the differences between the system limit reliability functions and the members of their reliability function sequences to be estimated. Next, an example of the speed of convergence evaluations of reliability function sequences for a homogeneous series-parallel system is given. Partial results of the investigation on the asymptotic approach to reliability evaluation of "m out of n"series, series-"m out of n" and hierarchical systems and on system reliability improvement are presented, These result applications are illustrated graphically as well.

Preface

xv

The analysis of large systems' reliability in their operation processes is given at the end of Chapter 8. The book is completed by the Summary that contains the evaluation of the presented results, the formulation of open problems concerned with large systems' reliability and the perspective of further investigations on the considered problems.

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NOTATIONS

Ei

- c o m p o n e n t s of series, parallel and "m out of n" systems

EU

- c o m p o n e n t s of series-parallel and parallel-series systems

-component lifetimes of two-state series, parallel and "m out of n" systems -component lifetimes of two-state series-parallel and parallel-series systems

T R(t) F(t)

- a two-state system lifetime

RCi)(t)

-component reliability functions of two-state non-homogeneous series, parallel and "m out of n" systems

F (i) (t)

-component lifetime distribution functions of two-state homogeneous series, parallel and "m out of n" systems

R(i,J)(t)

-component reliability functions of two-state non-homogeneous series-parallel and parallel-series systems

FCi,J)(t)

- a component reliability function of a two-state homogeneous system - a component lifetime homogeneous system

distribution

function

of

a

two-state

-component lifetime distribution functions of two-state homogeneous series-parallel and parallel-series systems

non-

non-

n.(t)

- a reliability function of a two-state homogeneous series system

R n(t)

- a reliability function of a two-state non-homogeneous series system

R.(t) R'. (t)

- a reliability function of a two-state homogeneous parallel system

R(m)(t)

- a reliability function of a two-state homogeneous "m out of n" system

R,(m) . (t)

- a reliability function of a two-state non-homogeneous "m out of n"

Rk.t (t)

system - a reliability function of a two-state homogeneous parallel-series

- a reliability function of a. two-state non-homogeneous parallel system

system

Notations

xviii l!

R k,,t. ( t )

- a reliability function of a two-state non-homogeneous parallel-series

Rk.~. (t)

system - a reliability function of a two-state homogeneous series-parallel

R'k.l. (t)

system - a reliability function of a two-state non-homogeneous series-parallel system

.9l ( t )

- a limit reliability function of two-state homogeneous series and parallel-series systems

R

91' (t)

gt(t) .~"(t)

- a limit reliability function of two-state non-homogeneous series and parallel-series systems - a limit reliability function of two-state homogeneous parallel and series-parallel systems - a limit reliability function of two-state non-homogeneous parallel and series-parallel systems

.~'(~

- a limit reliability function of a two-state homogeneous "m out of n" system

9i'(') (t)

- a limit reliability function of a two-state homogeneous "m out of n" system

~o>(t,.) E(T) ~(7) z

ri(u)

ro.(u)

- a limit reliability function of a two-state homogeneous "m out of n" system - a mean lifetime of a two-state system - a lifetime standard deviation of a two-state system number of reliability states of a multi-state component and a multistate system -multi-state component lifetimes of series, parallel and "m out of n" systems in a state subset - m u l t i - s t a t e component lifetimes of series-parallel and parallel-series

-a

systems in a state subset a multi-state system lifetime in a state subset

7(u)

-

R(t,.)

-a

F(t,.)

system - a multi-state component lifetime distribution homogeneous system in a state subset

multi-state component reliability function of a homogeneous function of a

R(i)(t,.)

-multi-state component reliability functions of homogeneous series, parallel and "m out of n" systems

F(i)(t,.)

- multi-state

component

lifetime

distribution

functions

of

homogeneous series, parallel and "m out of n" systems in a state subset

Notations

xix

R(i'J)(t,.)

-multi-state component reliability functions of homogeneous seriesparallel and parallel-series systems

F(i'J)(t,.)

- multi-state component lifetime distribution functions of homogeneous series-parallel and parallel-series systems in a state subset - a reliability function of a multi-state homogeneous series system

m

Rn(t,') R' (t,.) tl

Rn(t,') R' (t,.) n

- a reliability function of a multi-state non-homogeneous series system - a reliability function of a multi-state homogeneous parallel system - a reliability function of a multi-state non-homogeneous parallel system - a reliability function of a multi,state homogeneous "m out of n" system - a reliability function of a multi-state homogeneous "m out of n" system

R'T ) (t,.)

- a reliability function of a multi-state non-homogeneous "m out of n" system

(t,.) Rkn,l ~ (t,')

- a reliability function of a multi-state non-homogeneous "m out of n" system - a reliability function of a multi-state homogeneous parallel-series

R k.,t. (t,.)

system - a reliability function of a multi-state non-homogeneous parallel-

w

series system - a reliability function of a multi-state homogeneous series-parallel

R'k.,l ~ (t,')

system - a reliability function of a multi-state non-homogeneous seriesparallel system

m

.91(t,.)

- a limit reliability function of multi-state homogeneous series and parallel-series systems

!

.ql' (t,.) .91(t,.) .~i"(t,.) .~(~ (t,.)

- a limit reliability function of multi-state non-homogeneous series and parallel-series systems - a limit reliability function of multi-state homogeneous parallel and series-parallel systems - a limit reliability function of multi-state non-homogeneous parallel and series-parallel systems - a limit reliability function of a multi-state homogeneous "m out of n" system

Notations

xx

.~'(') (t,.)

- a limit reliability function of a multi-state homogeneous "m out of n" system

.~0) (t,.)

a limit reliability function of a multi-state homogeneous "m out of n" system - a critical reliability state of a system - a risk function of a multi-state system

-

r

r(t) Mi(u)

-a

multi-state component mean lifetime in a state subset

O'i(U )

-a

multi-state component lifetime standard deviation in a state subset

Mi(u)

-a

multi-state component mean lifetime in a state

M(u) o-(u)

a multi-state system mean lifetime in a state subset

-

- a multi-state system lifetime standard deviation in a state subset

M(u) 8

a multi-state system mean lifetime in a state

-

- a permitted level of a multi-state system risk function - a moment of exceeding a permitted multi-state system risk level

"t"

-domains of attraction of limit reliability functions .9i',(t) of two-state homogeneous series system

R kn(m),l1,12 ..... lk n (t) - a

reliability

function

of

a

homogeneous

two-state

series-

of

a

homogeneous

two-state

series-

"m out of kn" system (~)

k n ,11 ,l 2 ..... lk n

(t)-a

reliability

function

"m out of k." system R (m) ( t ) kn ,In

- a reliability function of a homogeneous and regular two-state series"m out of k." system

kn ,In

- a reliability function of a homogeneous and regular two-state series"m out of k~" system

( m l , m 2 . . . . . mk n ) R k n ,l 1,12 . . . . . lk n (t)

- a reliability function of a two-state " m i out of I i "-series system

mk n ) (t) n ,11,12..... lk n

- a reliability function of a two-state "m; out of l; "-series system

'k(ml ,m2 .....

e (m) ( t ) kn ,In

-a

reliability function of a homogeneous and regular two-state

"m out of k."-series system

-R(~) (t) kn ,In

-a

reliability function of a homogeneous and regular two-state

"m out of k,"-series system

.~m) (t)

- a limit reliability function of a homogeneous and regular two-state series-"m out of k." system - a limit reliability function of a homogeneous and regular two-state series-"m out of k," system

Notations ff~'~m) (t)

-a

xxi

limit reliability function of a homogeneous and regular two-state "m out of kn"-series system limit reliability function of a homogeneous and regular two-state "m out of kn"-series system

~ i (~ ) ( t )

-a

Rr,kn,l n (t)

- a reliability function of a two-state series-parallel system of order r

.91i (t)

limit reliability function of a two-state series-parallel system of

-a

order r m

Rr,kn ,ln ( t )

- a reliability function of a two-state parallel-series system of order r

.~l, (t)

- a limit reliability function of a two-state parallel-series system of

P

-

R(nO (t)

- a reliability function of a two-state series system with components

order r a factor reducing a component failure rate

improved by reducing their failure rates by a factor p -a

-a

reliability function of a two-state series system with a single hot reservation of its components reliability function of a two-state series system with a single cold reservation of its components

- a reliability function of a two-state series system with a single mixed reservation of its components - a reliability function of a two-state series system with its single hot reservation

R~6) (t)

- a reliability function of a two-state series system with its single cold .reservation

(t)

-a

limit reliability function of a two-state series system with

components improved by reducing their failure rates by a factor p g/(2) (t)

-a

limit reliability function of a two-state series system with a single hot reservation of its components

.gi'(3) (t)

- a limit reliability function of a two-state series system with a single cold reservation of its components

.~'(4) (t)

- a limit reliability function of a two-state series system with a single mixed reservation of its components

9i'(s) (t)

-

a limit reliability function of a two-state series system with its single hot reservation

.gi'(6) (t)

- a limit reliability function of a two-state series system with its single cold reservation

Notations

xxii T (1)

-a

lifetime mean value of a two-state series system with components improved by reducing their failure rates by a factor ,o

T (2)

-a

T (3)

- a lifetime mean value of a two-state series system with a single cold

lifetime mean value of a two-state series system with a single hot reservation of its components reservation of its components

T (4)

- a lifetime mean value of a two-state series system with a single mixed reservation of its components

T (5)

- a lifetime mean value of a two-state series system with its single hot reservation

T (6)

-a

z k

-a

system operational state

Z(t) okt

-a

process of changing system operational states

[H u (t)] vxv

- a matrix of conditional distribution functions of sojourn times 0 kt

[pk (0)],~

- a vector of probabilities of process Z(t) initial states

E[O hi ]

- m e a n values of sojourn times 0 kt

0k

-unconditional sojourn times of process Z(t) at states z h

Hk(t)

-unconditional distribution functions of sojourn times 0 k

E[Oh]

-mean

values of unconditional sojourn times 0 h

Mk

-mean

values of unconditional sojourn times 0 h

pk(t)

-transient

lifetime mean value of a two-state series system with its single cold reservation.

P

k

-

R(h)(t)

sojourn times of a process Z(t) at operational states

-conditional

probabilities of process Z(t) at states z k

limit values of transient probabilities p k (t)

-conditional reliability functions of a two-state system at operational states zh

R(t) ,,

- an unconditional reliability function of a two-state system -conditional

lifetimes

homogeneous states z h

of system components

two-state

series-parallel

system

E;j at

of a nonoperational

Notations [R(i,J)(t)] (k)

-conditional reliability functions of system components E,y of a nonhomogeneous states zk

T(k)

xxiii

two-state

series-parallel

system

at

operational

-conditional lifetimes of a non-homogeneous two-state series-parallel system at operational states z k

R(k.,t. (t) k).

-conditional reliability functions of a non-homogeneous two-state series-parallel system at operational states z k

R(t) m

O"

2

- a n unconditional lifetime of a non-homogeneous two-state seriesparallel system -unconditional reliability functions of a non-homogeneous two-state series-parallel system - a n unconditional mean value of a non-homogeneous two-state series-parallel system lifetime unconditional variance of a non-homogeneous two-state seriesparallel system lifetime

- a n

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LIST OF FIGURES

Fig. 2.1. Fig. 2.2. Fig. 2.3. Fig. 2.4. Fig. 2.5. Fig. 2.6. Fig. 2.7. Fig. 2.8. Fig. 2.9. Fig. 2.10. Fig. 2.11. Fig. 2.12. Fig. 3.1. Fig. 4.1. Fig. 4.2. Fig. 4.3. Fig. 4.4. Fig. 4.5. Fig. 4.6. Fig. 4.7. Fig. 4.8. Fig. 5.1. Fig. 5.2. Fig. Fig. Fig. Fig. Fig.

5.3. 5.4. 5.5. 5.6. 5.7.

Fig. 5.8.

The scheme of a series system The scheme of a parallel system The scheme of an "m out of n" system The scheme of a series-parallel system The scheme of a regular series-parallel system The scheme of a parallel-series system The scheme of a regular parallel-series system The scheme of a non-homogeneous series system The scheme of a non-homogeneous parallel system The scheme of a non-homogeneous "m out of n" system The scheme of a regular non-homogeneous series-parallel system The scheme of a regular non-homogeneous parallel-series system Illustration of states changing in system with ageing components The graphs of the exact and limit reliability functions of the gas piping system The cross-section of the energetic cable The graphs of the exact and approximate reliability functions of the energetic cable The graphs of the exact and approximate reliability function of the lighting system The graphs of the reliability function of the gas distribution system The model of a non-homogeneous regular series-parallel water supply system The graphs of the exact and approximate reliability functions of the water supply system The graphs of the exact and approximate reliability functions of the model parallel-series system The graphs of the piping system reliability function and risk function The graphs of multi-state reliability function and risk function of the piping system The graphs of the energetic cable reliability function and risk function The steel rope M-80-200-10 cross-section The graphs of the rope multi-state reliability function and risk function The graphs of the still rope multi-state reliability function and risk function The graph of the component u = 2 of the exact and approximate piping system reliability function The graph of the piping system risk function

List of Figures

xxvi Fig. 6.1. Fig. 6.2. Fig. 6.3. Fig. 6.4. Fig. 6.5. Fig. 6.6. Fig. 6.7. Fig. 6.8. Fig. 6.9. Fig. 6.10. Fig. 6.11. Fig. 6.12. Fig. 7.1. Fig. 7.2. Fig. 7.3. Fig. 8.1. Fig. 8.2. Fig. 8.3. Fig. 8.4. Fig. 8.5. Fig. 8.6. Fig. Fig. Fig. Fig. Fig.

8.7. 8.8. 8.9. 8.10. 8.11.

The scheme of the grain transportation system The graphs of the components of multi-state reliability functions and the risk function of the port grain transportation system The scheme of the oil transportation system The graphs of the multi-state reliability functions and the risk function of the port oil transportation system The scheme of the bulk cargo transportation system Graphs of the multi-state reliability function and the risk function of the port bulk cargo transportation system The scheme of the ship-rope transportation system The cross-section of the rope Graphs of the rope elevator exact and approximate reliability functions in the state subset u > 1 Graphs of the rope elevator exact and approximate reliability functions in the state subset u > 2 Graphs of the rope elevator exact and approximate reliability functions in the state u = 3 Graphs of the approximate rope elevator risk functions Graphs of the multi-state reliability function and the risk function of the piping system The graphs of the multi-state reliability function of the piping system and its risk function Graphs of the multi-state reliability function and the risk function of the bus transportation system The graphs of the limit reliability function and their lower and upper evaluations for a homogeneous series-parallel system (k, = 10, l, =4) The graphs of the limit reliability function and their lower and upper evaluations for a homogeneous series-parallel system (k, = 50, l, =4) The graphs of the limit reliability function and their lower and upper evaluations for a homogeneous series-parallel system (k, = 100, l, =4) The scheme of a series-"m out of k," system The scheme of a regular series-"m out of k," system The graphs of the reliability functions R(5) "'30,10 (t) and .~9O) (10 -1 x/3t)

The scheme of an "m~ out of/~"-series system The scheme of a regular "m out of/."-series system The scheme of a series-parallel system of order 1 The scheme of a series-parallel system of order 2 Graphs of exact and approximate reliability functions of a hierarchical regular series-parallel homogeneous system of order 2 Fig. 8.12. Graphs of exact and approximate reliability functions of a hierarchical regular parallel-series homogeneous system of order 2 Fig. 8.13. The scheme of a series system

List of Figures Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

8.14. 8.15. 8.16. 8.17. 8.18. 8.19. 8.20. 8.21.

The The The The The The The The

scheme scheme scheme scheme scheme scheme scheme scheme

xxvii

of a series system with components having hot reservation of a series system with components having cold reservation of a series system with components having mixed reservation of a series system with hot reservation of a series system with cold reservation of the grain transportation system structure at operation state 1 of the grain transportation system structure at operation state 2 of the grain transportation system structure at operation state 3

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LIST OF TABLES

Table 4.1. Table 4.2. Table 4.3. Table 4.4. Table 4.5. Table 4.6. Table 5.1. Table 5.2. Table 5.3. Table 5.4. Table 6.1. Table 6.2. Table 6.3. Table 6.4. Table 6.5. Table 6.6. Table 6.7. Table Table Table Table Table Table

6.8. 7.1. 7.2. 7.3. 7.4. 7.5.

The values and differences between the exact and limit reliability functions of the gas piping system The values of the exact and approximate reliability functions of the energetic cable The values of the exact and approximate reliability function of the lighting system The behaviour of the exact and approximate reliability function of the gas distribution system The behaviour of the exact and approximate reliability functions of the water supply system The values of the exact and approximate reliability functions of the model parallel-series system The values of the energetic cable risk function The values of the still rope multi-state reliability function and risk function The values of the component u = 2 of the exact and approximate piping system reliability function The values of the piping system risk function The values of the components of multi-state reliability functions and the risk function of the port grain transportation system The values of the multi-state reliability functions components and the risk function of the port oil transportation system The values of the multi-state reliability function and the risk function of the port bulk transportation system The rope elevator loading states characteristics The values of the rope elevator exact and approximate reliability functions in the state subset u > 1 The values of the rope elevator exact and approximate reliability functions in the state subset u > 2 The values of the rope elevator exact and approximate reliability functions in the state u = 3 The approximate values of the rope elevator risk functions Algorithm of reliability evaluation of a series system Algorithm of reliability evaluation of a parallel system Algorithm of reliability evaluation of an "m out of n" system Algorithm of reliability evaluation of a series-parallel system Algorithm of reliability evaluation of a parallel-series system

xxx Table 7.6. Table 7.7. Table 7.8. Table 7.9. Table 7.10. Table 7.11. Table 7.12. Table8.1. Table 8.2. Table 8.3. Table 8.4. Table 8.5.

List of Tables

Reliability evaluation of the piping system The values of the multi-state reliability function and the risk function of the piping system Reliability evaluation of the homogeneous parallel system Reliability evaluation of the piping system The values of the piping system multi-state reliability function and its risk function Reliability evaluation of the bus transportation system Values of the multi-state reliability function and the risk function of the bus transportation system The evaluation of the speed of convergence of reliability function sequences for a homogeneous series-parallel system (kn = 10, In =4) The evaluation of the speed of convergence of reliability function sequences for a homogeneous series-parallel system (kn = 50, I, =4) The evaluation of the speed of convergence of reliability function sequences for a homogeneous series-parallel system (kn = 100, In =4) Values of exact and approximate reliability functions of a hierarchical regular series-parallel homogeneous system of order 2 Values of exact and approximate reliability functions of a hierarchical regular parallel-series homogeneous system of order 2

CHAPTER 1

BASIC NOTIONS Basic notions and agreements, which are necessary to further considerations, are introduced. The asymptotic approach to the system reliability investigation and the system limit reliability function is defined

Considering the reliability of two-state systems we assume that the distributions of the component and the system lifetimes T do not necessarily have to be concentrated in the interval t), t ~ (-oo, oo),

does not have to satisfy the usually demanded condition R(t) = 1 for t ~ (-oo,0).

This is a generalisation of the normally used concept of a reliability function. This generalisation is convenient in the theoretical considerations. At the same time, from the achieved results on the generalised reliability functions, for particular cases, the same properties of the normally used reliability functions appear. From that assumption it follows that between a reliability function R(t) and a distribution function F(t) = P(T _t o .

Corollary 1.2 A function R(t) = 1 - e x p [ - V ( t ) ] ,

t ~ ( - ~ , oo),

is a reliability function if and only if a function V(t) is non-negative, non-increasing, right continuous, v( - ~ ) = 0% v ( + ~ ) = 0

and moreover V(t) can be identically equal to oo in an interval.

Corollary 1.3 A function m

R(t) = exp[-V (t)], t ~ (-0% ~ ) , m

is a reliability function if and only if a function V(t) is non-negative, non-decreasing, right continuous, m

u

v ( - o o ) = o, v ( + o o ) = ~ , m

and moreover V (t) can be identically equal to oo in an interval.

Corollary 1.4 A function m-I [V(t)] i , (-oo, N, R (~ (t) = 1 - Y, . . . . exp[-V(t)] t ~ oo), m E /=0 i! is a reliability function if and only if a function V(t) is non-negative, non-increasing, fight continuous,

Chapter 1 v(-~)

3

= ~ , v(+oo) = 0,

and moreover V(t) can be identically equal to oo in an interval. Corollary 1.5 A function 1

R (~) (t) = 1 -

X2 -ool e 2 dx, t ~(-~,r

-v(t)

0 < / 1 t ) = P(T >ant +bn) =Rn(ant + bn),

where Rn(t) is a reliability function of a system composed of n components, then the following definition becomes natural. D e f i n i t i o n 1.5

A reliability function .q/(t) is called a limit reliability function or an asymptotic reliability function of a system having a reliability function R,(t) if there exist normalising constants a, > 0, b, ~ (-0% oo) such that lim

R.(a.t +

b.) = 9?(0 for t ~ C~t.

n--~oo

Thus, if the asymptotic reliability function .ql(t) of a system is known, then for sufficiently large n, the approximate formula Rn(t) --- ~ ( ( t - bn)/an), t ~ (-oo, ~).

(1.1)

may be used instead of the system exact reliability function Rn(t). From the condition lim

R.(a.t +

b.) = .q?(t) for t ~ C~,

n--~oo

it follows that setting an = aa., fin = ban + b.,

where a > 0 and b ~ (-oo, oo), we get lira R . ( a . t + ft.) = lim R.(a.(at + b) + b.) = .qI(at + b) for t e Cgt. n-~Qo

n--~oo

Hence, if 97(0 is the limit reliability function of a system, then iR(at + b) with arbitrary a > 0 and b ~ (-0% oo) is also its limit reliability function. That fact, in a natural way, yields the concept of a type of limit reliability function.

Chapter 1

7

Definition 1.6 The limit reliability functions .q?o(t) and .qiP(t) are said to be of the same type if there exist numbers a > 0 and b e ( - ~ , Qo) such that

.9lo(t) = .9l(at + b) for t e (-0% oo). Agreement 1.2 In further considerations we assume the following notations:

x(n) r

x(n) ~ y(n) or x ( n ) = r( y ( n) ) , where x(n) and y(n) are either positive or negative functions, means that x(n) is of order y(n) in a sense lim x(n) / y(n) = 1,

x(n) >> y(n) or x(n) = O ( y ( n ) ) , where x(n) and y(n) are positive functions, means that x(n) is of order much greater than y(n) in a sense lim x(n) / y(n) = oo. n-.-~ oo

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CHAPTER 2

TWO-STATE SYSTEMS Two-state homogeneous and non-homogeneous series, parallel "m out o f n" seriesparallel and parallel-series systems are defined. Their exact reliability functions are determined.

We assume that

El, i = 1,2,...,n, n ~ N, are two-state components of the system having reliability functions

Ri(t) = l~ Ti > t), t e (-0% ~), where T/, i = 1,2,...,n, are independent random variables representing the lifetimes of components E~ with distribution functions

Fi(t) = P(T~ u l e,(O) = z) = P(Ti(u) > t), t ~ (-0% oo), u = 0,1 .... ,z,

is the probability that the c o m p o n e n t E~ is in the state subset

{ u , u + 1,...,z}

(3.1) at the

m o m e n t t, t ~ (-0% oo), while it was in the state z at the m o m e n t t = 0, is called the multi-state reliability function o f a c o m p o n e n t E~. U n d e r this definition we have R~(t,O) > R~(t,1) > _ . . . > R,(t,z), t ~ (-0% oo), i = 1,2 .... ,n.

Further, if we d e n o t e by p,(t,u) = P(e,(t) = u[ e,(0) = z), t ~ (-0% oo), u = 0,1,...,z,

the probability that the c o m p o n e n t E~ is in the state u at the m o m e n t t, while it was in the state z at the m o m e n t t = 0, then by (3.1) R~(t,O) = 1, R~(t,z) = p~(t,z), t ~ (-oo, oo), i = 1,2,...,n,

(3.2)

P i (t, u) = R i (t, u) - R i (t, u + 1), u = 0,1,..., z - 1, t ~ (-0% oo), i = 1,2 .... ,n.

(3.3)

and

Chapter 3

25

Moreover, if R~ (t, u) = 1 for t < 0, u = 1,2,...,z, i = 1,2 .... ,n, then oo

Mi(u) = ~ R i (t, u)dt, u = 1,2 .... ,z, i = 1,2 ..... n,

(3.4)

o

is the mean lifetime o f the component E~ in the state subset {u, u + 1,..., z}, dr, (u) = ~ N I (u) - [ M , (u)] 2 , u = 1,2,...,z, i = 1,2,...,n,

(3.5)

where co

N i (u) = 2 ~ tR i (t, u ) d t , u = 1,2 .... ,z, i = 1,2 .... ,n,

(3.6)

o

is the standard deviation o f the component E, lifetime in the state subset {u, u + 1,..., z} and m

oo

Mi(u) = ~ Pi (t, u)dt, u = 1,2,...,z, i = 1,2 ..... n,

(3.7)

o

is the mean lifetime o f the component E~ in the state u, in the case w h e n the integrals defined by (3.4), (3.6) and (3.7) are convergent. Next, according to (3.2), (3.3), ( 3 . 4 ) a n d (3.7), we have m

m

Mi(u)=Mi(u)-Mi(u+l

), u = 0 , 1 , . . . , z - 1 ,

Mi(z)=M~(z), i = 1,2,...,n.

(3.8)

D e f i n i t i o n 3.2

A vector

Rn(t ," ) : [Rn(t,O),Rn(t, 1), .... R.(t,z)], t ~ (-oo, oo), where

Rn(t,u) = P(s(t) >_u Js(O ) = z) = P(T(u) > t), t ~ (-oo,oo), u = O, 1,...,z, is the probability

(3.9)

that the system is in the state subset {u, u + 1,..., z} at the m o m e n t t,

t e (-oo, oo), while it was in the state z at the m o m e n t t = O, is called the multi-state reliability function o f a system.

Multi-State Systems

26 Un der this definition we have

R.(t,O) > R.(t, 1) > . . .

> R.(t,z), t ~ ( - ~ , ~),

and if

p(t,u) = P(s(t) = u Is(O) = z), t ~ (-0% ~ ) , u = 0,1 .... ,z,

(3.10)

is the probability that the system is in the state u at the moment t, t E (-m, m), while it was in the state z at the m o m e n t t = O, then R.(t,0) = 1, R.(t,z) = p(t,z), t ~ (-oo, oo),

(3.11)

p(t,u) = R.(t,u) - R . (t, u + 1), u = 0,1,..., z - 1, t ~ (-oo, oo).

(3.12)

and

Moreover, if

R.(t,u) = 1 for t _< 0, u = 1,2 ..... z, then oo

m(u) = f R,(t,u)dt,

u = 1,2,...,z,

(3.13)

o

is the m e a n lifetime o f the system in the state subset {u, u + 1.... , z},

or(u) = 4 N ( u ) - [ M ( u ) ] 2

, u = 1,2,...,z,

(3.14)

where oo

N ( u ) = 2 ~ t R,(t,u)dt, u = 1,2,...,z,

(3.15)

o

is the standard deviation o f the system sojourn time in the state subset {u, u + 1,..., z} and m o r e o v e r m

oo

M(u) = ~ p(t, u)dt, u = 1,2, .... z,

(3.16)

0

is the m e a n lifetime o f the system in the state u while the integrals (3.13), (3.14) and (3.15) are convergent.

Chapter 3

27

Additionally, according to (3.11), (3.12), (3.13)and (3.16), we get the following relationship M(u)=M(u)-M(u+l),

u=O,1,...,z-1,

M(z)=M(z).

(3.17)

D e f i n i t i o n 3.3 A probability

r(t) = P(s(t) < r I s(0) = z) = P(T(r) < t), t ~ (-oo, oo), that the system is in the subset of states worse than the critical state r, r e { 1, .... z} while it was in the state z at the moment t = 0 is called a risk function of the multi-state system or, in short, a risk. Under this definition, from (3.1), we have r(t) = 1 - P(s(t) >_r I s(0) = z) = 1 - R,(t,r), t ~ (-oo, oo).

(3.18)

and if r is the moment when the risk exceeds a permitted level 6, then r "- r-1

(6),

(3.19)

where r - l (t), if it exists, is the inverse function of the risk function r(t). D e f i n i t i o n 3.4 A multi-state system is called series if its lifetime T(u) in the state subset {u, u + 1.... , z}

is given by

T(u) =

min {T, ( u ) } , u = 1,2,...,z.

The above definition means that a multi-state series system is in the state subset {u, u + 1,..., z} if and only if all its components are in this subset of states. It is easy to work out that the reliability function of the multi-state series system is given by m

m

a

R n (t,.) = [ 1, R n (t,1) ,..., R n (t, z) ], where tl

R n (t, u) = I-I Ri (t, u ) , t ~ (-0% oo), u = 1,2,...,z. i=l

28

Multi-State Systems

Definition 3.5

A multi-state series system is called homogeneous if its component lifetimes T~(u) in the state subsets have an identical distribution function F,(t,u) = F(t,u), u = 1,2 ..... z, t ~ (-co, oo), i = 1,2 ..... n,

i.e. if its components Ej have the same reliability function R~(t,u) = R ( t , u ) = 1 - F ( t , u ) ,

u = 1,2,...,z, t e (-0% oo), i = 1,2,...,n.

The reliability function of the homogeneous multi-state series system is given by m

m

m

R n (t,-) = [1, R n(t,1), .... R n (t, z) ],

(3.20)

where m

R n (t, u) = [R(t,u)]", t ~ (-oo, oo), u = 1,2,...,z.

(3.21)

Definition 3.6

A multi-state system is called parallel if its lifetime T(u) in the state subset {u, u + 1,..., z} is given by T(u) = m a x { T / ( u ) } , u = 1,2,...,z. l O, b,(u) ~ (-0% oo), u = 1,2,...,z,

are some suitably chosen numbers, called normalising constants. And, since P ( ( T ( u ) - b n (u)) / a n (u) > t) = P(T(u) > an(U)t + bn(u)) = R.(a.(u)t + b.(u),u), u = 1,2 ..... z,

where Rn(t ," ) = [Rn(t,O),R.(t, 1),...,R.(t,z)], t ~ ( - ~ , oo),

is the multi-state reliability function of the system composed of n components, then we assume the following definition.

Definition 3.22 A vector .qi~(t,. ) = [1,.qf(t, 1),...,.q/(t,z)], t ~ (-oo, oo), is called the limit multi-state reliability function of the system with reliability function R.(t,. ) if there exist normalising constants a.(u) > O, b.(u) ~ (-0% oo) such that

Chapter 3

37

lim R,,(a.(u)t + b.(u),u)= 9f(t,u) for t ~ C~.), u = 1,2 ..... z,

n-.~oo

where C~u) is the set of continuity points of gl(t,u). Knowing the system limit reliability function allows us, for sufficiently large n, to apply the following approximate formula

t-b.(u) ,. ), a.(u)

R.(t,.)---~~ i.e.

[ 1,R,(t, 1),...,Rn(t,z)] ~ [ 1,.q/(

t - bn(1) an(l)

t-bn(z) ,z)], t ~ (-oo, ~). ,1) ..... ~ ( ~

a.(z)

(3.49)

This Page Intentionally Left Blank

CHAPTER 4

RELIABILITY OF LARGE TWO-STATE SYSTEMS Auxiliary theorems on limit reliability functions of large two-state systems, which are necessary for their approximate reliability evaluation, are formulated. The classes of limit reliability functions for homogeneous and non-homogeneous series, parallel, series-parallel and parallel-series systems and for a homogeneous "m out of n" system are fixed. Applications of the asymptotic approach to reliability evaluations of model systems are presented. Six corollaries are formulated and proved on the basis of the auxiliary theorems and applied to finding limit reliability functions of the considered systems and approximate evaluations of their reliability functions, lifetime mean values and lifetime standard deviations. The reliability evaluation is done for the following systems: the model non-homogeneous series system, the homogeneous parallel system of an energetic cable, the "16 out of 35" lighting system, the homogeneous regular series-parallel gas distribution system, the non-homogeneous regular series-parallel water supply system and the model homogeneous regular parallel-series system. The accuracy of the performed evaluations is illustrated in tables and figures. The reliability data of components are assumed either to be arbitrary or to come from experts. These reliability evaluations of the considered systems' characteristics are an illustration of the possibility of applying the asymptotic approach in system reliability analysis of large real technical systems.

4.1. Reliability evaluation of two-state series systems The investigations of limit reliability functions of homogeneous two-state series systems are based on the following auxiliary theorem. L e m m a 4.1

If m

(i) .~ (t) = exp[-V (t)] is a non-degenerate reliability function,

40

Reliability o f Large Two-State Systems

(ii) R. (t) is the reliability function of a homogeneous two-state series system defined by (2.1), (iii)

a n >0, b n ~ (-oo, ~),

then m

lim R . (ant + b.) = .qi'(t) for t e C~

(4.1)

if and only if m

lim nF(a.t + b.) = g (t) for t e C~

(4.2)

n - - . } o o

Lemma 4.1 is an essential tool in finding limit reliability functions of two-state series systems. Its various proofs may be found in [7], [36] and [71]. It also is the basis for fixing the class of all possible limit reliability functions of these systems. This class is determined by the following theorem proved in [7], [36] and [71]. Theorem 4.1

The only non-degenerate limit reliability functions of the homogeneous two-state series system are: 9/l (t) = exp[-(-t) -a ] for t < 0, .9?1(t) = 0 for t > 0, a > 0,

(4.3)

gi'-2 (t) = 1 for t < O, gi'--2 (t) = exp[-t ~ ] for t > O, a > O,

(4.4)

m

~3 (t)=

exp[-exp[t]] for t ~ (-0% oo).

(4.5)

The next auxiliary theorem is an extension of Lemma 4.1 to non-homogeneous twostate series systems. Lemma 4.2

If m

(i) ~ ' ( t ) - exp[- V' (t) ] is a non-degenerate reliability function, m

(ii) R' n (t) is the reliability function of a non-homogeneous two-state series system def'med by (2.8), (iii) then

a n > O, b,, e ( - ~ , oo),

Chapter 4 lim R' n (ant + b , ) = ~ ' ( t )

41

for t e C~,

if and only if lim n ~ q i F ( O ( a n t + b n )

=V"(t) fort ~ C~,.

n---~oo i=1

The proof of Lemma 4.2 is given in [56] and [71]. This lemma is a particular case of Lemma 1 proved in [60]. In [60] Lemma 2 is also proved. From the latest lemma, as a particular case, it is possible to derive the next auxiliary theorem that is a more convenient tool than Lemma 4.2 for finding limit reliability functions of nonhomogeneous series systems and the starting point for fixing limit reliability functions for these systems. Lemma 4.3 If

(i) gi" (t) = e x p [ - V' (t) ] is a non-degenerate reliability function, (ii)

R' n (t) is the reliability function of a non-homogeneous two-state series system defined by (2.8),

(iii)

a n > 0, b n e (-oo, oo),

(iv) F(t) is one of the distribution functions ~~ such that

F(2)(t),...,F(a)(t) defined by (2.7)

(v) 3 N Y n > N F(ant + bn) = 0 for t < to and F(a,t + bn) r 0 for t _ to, where t0 e < -o% oo),

(vi) lim F(i) (ant + bn ) < 1 for t > to, i = 1,2,...,a, n--,| F(ant + bn) and moreover there exists a non-decreasing function

_

[

0

(vii) d(t)= 1 lim ~ q i d i ( a n t + b n ) [ n---~oo i=1

where

for t < t o for t>to,

(4.6)

42

Reliability o f Large Two-State Systems F (i) (ant + b n )

m

(viii) d i (ant + b n ) =

F ( a , t + b, )

then lim R ' , ( a , t + b , ) = g l ' ( t )

fort e C~.

(4.7)

if and only if m

m

lim n F ( a n t + b n )d (t) = V ' (t) for t ~ C~..

(4.8)

On the basis of Theorem 4.1 and Lemma 4.3 in [56] and [71] the class of limit reliability functions for non-homogeneous two-state series systems has been fixed. The members of this class are specified in the following theorem ([71 ]).

Theorem 4.2 The only non-degenerate limit reliability functions of the non-homogeneous two-state series system, under the assumptions of Lemma 4.3, are: m

m

m

.O/'l (t) = exp[-d (t)(-t) -a ] for t < O, fli" 1 (t) = 0 for t > O, cr > O,

(4.9)

~'2 (t) =

(4.10)

"~'3 (t)

1 for t < O, gi" 2 (t) = exp[-d (t)t a ] for t >_O, a > O,

= exp[-d(t)exp[t]]

for t ~ (-oo, oo),

(4.11)

where d ( t ) is a non-decreasing function dependent on the reliability functions of particular system components and their fractions in the system defined by (4.6). The above theorem is a particular case of Theorem 2 proved in [60].

Corollary 4.1 If the ith type components of the non-homogeneous two-state series system have Weibull reliability functions R(~

= 1 for t < O, R(~

= exp[-fli tai ] for t > O, ai > O, fl~ > O, i = 1,2 .... ,a,

(4.12)

and a, = 1/(fin) t/a , b, = O,

where

(4.13)

Chapter 4 a = min{a/}, l 0.

Since a I = 1, ,81 = 0.025, a 2 = 1, f12 = 0.02, ct3 = 2, f13 = 0.0015, a 4 = 2,184 = 0.001, then a = min {1,1,2,2} = 1, ,8 = m a x {0.025, 0.02} = 0.025. Assuming normalising constants a, = 1~fin=0.4, b n = 0

and according to (4.15), after determining

d(t)= 5-'. qifl/=0.56, (i:ai=a)

fl

from Corollary 4.1 it follows that the gas piping system limit reliability function is gi" 2 (t) = exp[-0.56t] for t >__ 0. Hence, according to (1.1), the exact reliability of the considered system may be approximated by the formula m

g'~o o (t) ~ exp[-0.56(t / 0.4)] = exp[-1.4t] for t > 0.

(4.17)

Reliability of Large Two-State Systems

46

The mean values of particular system component lifetimes in years are as follows:

E(T~) = 1/0.025 = 40, E(T2) = 1/0.020 = 50, E(T3)=/(3/2)(0.0015)-~/2 ~ 23, E(T4) = /'(3/2)(0.001) -~/2 -_- 28. The approximate mean value of the gas piping lifetime and its standard deviation calculated on the base of the formula (4.17), are: E(T) ~ 1/1.4 ~ 0.71 years, o(7) ~ 1/1.4 ~ 0.71 years. The behaviour of the exact and approximate reliability functions of the gas piping is illustrated in Table 4.1 and Figure 4.1. Moreover, in Table 4.1, the differences between the values of these functions are also given. These differences testify that the approximation of the system's exact reliability function by its limit reliability function is good enough.

Table 4.1. The values and differences between the exact and limit reliability functions of the gas piping system

t '

- ((t - b n ) / a n ) ! "~'2

loo (t)

o.oo

'

i

1.ooo

'

A = R- ' l o o

1.ooo

--'.9t'

0.0oo

i

i

i |

i

i

i

I

i L i

i

i

l

I

0.10 0'20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.20 1.40 1.60 1.80

,

i

!

i

l

,,

i

i

i

I

i

i

i

l

l

i i

........

l

,

0.869 .........I,:..... 0.754 0.654 0.567 0.491 0.425 0.367 0.317 0.274 0.236 0.175 0.129 I 0.095 0.070 ,

,

i

i

i

i

i

,

!

1

,

!

!

0.869 0.756 0.657 0'571 0.497 0.432 0.375 0.326 0.284 0.247 0.186 0.141 0.106 0.080,

'i ,

i

i

i

i

........

i

,

l

i

l

,

0.000 -0.001 -0.003 -0.004 -0.006 -0.007 -0.008 -0.009 -0.010 -0.011 -0.012 -0.012 -0.012 -0.0, 11

i

i

,i,

.

2

Chapter 4

47

m

R'loo(t), .qi~2((t- bn)/an) 1.0 0.8 0.6 0.4 0.2 0.(

i

0.0

0.5

[

1.0

~......

1.5

i

2.0

i

2.5

r

i

3.0

t

Fig. 4.1. The graphs of the exact and limit reliability functions of the gas piping system

4.2. Reliability evaluation of two-state parallel systems The class of limit reliability functions for homogeneous two-state parallel systems may be determined on the basis of the following auxiliary theorem proved for instance in [7], [36] and [71].

Lemma 4.4 If .9/(t)is the limit reliability function of a homogeneous two-state series system with reliability functions of particular components R(t), then m

.q?(t) = 1 - .q?(-t) for t ~ C~is the limit reliability function of a homogeneous two-state parallel system with reliability functions of particular components m

R(t) = 1 - R (-t) for t ~ C~. At the same time, if (an,bn) is a pair of normalising constants in the first case, then

(a n ,-b n) is such a pair in the second case.

48

Reliability of Large Two-State Systems

Applying the above lemma it is possible to prove an equivalent of Lemma 4.1 that allows us to justify facts on limit reliability functions for homogeneous parallel systems. Its form is as follows ([7], [36], [71]). Lemma 4.5 If

(i) .0?(t) = 1-exp[-V(t)] is a non-degenerate reliability function, (ii) Rn(t) is the reliability function of a homogeneous two-state parallel system defined by (2.2), (iii) a , >0, b, ~(-oo, oo), then lim

R.(a.t + b . ) = 91(0 for t ~ C ~ ,

(4.18)

n--~oo

if and only if lira nR(a,t + b , ) = V(t) for t ~ C v .

(4.19)

n - - t , ao

By applying Lemma 4.5 and proceeding in an analogous way to the case of homogeneous series systems it is possible to fix the class of limit reliability functions for homogeneous two-state parallel systems. However, it is easier to obtain this result using Lernma 4.4 and Theorem 4.1. Their application immediately results in the following issue. Theorem 4.3 The only non-degenerate limit reliability functions of the homogeneous parallel system are:

9i'l(t) = 1 for t ___0, .qi'l(t) = 1 - exp[-C a] for t > 0, a > 0,

(4.20)

.~'2(t) = 1 - e x p [ - ( - 0 a] for t < 0, gi'2(0 = 0 for t >__0, a > 0,

(4.21)

~3(t) = 1 - e x p [ - e x p [ - t ] ] fort ~ (-oo,oo).

(4.22)

Corollary 4.2 If components of the homogeneous two-state parallel system have Weibull reliability functions R(t) = 1 for t < 0, R(t) -- exp[-flt" ] for t ___0, a > 0,/3 > 0,

Chapter 4

49

and a. = b./(alog n), b. = (log n/fl)v%

then ,9/3(t) = 1 - e x p [ - e x p [ - t ] ] , t ~ (-oo,oo), is its limit reliability function. Motivation: Since for sufficiently large n and all t e (-0%00) we have a.t + b. = b.(t/( alog n) + 1) > O,

then R(ant + bn) = exp[-fl(ant + b.) a] for t ~ (-0%00).

Hence n R(a.t + b.) = n e x p [ - ~ a . t + b.) '~] = n exp[-P(bn)a(t/(alog n) + 1)a]

= n exp[-log n(t/(alog n) + 1)a]. Further, applying the equality (t/(alog n) +1) a = 1 + t/log n + o(1/log n) for t ~ (-oo,oo),

we obtain V(t) = lim n R (a.t + b.) n -.~ oo

=

lim n exp[-log

n - t - o(1)]

n--.-koo

=

lim exp[-

t - o(1)] =

exp[-t] for

t ~ (-oo,oo),

n.-~oo

which from Lemma 4.5 completes the proof. 4 . 2 (an energetic cable) Let us consider an energetic cable composed of 36 wires of the type A1Si used in overhead energetic nets and assume that it is able to conduct the current if at least one of its wires is not failed. Under this assumption we may consider the cable as a homogeneous parallel system composed of n = 36 basic components. The cross-section of the considered cable is presented in Figure 4.2.

Example

Reliability of Large Two-State Systems

50

Fig. 4.2. The cross-section of the energetic cable Further, assuming that the cable wires have Weibull reliability ftmctions with parameters a = 2, fl = (7.07) '6, by (2.2), the cable's exact reliability function takes the form R36(t) =

1 for t < 0,

R36(t) =

1 - [1 - exp[-(7.07)-6t2] 36 for t > 0.

Thus, according to Corollary 4.2, assuming

an = (7.07)3/(2 x/log 36 ), bn = (7.07) 3 x/log 36, and applying (1.1), we arrive at the approximate formula for the cable reliability function of the form

R36(0 ~ ~?3((t- b,,)/a,,) = 1 - exp[-exp[-0.01071t + 7.167]] for t e (-oo,oo). The expected value of the cable lifetime T and its standard deviation, in months, calculated on the basis of the above approximate result and according to the formulae ([13])

E[ 7] =Ca. + b., cr= xa n / x[-6, where C --__0.5772 is Euler's constant, respectively are: E[T] =__723, or-_- 120. The values of the exact and approximate reliability functions of the cable are presented in Table 4.2 and Figure 4.3. Moreover, Table 4.2 also shows the differences between those values. The differences are not large, which means that the mistakes in replacing

Chapter 4

51

the exact cable reliability function by its approximate form are practically not significant. Table 4.2. The values of the exact and approximate reliability functions of the energetic cable

t

.

R 3 6 ( 0

.

.

0 400 "' 500 9 550 600 650 700

.

1,

.

.

.

.

.

'

75o 800 9 '900 9 1000 1100 1200

9

.

,,

' " .

.

.

.

.

.

i

.

. 1.000 1.000

L

0.193 0.053 0.012 0.002 0.000

,

,,1

.... " .

.

o.33o

.....

,

0.995 01965 0.874 0.712 0.513

L

( ~:=:bn ) an 1.000 1'000

~ 3

.

.

.

L

.

0.988 0.972 0.877 0.707

.

.

.

.

.

.

,

,

.

.

.

.

.

.

.

.

o.513

.

.....

.

.

A =R36 _ .~i'3

.

.

0.344 0.218 0.081 0.029 0.010 0.003

2

,

.

.

.

.

.

0.000 0.000 -0.003 -0.007 -0.003 0.005

.

.

.

o.600

,

,,,

-0'014 -0.025 -O.02S -0.017 -0.008 -0.003

,

,

.

.

.

.

.

.

.

.

.

,

,

,,

' ,

,

J R36(t), .~'3((t- bn)/an) !

i

1.0~ 0.80.6).4

-

0.2-

0.0

0

,

-

200

400

,

600

800

_

~--

1000

t

Fig. 4.3. The graphs of the exact and approximate reliability functions of the energetic cable

52

Reliability of Large Two-State Systems

The next lemma is a slight modification of Lemma 4.5 proved in [56] and [71 ]. It is also a particular case of Lemma 2, which is proved in [63]. L e m m a 4.6

If 91' (t) is the limit reliability function of a non-homogeneous two-state series system with reliability functions of particular components /~0(t), i = 1,2,...,a, then .qi" (t) = 1 - gi" (-t) for t ~ C~. is the limit reliability function of a non-homogeneous two-state parallel system with reliability functions of particular components

R(~

= 1 - /~o (-t) for t e C~( 0 , i = 1,2,...,a.

At the same time, if (an,b .) is a pair of normalising constants in the first case, then (an ,-b n ) is such a pair in the second ease. Applying the above lemma and Theorem 4.2 it is possible to arrive at the next result ([56], [63], [71]). Lemma 4.7

If (i) .qi"(t)= 1 - exp[-V'(t)] is a non-degenerate reliability function, (ii) R'~ (t) is the reliability function of a non-homogeneous two-state parallel system defined by (2.10), (iii) a, > 0, b, e (-oo,oo), then lim R' n (ant + bn) = 91' (t) for t e Cgt, n ....~ oo

if and only if a

lim n~,qiR(O(ant+bn) = V'(t) fort ~ Cv,. n-+oo

i=1

Chapter 4

53

The next lemma motivated in [56] and [71] that is useful in practical applications is a particular case of Lemma 3 proved in [63]. L e m m a 4.8 If

(i)

.qi"(t) = 1 - e x p [ - g ' (t)] is a non-degenerate reliability function,

(ii)

R' n (t) is the reliability function of a non-homogeneous two-state parallel system defined by (2.10),

(iii) a. > O, b. ~ ( - ~ , ~ ) , (iv) R(t) is one of the reliability functions R(t)(t), R(2)(t),...,R(a)(t) defined by (2.9) such that (v) 3 N V n > N R(a.t + b.) r 0 for t < to and R(a.t + b.) = 0 for t >_to, where to e (-~,oo>,

(vi)

R (i) (a.t + b n ) lira .... _to,

where

(viii) di(a.t + b.) =

R (i) (ant + b n ) R(a.t+b.)

then lim R' n (a.t + b . ) =

~'(t)

for t E C~,,

(4.24)

n---~oo

if and only if lira nR(a.t + b n )d(t) = V' (t) for t E Cv, . n -.-~ ao

(4.25)

Reliability of Large Two-State Systems

54

Starting from this lemma it is possible to fix the class of possible limit reliability for non-homogeneous two-state parallel systems ([56], [63], [71]). T h e o r e m 4.4 The only non-degenerate limit reliability functions of the non-homogeneous two-state parallel system, under the assumptions of Lemma 4.8, are:

gi" 1 (t) = 1 for t < 0, .$i"t (t) = 1 - exp[-d(t)t -'~] for t > 0, a > 0,

(4.26)

.K" 2 (t) = 1 - exp[-d(t)(-t) a] for t < 0, 9/'2 (t) = 0 for t > 0, c~ > 0,

(4.27)

~'3 (t)= 1 -exp[-d(t)exp[-t]] fort e (-oo,oo),

(4.28)

where d(t) is a non-increasing function dependent on the reliability functions of particular system components and their fractions in the system defined by (4.23). Theorem 4.4 is a particular case of Theorem 1 proved in [63].

4.3. Reliability evaluation of two-state "m out of n" systems The class of limit reliability function for homogeneous two-state "m out of n" systems may be established by applying the three following auxiliary theorems proved in [ 108] and [71 ]. L e m m a 4.9 If

(i) m = constant (m / n --->0 as n ~ oo),

m-l[V(t)] i

(ii) .qP)(t) = 1 - ~ .......... exp[-V(t)] is a non-degenerate reliability function, /--o i! (iii) R (,n) (t) is the reliability function of a homogeneous two-state "m out of n" system n defined by (2.3), (iv) a. > O, b. e (-oo,oo), then lim R Cnm)(ant + b n ) = .q#)(t) for t e Cgt(o ) tl---~o0

if and only if

(4.29)

Chapter 4

lim nR(ant + bn) = V(t) for t ~ Cv.

55 (4.30)

n --9, oo

L e m m a 4.10

If (i) m / n - - - > l z ,

0 O , c>O, ct>O, (4.38)

1 9i'} ~) (t)= 1 - 2 ~

-cltl ~ -x~-~o e 2 dx fortO, c > O , a > O ,

1

-r

.Oi'~") (t) = 1 - - - ~

x2 2 dx

for t < O, c 1 > O, a > O,

(4.40)

x2

1

9i'~a)(t) - 2

~

-0o~ e

(4.39)

1 c2t'~ - - ~

oI e 2 dx fort>_O, c2 > O , a > O ,

9i'~a) (t)= 1 for t < -1, .~i'~a) (t)=

_1 for 2

(4.41)

-1 < t < 1, 9i'~a) (t) = 0 for t >_.O. (4.42)

Chapter 4

57

Case 3. n - m = ~ = constant (m / n ~ 1 as n --->oo). ~s O)(t) = ~ (-t)-ia e x p [ - ( - t ) -a ] for t < O, .~8O) (t) = 0 for t > O, a > O, i=0

(4,43)

i!

t ia

.~9(l) (t) = 1 for t < 0, .~9O) (t) = ?0"~-'ffi . exp[-t a ] for t > 0, a > 0,

(4.44)

,~(1) (t) = ~ exp[!t] exp[-exp[t]] for t ~ (-oo,oo). i=0 i!

(4.45)

C o r o l l a r y 4.3 If components of the homogeneous two-state "m out of n" system have exponential reliability functions

R(t) = 1 for t < O, R(t) = e x p [ - g t ] for t > O, A > O, m tends to infinity in such a way that

m/n---~,u, O < p < l , a s

n~oo,

and

1 I n-m+l

a " = "A

1

n+l ' b " = ~ l o g - - - m-

(n+l)m

then x2

.qi'(~) ( t ) = 1 - ~ .1 -|i - T d r , t ~ (-oo,oo),

is its limit reliability function. Motivation: Since for sufficiently large n and all t ~ (-~,oo) we have

t ]n-m+l a,t + bn = -- ~

(n

+

1 n+l > 0, + - " log 1) m m

then for sufficiently large n

R(ant + b,) -- exp[-A (a,t + b,)]

58

Reliability of Large Two-State Systems

= exp[- t . / n - m + 1 V( n + l ) m - l ~

n +1

m ]

=[1-tln-m+l' +o( 1 )] m fort ~ (-oo,oo). (n + 1)m 4n n + l Hence v(t) = lim (n +,l)R(ant + b n ) - m n-+oo [ m(n - m + 1)

n+l

= n~oolim[ - t + O(~nn ) I ( n + l ) ( n - m + l ) ] m = - t fort ~ (-oo,oo), which from Lemma 4.10 completes the proof.

Example 4.3 (a lighting system) We consider a lighting system composed of n = 35 identical lighting points that is not failed if at least m = 16 of the points are not failed. Assuming that the lighting points have exponential reliability functions with the failure rate A = 1/year, from (2.3), the exact system reliability function is given by 35 ( t ) = l f o r t < 0 ,

35 (t) R o6,

=

1- E

(3:) exp[-it][1-exp[-t]]

35-i for t > 0.

i=O

According to Corollary 4.3, assuming

a n -

_0.1863, bn = log9- ~0.8109 -" 4

12

after applying (1.1), we get the approximate formula for the reliability function of the lighting system in the form R(~ ) (t) --- .q~/~) ((t - b n ) / a n )

=

1 .rz'1----'-----qz~r

5.37t-4.35

~ --00

x2 ---

e 2 dx fort ~ (-oo,oo).

Chapter 4

59

The mean value of the lighting system lifetime T and its standard deviation, in years, calculated from the above formula are: E[T] = 0.811, r

0.186.

The values of the exact and approximate reliability functions of the lighting system are presented in Table 4.3 and their graphs in Figure 4.4. The differences between the exact and approximate reliability functions of the system given in Table 4.3 testify good accuracy of the approximation. Unfortunately, there are no generalisations of Lemmas 4.9-4.11 for the nonhomogeneous two-state "m out of n" systems. Each particular case of a nonhomogeneous two-state "m out of n" system has to be considered separately and a suitable auxiliary theorem and corollary have to be formulated and proved and then applied to the reliability evaluation of a real system.

Table 4.3. The values of the exact and approximate reliability function of the lighting system R (3156)(t)

d = R ~156) .9i'~#)

a n

0.0 O. 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ' 111' 1.2 .. 1.3 .L ... 1 . 4

.

........

.

.

.

.

0.99999 0.99994 r 0.99948 0.99695 0.98629 0.95241 ' 0.87118 0.72419 0.52339 0.31633 0.15513 ! ' 0'.06041 0.01840 . 0.00433 ~ 0.00078 .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

i

.

.

.

.

.

Ii

.

1.00000 1.00000 1.00000 0.99980 0.99507 0.96253 i 0.86178 0.68205 0.46713 0.27703 0.14394 0.06653 F 0.02777 0.01062 ~ 0.00376. j......

.

.

~ .

'

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

I

I. .

.

.

.

.

.

.

0.00001 0.00006 0.00052 0.00285 0.00878 0.01012 -0.00940 -0.04214 .

.

.

.

.

.

.

.

.

.

.

-0.05626 -0.03929 -0.01119 0.00612 0.00937 0.00628 0.00298 ......

Reliability of Large Two-State @stems

60

k R[~6) (t), .9i'~") ((t -

bn) / a n)

t 0

0.2

0.4

0:6

0.8

1.0

1.2

f

Fig. 4.4. The graphs of the exact and approximate reliability function of the lighting system

4.4. Reliability evaluation of two-state series-parallel systems Prior to the formulation of the overall results for the classes of limit reliability functions for two-state regular series-parallel systems we introduce some assumptions for all cases of the considered systems shapes. These assumptions distinguish all possible relationships between the number of their series subsystems k, and the number of components l, in these subsystems. In the assumptions for two-state regular parallelseries systems, considered in the next section, k, is the number of parallel subsystems and l, is the number of components in these subsystems.

Assumption 4.1 Here are considered the relationships between k, and In of the form k, = n, l, = c(log n) pCn),n ~_ (0,oo), c > O, with the following cases distinguished:

(4.46)

Chapter 4

61

Case 1. k~ = n, Iln - c logn I >> s, s > O, c > O. 1~ 1~ > log(logn), 8. log v [p(r~) - p(n)[ 1,

(l-p(n)).log(Iog n) --'~ V

n

4 0 l n >> c l o g n and p(n) O ,

Ipfr~) - p(n)l > (logn)'t,~ > O,

1-p(n)

for all natural v > 1,

62

Reliability o f Large Two-State Systems

I,o(rv) - ,o(n)t <
1, n--t, oO 1

where 8 > 0 and rv

(l-p(n))A(t)

n

A(n)~

~ f/(p(n)), i=I

wherein(n) for i = 1,2,...,v, is the ith superposition of a function log n, and v is such that f~+l(p(n)) 0. C a s e 3 . kn---> k, k > O , ln --~ oo.

The proofs of the theorems on limit reliability functions for homogeneous regular series-parallel systems and methods of finding such functions for individual systems are based on the following essential lemma. Lemma 4.12 If (i) k. ~ oo, (ii) .O?(t)= 1 - exp[-V(t)] is a non-degenerate reliability function,

(iii) R k,,t, (t) is the reliability function of a homogeneous regular two-state seriesparallel system defined by (2.5), (iv) an > 0, bn ~ (-oo,oo), then lim R k.,t. (ant + bn) = .91(t) for t ~ C ~ ,

(4.47)

n-.-~oo

if and only if lim k,[R(ant + b,) ]t, = V(t) for t ~ Cg . n --), oo

The proof of Lemma 4.12 is given in [53], [56] and [71].

(4.48)

Chapter 4

63

The justification of the next auxiliary theorem that follows from Lemma 4.12 may be found in [56] and [71 ]. L e m m a 4.13

If (i) k . ~ k ,

k>O,l.~,

(ii) .q/(t) is a non-degenerate reliability function, (iii) R kn,tn (t) is the reliability function of a homogeneous regular two-state seriesparallel system defined by (2.5), (iv)

a. > 0, b. e (-or

then lim R k.,t. (a.t + b.) = gl(t) for t ~ C ~ ,

(4.49)

n--~oo

if and only if lim [R(ant + bn) ] In

-" ~ 0 ( 0

for t ~ Cgto ,

(4.50)

n.-~oo

where 9i'o(t) is a non-degenerate reliability function and moreover .qT(t) = 1 - [ 1 - ~0(t)] k fort ~ (-0%00).

(4.51)

The results achieved in [53]-[56], [59] and based on Lemma 4.12 and Lemma 4.13 may be formulated in the form of the following theorem ([54], [57], [63]). T h e o r e m 4.6 The only non-degenerate limit reliability functions of the homogeneous regular twostate series-parallel system are:

Case 1. kn = n, [In - c log n[ >> s, s > 0, c > 0 (under Assumption 4.1). .~/~(t) = 1 for t _ O. 9?4(0 = 1 for t < O, .014(0 = 1 - e x p [ - e x p [ - t a - s/c]] for t > O, a > O,

(4..55)

.$i's(O = 1 - e x p [ - e x p [ ( - t ) a - s/c]] for t < O, .Oi's(t) = 0 for t > O, a > O,

(4.56)

.~'6(t) = 1 - e x p [ - e x p [ f l ( - t ) ~ - s/c]] for t < O, (4.57)

.016(0 = 1 - e x p [ - e x p [ - t ~ - s/c]] for t _> O, a > O, fl > O, ~7(t) = 1 for t < tt, gi'7(t) = I - e x p [ - e x p [ - s / c ] ] for tl < t < t2,

(4.58)

.ql7(t) = O for t > t2, tt < t2, C a s e 3 . k~ ---> k, k > O, l~ ---> oo. gi's(t) = 1 - [1 - e x p [ - ( - t ) -a ]]k for t < O, .O/s(t) = 0 for t > O, a > O,

(4.59)

gi'9(t) = 1 for t < O, .0?9(0 = 1 - [ 1 - exp[-ta]] k for t > O, a > O,

(4.60)

.Oi'lo(O = 1 - [1 - e x p [ - e x p t]] k for t ~ (-0%0o).

(4.61)

C o r o l l a r y 4.4 I f c o m p o n e n t s o f the h o m o g e n e o u s W e i b u l l reliability functions

regular two-state

series-parallel

system have

R(t) = 1 for t < O, R(t) = e x p [ - f l t ~ ] for t > O, a > O , f l > O ,

(4.62)

k, ~ k, l, > O,

(4.63)

an = 1 / ( ~ l n ) 1/a , bn=O,

(4.64)

and

then .~'9(t) = 1 for t < O, ~9(t) = 1 - [1 - exp[-t'~]] k for t > O, is its limit reliability function. Motivation: Since, a c c o r d i n g to (4.63) and 0 . 6 4 ) , we h a v e

ant + b. = (ill n ) - l / a t < 0 for t < 0

and

Chapter 4 ant + bn = (ill n )-l/a

65

t > 0 for t > 0,

then from (4.62) the equalities R(a.t + bn) = 1 for t < 0

and R(ant + bn) = exp[-fl(a.t + bn) ~] = exp[-ta/ln] for t > 0

are satisfied. Further, we have lim [R(ant + b.)] In

=

1 for t < 0

n---),oo

and lim [R(a,t + bn)] t, = exp[_t ~] for t > 0. n.-~oo

Thus, from Lemma 4.13, .OI9(t) is the limit reliability function of the system. Example 4.4 (a gas distribuaon system) The gas distribution system consists of k n = 2 piping lines, each of them composed of 1, = 1000 identical pipe segments having Weibull reliability functions R(t) = exp[-O.OOO2t 3] for t > 0. The system is a homogeneous regular series-parallel system and according to (2.5) its exact reliability function is given by R2,1ooo(t) = 1 - [ 1 - exp[-0.2t3]] 2 for t > 0. Assuming, according to (4.64), the normalising constants an = (0.0002-1000) -v3 = 1.71, bn = 0, from Corollary 4.4, we conclude that the limit reliability function of the system is given by the formula 9/'9(0 = 1 - [1 - exp[-t3]] 2 for t > 0. Thus, according to (1.1), the approximate formula (it is easy to check that it is exact in this case) takes the form

Reliability of Large Two-State Systems

66 R2,1ooo(t) ~ g?9((t-

bn)/an) = 1 - [1 - exp[-0.2t3]] 2 for t > 0.

The expected values of the lifetimes of pipe segments are ([71 ]) E(T/) =/-(4/3)(0.0002) -v3 -___15.3 years, while the mean value of the system lifetime is ([71]) E(T) = 2/-(4/3)(0.2) -v3 -/'(4/3)(0.04) -u3 ~ 2.4 years. The behaviour of the reliability function of the gas distribution system is illustrated in Table 4.4 and Figure 4.5.

Table 4.4. The behaviour of the exact and approximate reliability function of the gas distribution system R2,1ooo(t) = 9/9((tt 0.0 1.0000 1.0000 0.2 0.4 0.9998 0.6 0.9982 0.8 0.9905 1.0 0.9671 1.2 0.9146 1.4 0.8216 1.6 0.6873 1.8 " 0.5259 2.0 0.3630 2.2 0.2236 2.4 0.1220

bn)/an)

|

,

Chapter 4

67

R2,1ooo(t) = 9/9((t- b.)/a.) 1.0 D.8 0.60.40.20.0 0.0

, 0.5

....

1'.0

115

i

2.0

1

2.5

i

3.0

----~" t

Fig. 4.5. The graphs of the reliability function of the gas distribution system

The proofs of the facts concerned with limit reliability functions of non-homogeneous two-state series-parallel systems are based on the following auxiliary theorems formulated and proved in [56], [60], [63] and [71]. Lemma 4.14 If

(i)

k, --->0%

(ii)

9/' (t) = 1 - exp[- V' (t) ] is a non-degenerate reliability function,

(iii)

R'k.,t. (t) is the reliability function of a non-homogeneous regular two-state series-parallel system defined by (2.14)-(2.15),

(iv)

a. > 0, b. ~ (-oo,oo),

then lira R'k.,t" (ant +bn) = ~'(t) for t e C~,, n .-.~ ~t~

if and only if

Reliability of Large Two-State Systems

68

lira k n za qi [ R(0 (ant + bn )] In

n--~oo

= V(t)

i=l

for t e C v,

Lemma 4.15 If (i)

k, ---> ~,

(ii)

.qi" (t) = 1 - exp[- g' (t) ] is a non-degenerate reliability function,

(iii)

R'k.,t" (t) is the reliability function of a non-homogeneous regular two-state series-parallel system def'med by (2.14)-(2.15),

(iv)

a. > 0, b. e (-oo,oo),

(v) R(t) is one

of the reliability functions R~

R(2)(t),...,R(a)(t) defined by (2.15)

such that

3 N V n > N R(ant + b.) r 0 for t < to and R(a.t + b.) = 0 for t > to,

(vi)

where to e (-oo,oo>,

R (i) (ant + b n ) n ~ R(ant +bn)

(vii)

lim

1 for t < to, i = 1,2,...,a,

and moreover there exists a non-increasing function

i m a ~ qidi(ant +bn) for t < t o (viii) d(t) =

(4,65)

i=l

for t > t o , where

di(ant + bn) = [ R(i) (ant + bn ) R(ant+bn)

] In

then lim R' k.,t. (ant + b n ) = .q~'(t) for t ~ Cgi,, n..-~oo

if and only if

(4.66)

69

Chapter 4

lira k n [R(ant + b n)]t" d(t) = g' (t) for t ~ Cv,.

(4.67)

n.-~

Lemma 4.16

If (i) k. --->k, k > O , l. --->~, (ii)

.qi"(t) is a non-degenerate reliability function,

(iii)

R'k.,t" (t) is the reliability function of a non-homogeneous regular two-state series-parallel system defined by (2.14)-(2.15),

(iv) a. > 0, b. ~ (-~,~), (v) R(t) is one of reliability functions Rf0(t), R(2)(t),...,R(a)(t) defined by (2.15) such that (vi) 3 N V n > N R(a.t + b.) ~: 0 for t < to and R(a.t + b.) = 0 for t > to, where to ~ (-~,~>, (vii)

R (i) (a.t + b. ) lim _> s, s > 0, c > 0 (under Assumption 4.1 and the assumptions of L e m m a 4.15). -~"1 (t) = 1 for t < 0, 9/' 1 (t) = 1 - e x p [ - d ( t ) t -a] for t > 0, ct > 0,

(4.72)

.01' 2 (t) = 1 - e x p [ - d ( t ) ( - t ) a] for t < 0, .01'2 (t) = 0 for t > 0, a > 0,

(4.73)

gl' 3 ( t ) = 1 - e x p [ - d ( t ) e x p [ - t ] ]

(4.44)

f o r t ~ (-oo,oo),

Case 2. kn = n, In - c log n ~ s, s ~ (-0%00), c > 0 (under the assumptions o f L e m m a 4.15). gi" 4 (t) = 1 for t < 0, gi" 4 (t) = 1 - e x p [ - d ( t ) e x p [ - f

- s/c]] for t > 0, cr > 0, (4.75)

~i" s (t) = 1 - e x p [ - d ( t ) e x p [ ( - t ) ~ - s/c]] for t < 0, .0/' s (t) = 0 for t > 0, a > 0,

(4.76)

91' 6 (t) = 1 - e x p [ - d ( t ) e x p [ f l ( - t ) a - s/c]] for t < 0, ~ ' 6 (t) = 1 - e x p [ - d ( t ) e x p [ - f -

s/c]] for t > 0, a > 0, fl > 0,

-~'7 (t) = 1 for t < ti, .0?'7 (t) = 1 - exp[-d(t)exp[-s/c]]

~'7 (t) = 0 for t > t2, tt < t2,

(4.77)

for tl -< t < t2, (4.78)

Chapter 4

71

Case 3. k, --->k, k > O, 1, --~ oo (under the assumptions of Lemma 4.16). 91' 8 (t) = 1 - f i [1 - d i (t) exp[-(-t) -a ]] qik for t < O, i=l

tips (t) = 0 for t > O, ct > O,

(4.79)

*~?'9 (t) = 1 f o r t < O , 91' 9 (t) = 1 - fi[1-di(t)exp[-ta]] qik f o r t > O , cr>O, (4.80) i=l

Pl'lo (t) = 1 - f l [ 1 - d i ( t ) e x p [ - e x p t ] ]

qik fort e (-oo,oo),

(4.81)

i=l

where d(t) and di(t) are non-increasing functions dependent on the reliability functions of the system's particular components and their fractions in the system defined by (4.65) and (4.68) respectively. Corollary 4.5 If components of the non-homogeneous regular two-state series-parallel system have Weibull reliability functions

R(tJ)(t) = 1 for t < O, R(~Jg(t) = exp[-fl0, t % ] for t > O, cr0. > O, fl# > O,

(4:82)

i = 1,2,,..,a, j = 1,2,...,ei, and kn --->k, 1, --->~ , a.

=

(4.83)

1/(flln) -l/a ,

b,, =

(4.84)

O,

where Pij~O''

(4.85)

,8 = rrdn{,fli'ai = a } ,

(4,86)

a i ---- rain {a~]}, ~ i - l~j~ei

a = max{a/}, l_.0, and 50 segment pipes with a reliability function R(2'2)(t) = exp[-0.2 ~ ] for t > 0. Thus, according to Definition 2.16, the water supply system is a non-homogeneous regular series-parallel system with the following parameters k. = k = 3, I. = 100, a = 2, ql = 2/3, q2 = 1/3. Considering (2.14), we have 2

S t 3,100

(t) = 1

I'I[1-(R(i)(t))l~176q'3 i=1

= 1 - [1 - (R(l)(t))l~1762 [1 - (R(2)(t))l~176 where after considering the substitutions: el = 2, Pll = 0.4, PI2 = 0.6, all = 1, P l l

= 0 . 0 5 , a12 =

2, ,312 = 0.0015,

e2 = 2, P21 = 0.5, P22 = 0.5, ~21 =

3, f121 = 0.0007, a22 = 0.5, ,322 = 0.2,

and the formula (2.15) el

R(~)(t) = I-I (R 0'j) (t)) p*j j-1

= (R(l'l)(t))~

and e2

R(2)(t) = I-I (R (2,j) (t)) p2j j=l

0'6 =

exp[-0.02t- 0.0009t2]

Chapter 4 = (Rf2'l)(t))~176

75

= exp[-0.00035t 3 - 0.1 x~ ].

From the above it follows that the exact reliability function of the system is given by R'3,1o0 (t) = 1 - [1 - e x p [ - 2 t - 0.09t2]] 2. [1 - exp[-0.035t 3 - 1 0 , ~ ]] for t >__0. Further, according to (4.85), (4.86) and (4.84), we have al = min { al l, a12 } = min { 1,2 } = 1,

181 =Pll/311 = 0.4. 0.05 = 0.02, a2 = min{ a2t,

t~22} --"

min{3, 0.5} = 0.5,

f12 = P22f122 = 0.5" 0.2 = 0 . 1 ,

a = max { al, a2} = max { 1,0.5 } = 1, ,8= min{,fll} = min {0.02}= 0.02,

a,, = (0.02.100) ~ = 0.5, b, = 0, and from Corollary 4.5 the limit reliability function of the system is given by

~'9

(t) = 1 - [1 - exp[-t]] 2 for t > 0.

Hence, after considering (1.1), the reliability function of the system is approximately given by

R' 3,100 (t) = ;9?'9 ( ( t - b,)/a,) = 1 - [1 - exp[-2t]] 2 for t > 0.

(4.87)

The reliability data o f the system components have come from experts. According to their opinions the mean lifetimes o f the pipe segments, depending on their types, vary in a range from 10 up to 50 years and are as follows ([71 ])"

E(Tll) = 1/0.05 = 20, E(T12) = / ( 3 / 2 ) ( 0 . 0 0 1 5 ) -1/2 = 23,

E(T21) =/-(4/3)(0.0007) -1/3

~ 10, E(T22) =/-(3)(0.2) -2 ~- 50.

The water supply system lifetime and its standard deviation calculated on the basis of the approximate formula (4.87) are" E(T) ~ 0.75 years, o(7) ~ 0.56 years.

Reliability of Large Two-State Systems

76

The values of the exact and approximate reliability functions of the system and the differences between them are presented in Table 4.5 and Figure 4.7.

Table 4.5. The behaviour of the exact and approximate reliability functions of the water supply system

(t)

R'3,1o 0 m

|

t

m

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ' 1.6 1.8 2.0 2.2 2.4

i

9

1.0000 0.8910 0.6902 0.4984 0.3449 0.2321 0.1530 0.0994 '~ 0.0637 0.0404 0.0254 ' 0.0158 0.0098

m

m

i

~ . . . . . . . . . . . . . . . . . . . . . . . . . .

i

i

i

|

.0/,, 9 m

l

(t-bn,)

A = R'3,1o 0 - .Oi"

an

,

9

u

1.0000 0,8913 0.6968 0.'5117 0.3630 0.2524 . . . . 011732 0,1179 0.0799 '. 0.0539 0.0363 0.0244 0.0164

l

m

m

9

i

i

i

i

! i i

,

,,

0,0000 -0.0003 -0.0066 -0.0133 -0.0181 -0.0202 -0.0202 -0.0186 -0.0162 -0.0135 -0.0109 -0.0086 -0.0066

,,,

! R'3.1oo(t), 9i~9((t- b,)/a,,) 1.00.80.60.40.2 0.0

.

.

0.0

.

.

.

.

.

.

.

f

0.5

--

I

1.0

1

1.5

-

-

[

2.0

2.5

3.0

t

Fig. 4.7. The graphs of the exact and approximate reliability functions of the water supply system

Chapter 4

77

4.5. Reliability evaluation of two-state parallel-series systems The class of limit reliability functions for homogeneous regular two-state parallel-series systems is successively fixed in ([53]-[56], [59]) on the basis of the following lemmas. Lemma 4.17

If .q?(t) is the limit reliability function of a homogeneous regular two-state series-parallel system composed of components with a reliability function R(t), then m

.qi' (t) = 1 - .q~(-t) for t ~ Cgi, is the limit reliability function of a homogeneous regular two-state parallel-series system composed of components with a reliability function m

R(t) = 1 - R(-t) for t e C R . At the same time, if (a n ,b n ) is the pair of normalising constants in the first case, then (a. ,-bn ) is such a pair in the second case. Lemma 4.18

If (i) k~ --> oo, m

(ii) .qi'(t) = exp[-V (t)] is a non-degenerate reliability function,

l~knln (t)

(iii)

is the reliability function of a homogeneous regular two-state parallel-

series system defined by (2.6), (iv) a. > 0, b. ~ (-oo,oo), then m

lira Rknl. (ant + b n ) = ~ ( t ) for t ~ C ~

n--),~

(4.88)

if and only if lim kn[F(ant +bn)] In = V(t) for t e Cg. n - - ~ o o

(4.89)

78

Reliability o f Large Two-State Systems

Lemma 4.19 If (i) kn ---> k, k > O , l~---> oo, m

(ii) ~ (t) is a non-degenerate reliability function, (iii) Rk.t. (t) is the reliability function of a homogeneous regular two-state parallelseries system defined by (2.6), (iv)

an > O, bn ~ (-~,oo),

then lim Rknln (ant + b n ) = .91 (t) for t ~ C ~ ,

(4.90)

if and only if lim [F(ant + bn) ]tn = ~o(t) for t ~ Czro, n-.-t,

(4.91)

oo

where ,fro(t) is a non-degenerate distribution function and moreover ,~(t) = [1 -.fro(t)]* fort ~ (-oo,oo).

(4.92)

By applying Lemma 4.18 and Lemma 4.19 and proceeding in the same way as in the case of homogeneous regular series-parallel systems it is possible to fix the class of limit reliability functions for homogeneous regular parallel-series systems. This class is presented in [54] and [56] as the successive results given in [53], [55], [59] and [61]. However, it is much easier to obtain this result by applying Lemma 4.17 and Theorem 4.6. Their direct application immediately results in the following theorem ([54], [56], [71]). Theorem 4.8 The only non-degenerate limit reliability functions of the homogeneous regular twostate parallel-series system are: Case I. kn = n, Iln - c log n ! >> s, s > 0, c > 0 (under Assumption 4.1).

gi'1(t) = exp[-(-t) -a] for t < O, ~1 (t)= O, for t > O, a > O, m

(4.93)

m

.qi'2 (t) = 1 for t < O, gi'2 (t) = exp[-t~, for t >_O, a > O,

(4.94)

Chapter 4

79 (4.95)

.qi'3 (t) = e x p [ - e x p [ t ] ] for t ~ (-oo,oo),

Case 2. k, = n, In - c log n ~ s, s ~ (-~,oo), c > O; . . . . .

~4 (t)

m

= e x p [ - e x p [ - (-t) a - s/c]] for t < O,

~4 (t)

= 0 for t > O, a > O,

(4.96)

m

gi' 5 (t) = 1 for t < O, .ql s (t) = e x p [ - e x p [ t a - s/c]] for t > O, a > O,

(4.97)

"~6 (t) = e x p [ - e x p [ - (-t) a - s/c]] for t < O, (4.98)

916 (t) = e x p [ - e x p [ f l t a - s/c]] for t > O, a > O, fl > O, m

*~77(t)

= 1 for t < h,

*~7 (t)

= exp[-exp[-s/c]] for tt < t < t2,

m

~7 ( t ) =

(4.99)

0 for t > t2, t~ < t2,

C a s e 3 . k,---> k, k > O, l, ---~ oo. -~s (t) = 1 for t < O, .0/--8 (t) = [ 1 - exp[-t-a]] k for t > O, a > O,

(4.100)

~9 9 (t) = [1 - e x p [ - (-t)a]] k for t < O, -~9 (t) - 0 for t > O, a > O,

(4.101)

.qi'-'lo(t) = [1 - e x p [ - e x p [ - t]]] k for t e (-0%00).

(4.102)

C o r o l l a r y 4.6 If c o m p o n e n t s o f the h o m o g e n e o u s Weibull reliability functions

regular two-state

parallel-series

R(t) = 1 for t < 0, R(t) = e x p [ - f l t a ] for t > 0, a > 0, fl > 0, and k ~ = n , 1~ - c l o g n > > s ,

c>O,s>O,

a# = bff(afl(bn)alog n), b,, = [(1/fl)log(ln/log n)] l/'~, then m

gi' 3 (t) = r

t ~ (=oo,oo),

is its limit reliability function.

s y s t e m have

Reliability o f Large Two-State Systems

80

Motivation: Since for sufficiently large n and all t ~ (-oo,oo), we have a~t + bn > 0 and

a./b. ~ 0 as n ~ 0% then

F(ant + bn) = 1 - exp[-fl(ant + b.) a] = 1 - exp[-~(bn)a(1 + (an/b.)t) a] = 1 - exp[[-fl(bn)a(1 + a(a./bnt) + o(a./b.)]. Moreover

afl(b.) '~ a./b. = 1/log n -~ 0 as n ~ 0% and therefore

F(a.t +b.) = 1 - exp[-log(l~/log n) - t/log n + o(1/log n)] = 1 - (log n)/l.(1 - t/log n + o(1/log n) = 1 - (log n)/l. + t/l. - o(1/l.) for t ~ (-oo,oo), and

V ( t ) = l i m k . [ F ( a . t + bn)] t.

= lim n[ 1 - ( l o g n)/l. + t/t.- o(1//~)]/" n....~oo

= lim e x p [ t - In o(1/1.)1 = exp[t] for t e (-oo,oo), n---~oo

which from L e m m a 4.18 completes the proof.

Example 4.6 (a model parallel-series system) If the shape o f a h o m o g e n e o u s regular parallel-series system is such that kn = 3 0 , / . = 6 0 , and its components have Weibull reliability functions with parameters

Chapter 4

81

f l = 1/100, a = 1,

then, according to (2.6), its exact reliability function is given by R"-3o,6o(t) = 1 for t < 0, R--3o,6o(t)= [1 - [1 - exp[-0.01t]]6~ 3~ for t > 0. Applying Corollary 4.6 with the normalising constants a,, = 100/log30 ___-29.4, b,, = 1001og(60/log30) --- 287, from (1.1), we get the following approximate expression for the reliability function of the system u

m

R3o,6 0 (t) -=9? 3 ((t - b n ) / a n )= exp[-exp[0.034t - 9.76]] for t e (-oo,oo).

(4.103)

The mean values of the system component lifetimes T~are here E[T~]=I/fl

=100h,

while the expected value and the standard deviation of the system lifetime calculated from (4.103) are ([13], [71]): E[T] _=_-0.5772an + b,, --__270 h, o'(T) = xa n / ~

= 38 h.

The behaviour of the exact and approximate reliability functions of the considered system is illustrated in Table 4.6 and Figure 4.8.

Table 4.6. The values of the exact and approximate reliability functions of the model parallel-series system m

t .

,,

R30,60 (t)

0 1.0000. 100 1.0000 150 1.0000 200 0.9961 220 0.9742 240 0.9049 260'0.7453 280 0.4947 300 0.2382 320 0.0760 340 0.0151 360 0.0018

.ql3 ((t - bn ) / a n)

h = R3o,6 0 - ~ 3

1.0000

0.0000 0.0017 0.0084 0.0456 0.0714 0.0877 0.0742 0.0974 0.0265 0.0293 0.0127 0.0018

o.9983 0.9906 0.9495 0.9028 0.8172 ' 0.6713 0.3973 0.2117 0.0467 0.0024 0.0000

,,

.......

. . . . . .

82

Reliability o f Large Two-State Systems

R3o,6o(t), .9i~3((t- b,)/a,) 1.0 0.8 0.6 0.4 0.2 0.0

T

0

100

200

7--

300

"

400

t

Fig. 4.8. The graphs of the exact and approximate reliability functions of the model parallel-series system The generalisations of Lemmas 4.18-4.19 are the next lemmas proved in [56], [60][61 ] and [63], giving the way of finding limit reliability functions for non-homogeneous regular parallel-series systems. Lemma 4.20 If .qi"(t) is the limit reliability function of a non-homogeneous regular two-state seriesparallel system composed of components with reliability functions R(iJ)(t), i = 1,2,...,a,j = 1,2,...,ei,

then m

u

9?'(t) = 1 - 9 ? ' ( - 0 for t ~ Ca,, is the limit reliability function of a non-homogeneous regular two-state parallel-series system composed of components with reliability functions l~iJ)(t) - 1 - R (tJ) (-t) for t ~ CR(~.j), i - 1,2,...,a, j = 1,2,...,el.

At the same time, if (a., b. ) is the pair of normalising constants in the first case, then (a. ,-b n ) is such a pair in the second case.

Chapter 4

83

Lemma 4.21

If (i)

.qi"(t) = exp[- V'(t) ] is a non-degenerate reliability function,

(ii)

R knln (t) is the reliability function of a non-homogeneous regular two-state parallel-series system defined by (2.16)-(2.17),

(iii)

a. > O, b. e (-oo,oo),

then m !

m

lim R k.t. ( a . t + b . ) = g ? ' ( t )

for t e C~,

n---~oo

if and only if lim k n ~ q i [ F ( i ) ( a n t + b n ) ] t" = V'(t) fort e CV, . n---~oo

i=1

Lemma 4.22

If (i)

k, --+ oo,

(ii)

.qi" (t)= e x p [ - V ' ( t ) ] is a non-degenerate reliability function,

(iii)

R'k.t. (t) is the reliability function of a non-homogeneous regular two-state

u

m

parallel-series system defined by (2.16)-(2.17), (iv)

a. > 0, b. e (-oo,oo),

(v) F(t) is one of the distribution functions F(l)(t), F(2)(t),...,F(a)(t) defined by (2.17) such that (vi)

(vii)

:! N V n > N F(a.t + b.) = 0 for t < to and F(a.t + b.) ~ 0 for t > to, where to e to, i = 1,2,...,a, lim .~oo F ( a . t + b. ) and moreover there exists a non-decreasing function

Reliability of Large Two-State Systems

84

for t < t o

f

(viii) d(t) =

a

~

lim Y. qidi(an t+bn) n-.+oo i=l

(4.104)

for t >_to,

where F(i) (ant + bn ) ]t. di(a.t + b.) = [ F(ant + bn)

then lim R k.t. (a.t + b . ) = ~ ' ( t )

for t e C~,

(4.105)

n--~oo

if and only if lim k . [ F ( a . t + bn)] t" d(t) = V"(t) for t e CV, .

(4.106)

n ---~ oo

L e m m a 4.23 If (i)

k,, --->k, k > O , l,, ---~ o%

(ii)

~ ' ( t ) is a non-degenerate reliability function,

(iii)

Rk.t. (t) is the reliability function of a non-homogeneous regular two-state parallel-series system defined by (2.16)-(2.17),

(iv)

a. > O, b. e (-oo,oo),

(v) F(t) is one of the distribution functions FC~

defined by (2.17)

such that (vi)

3 N ~' n > N F(a.t + b.) = 0 for t < to and F(a.t + b.) r 0 for t > to, where to e O, a > O, fl > O, m

m!

(4.116)

m

~ ' 7 (t) = 1 for t < tl, ~ 7 (t) = e x p [ - d ( t ) e x p [ - s / c]] for tl < t < t2, m!

(4.117)

7 (t) = 0 for t > t2, tt < t2,

Case 3. k, ~ k, k > O, In ~ oo (under the assumptions o f L e m m a 4.23). "~'8 (t) = 1 f o r t < O ,

~ " 8 (t) = ~ [ 1 - d i ( t ) e x p [ - t - a ] ]

qik f o r t > O , a > O ,

(4.118)

i=1

-~'9 (t) = [qI [1 - di (t) e x p [ - ( - t ) = ]]qik for t < O, i-1 !

.~i" 9 ( t ) = 0 for t > O, a > O ,

(4.119)

"~'1o (t) = (-I[1-di(t)exp[-exp(-t)]] qik f o r t e (-oo,oo),

(4.120)

i=1 m

where d(t) and dr(t) are non-decreasing functions dependent on the reliability functions o f particular system components and their fractions in the system defined by (4.104) and (4.107) respectively.

CHAPTER 5

R E L I A B I L I T Y OF LARGE MULTI-STATE SYSTEMS Auxiliary theorems on limit reliability functions of multi-state systems, which are necessary for their approximate reliability evaluation, are formulated and proved. The classes of limit reliability functions for homogeneous and non-homogeneous series, parallel, series-parallel and parallel-series multi-state systems and for a homogeneous multi-state "m out of n'" system are fixed. Practical applications of the multi-state asymptotic approach to reliability evaluation of real technical systems are presented. On the basis of auxiliary theorems some corollaries are formulated and proved and then applied to approximate reliability and risk characteristics determination of real technical multi-state systems having series, parallel, "m out of n", series-parallel and parallel-series reliability structures. Evaluations are given of multi-state reliability functions, mean sojourn times in the state subsets and their standard deviations, mean lifetimes in the states, risk functions, and exceeding moments of a permitted risk level for selected real systems. The homogeneous series piping transportation system, the model, homogeneous series telecommunication network, the homogeneous series bus transportation system, the non-homogeneous series piping transportation system, the homogeneous parallel system of an electrical cable, the non-homogeneous parallel rope system, the "10 out of 36" homogeneous steel rope system, the model homogeneous series-parallel system, the homogeneous and non-homogeneous series-parallel pipeline systems and the homogeneous parallel-series electrical energy distribution system are analysed and their reliability characteristics are evaluated. Necessary data on system components reliability and system operation processes come from experts, from trade norms and from certificates of the system component producers. Component reliability and system operation processes data by necessity are approximate and concerned with the components' mean lifetimes in the reliability state subsets and the hypothetical distributions of these lifetimes. The accuracy of the asymptotic approach to the reliability evaluation of the considered systems is illustrated in tables and figures.

88

Reliability o f Large Multi-State Systems

5.1. Reliability evaluation of multi-state series systems In proving facts on limit reliability functions of homogeneous multi-state series systems we apply the following obvious extension of Lemma 4.1 ([74]). L e m m a 5.1

If m

(i) .qi'(t, u) = exp[-V (t,u) ], u = 1,2,...,z, is a non-degenerate reliability function, m

m

u

(ii) R n (t,.) = [ 1, g n (t,1) ..... R n (t, z) ], t e (-oo,oo), is the reliability function of a homogeneous multi-state series system defined by (3.20)-(3.21), (iii) an (u) > 0, b,(u) e (-oo,oo), u = 1,2,...,z, then m

m

.qi' (t,.) = [ 1, .q? (t,1) ,..., .qi'(t, z) ], t ~ (-~,oo), is the multi-state limit reliability function of this system, i.e. m

lim R n (a,(u)t + b,(u),u) = ~ ( t , u )

for t ~ C~(u), u = 1,2,...,z,

(5.1)

if and only if D

lim nF(an(U)t + b,(u),u) = V (t, u) for t ~ C~(u) , u = 1,2,...,z.

(5.2)

? l - - ~ o o

Motivation: For each fixed u, u = 1,2,...,z, assumptions (i)-(iii) of Lemma 5.1 are

identical to assumptions (i)-(iii) of Lemma 4.1, condition (5.1) is identical to condition (4.1) and moreover condition (5.2) is identical to condition (4.2). Since, from Lemma 4.1, condition (4.1) and condition (4.2) are equivalent, then conditions (5.1) and (5.2) are also equivalent. Fq Lemma 5.1 and Theorem 4.1 from Chapter 4 allow us to fix the class of all limit reliability functions for homogeneous multi-state series systems. Their application results in the following theorem ([74]). Theorem 5.1

The class of limit non-degenerate reliability functions of the homogeneous multi-state series system is composed of 3z reliability functions of the form .qi' (t,.) = [1, .qi' (t,1) ,..., ~ (t, z) ], t ~ (-~,oo),

(5.3)

Chapter 5

89

where (5.4)

.q/(t, u) ~ { .qi'l (t), .gi'2 (t), .q/3 (t) }, u = 1,2,...,z, m

and .gli(t), i = 1,2,3, are defined by (4.3)-(4.5). Motivation:

For each fixed u, u = 1,2,...,z, the co-ordinate ~ ( t , u )

of the vector

.qi'(t, .) def'med by (5.3), from Theorem 4.1 that is the consequence of L e m m a 4.1, may be one of the three types of reliability functions defined by (4.3)-(4.5). Thus the number of different multi-state limit reliability functions of the considered system is equal to the number of z-term variations of the 3-component set (5.4), i.e. 3 z, and they are of the form (5.3). E] C o r o l l a r y 5.1 If the homogeneous multi-state series system is composed of components having Weibull reliability functions

R(t,.) = [ 1,R(t, 1),...,R(t,z)], t ~ (-oo,oo), where

R(t,u) = 1 for t < O, R(t,u) = exp[-fl(u) t a(") ] for t > O, a(u) > O, fl(u) > O, u = 1,2,...,z,

and

a,(u) = (nil(u)) -1/at"), b,(u) = O, u = 1,2 .... ,z, then .012 (t,.) = [1, .q/2 (t,1) ,..., ill2 (t, z) ], t ~ (-o0,o0), where 81--2 (t, u) = 1 for t < O, 9i'--2 (t, u) = exp[-t a(") ] for t > O, u = 1,2,...,z, is its limit reliability function. M o t i v a t i o n : Since for each fixed u we have

a,(u)t + b,(u) < 0 for t < 0 and

a,(u)t + b,(u) > 0 for t >_ O,

Reliability of Large Multi-State Systems

90 then

F(an(U)t + bn(u),u) = 0 for t < 0 and

F(an(u)t + bn(u),u)= 1 -exp[-fl(u)(an(U)t + bn(u)) atu)] = 1 - exp[-tatU)/n ] = tatU)/n - o(1/n) for t > O. Hence

V(t,u)

=

lim nF (an(U)t + bn(u),u) = 0 for t < 0 n .--~ oo

and

V(t,u)

=

lim nF (an(U)t + bn(u),u) = t atu) for t > O, n ---) oo

which from Lemma 5.1 completes the proof. E x a m p l e 5.1 (a piping transportation system) The piping system is composed of n = 1000 identical pipe segments with reliability functions

R(t,u) = exp[-0.0002ut a] for t > 0, u = 1,2,3,4. Since it is a homogeneous five-state system, then according to (3.20) and (3.21) its multi-state reliability function is given by Rlooo (t, .) = [1,exp[-0.2t 3 ],exp[-0.4t 3],exp[-0.6t3],exp[-0.8t 3 ]] for t _ 0. Assuming normalising constants

an(u) = (0.0002.1000u)l/3, bn(u) = 0, u = 1,2,3,4, on the basis of Corollary 5.1, we conclude that the system limit reliability function is ~--2 9 (t,.) = [1,exp[-tal,exp[-t3],exp[-t3],exp[-t3]] for t > 0. Hence, considering (3.49), we arrive at the following approximate formula (it is exact in this case) m

Rlooo(t,

.) ~- "~2 ((t - b.(u))/a.(u),.)

Chapter 5

91

= [1,exp[-0.2t3],exp[-0.4t3],exp[-0.6ta],exp[-0.8t3]] for t > 0. The mean values of the sojourn times T~(u) in the state subsets in years, according to (3.4), are:

M~(u) = E[Tt(u)] =/(4/3)(0.0002u) -1/3, u = 1,2,3,4, i.e.

Mi(1) = 15.3, Mi(2) = 12.1,/14.(3) = 10.6, Mi(4) = 9.6, and according to (3.8), their mean sojourn lifetimes in the particular states are: Mi(1 ) = 3.2, M i ( 2 ) = 1.5, M;(3) = 1.0, M i ( 4 ) =9.6. The expected values of the system sojourn times T(u) in the state subsets, according to (3.13), are:

M(u) = E[T(u)] = 1-(4/3)(0.2u) -1/3, u = 1,2,3,4, i.e.

M(1) --- 1.53, M(2) --- 1.21, M(3) _= 1.06, M(4) ~ 0.96. Thus, from (3.17), the expected values of the system sojourn times in the particular states are: M(1) ~ 0.32,M(2) ~ 0.15,M(3) _~0.10, M(4) ~ 0.96. If the critical state is r = 2, then from (3.18), the system risk function is given by r(t) ~ 1 - exp[-0.4t 3] for t > 0. The moment when the risk exceeds a permitted level 8 = 0.05, calculated according to (3.19), is r = r-l(b") ~ (-log(1 - 8)/0.4) 1/3 = 0.5 years. The graphs of the piping system limit multi-state reliability function and its risk function are plotted using a computer program ([71 ]) and presented in Figure 5.1.

Reliability of Large Multi-State Systems

92

Fig. 5.1. The graphs of the piping system reliability function and risk function Corollary 5.2 If the homogeneous multi-state series system is composed of components having Erlang's reliability function of order 2 given by R(t,.) = [1,R(t,1),...,R(t,z)], t ~ ( - ~ , ~ ) ,

where

R(t,u) = 1 for t < O, R(t,u) = [ 1+ 2(u)] exp[ - 2 ( u ) t] for t > O, 2(u) > O, u = 1,2 .... ,z, and

G

an(U) = 2(u)x/-nn' bn(u)= O, u = 1,2,...,z, then .qi'2 (t,.) = [ 1, -q?2 (t,1) ,..., .qi'2 (t, z) ], t e (-oo,oo), where

.91--2 (t,u) = 1 for t < O, .9?"-2 (t,u) = e x p [ - t 2 ] for t > O, u = 1,2,...,z, is its limit reliability function.

Chapter 5

93

Motivation: Since for each fixed u, we have

a.(u)t + b.(u) < 0 for t < 0 and

an(U)t + bn(u) >-0 for t > O, then

F(a.(u)t + bn(u),u) = 0 for t < 0 and

F(an(U)t + bn(u),u) = 1 - [1 + A ( u ) a n ( u ) t ] e x p [ - 2 ( u ) (an(U)t)] = 1 - [1 + ,f2tl ~n ] e x p [ - 4r2tl ~n ]

= 1 - [ 1 + ,~r

][1-4~t14~n

+ t 2 in -o(lln)]

= t 2 / n + o(1/n) for t >__O. Hence V (t, u) = lim nF(a.(u)t + b.(u),u) = 0 for t < 0 n - - ~

and m

V (t, u) = lim nF(an(u)t + b.(u),u) = lim n[t n..-~ct~

2

/ n -- o(1/n)] = t 2 for t > 0,

n-.~oo

which from L e m m a 5.1 completes the proof.

Example 5.2

(a model telecommunication network)

The telecommunication network operating for telephone subscribers is c o m p o s e d o f n = 2000 subscriber terminals, subscriber cables and one head linking subscriber cables with distributing cables. We analyse the reliability of the network part that consists of subscriber cables only. Thus the considered system is c o m p o s e d o f n = 2000 double cables that consist o f one basic cable and one cable in a cold reserve. The cables are five-state (z = 4) components o f the system having exponential reliability functions with the following transition rates between the state subsets

1

2(u) = ~ , 60 - 10u

u = 1,2,3,4.

94

Reliability of Large Multi-State Systems

Under this assumption, since the sojourn time of a double cable in the state subsets is the sum of two lifetimes having exponential distributions, then it has Erlang's distribution of order 2, i.e. its reliability function is given by 1

1

R(t,u)= 1 fort 0, u = 1,2,3,4. 60 - 10u 60-10u Thus the considered part of the telecommunication network is a homogeneous five-state series system and according to Corollary 5.2, assuming the normalising constants a n (u) = 0.6 - 0. l u , b n (u) = 0 for u = 1,2,3,4, we conclude that its limit reliability function is .~i"-2(t,.) = [1 ,exp[-t2],exp[-t2],exp[-t2],exp[-t 2] for t > 0. Hence, from (3.49), we get the following approximate formula m

R2oooo (t,-) ~ .qi'2 ((t - bn(u))/an(u),.) = [1,exp[-0.781t2],exp[-1.389t2],exp[-3.125t2],exp[-12.5t2]] for t > 0. The mean lifetimes T,(u) of the system components in the state subsets in years, according to (3.4), are: M,.(u) = E[ T,.(u)] = 60 - 10u, u = 1,2,3,4,

i.e. M i ( 1 ) = 5 0 , M i ( 2 ) = 4 0 , Mi(3)= 30, Mi(4)=20, and from (3.8), the system component mean lifetimes in particular states are: M~(1) = 10, M~(2) = 10, M~(3) = 10, M~(4) = 20. The expected values of the network sojourn times T(u) in the state subsets, according to (3.13), are" M(u) = E[T(u)] =/-(3/2)(0.6-0.1u), u = 1,2,3,4,

i.e. M(1) _~ 0.44, M(2) = 0.35, M(3) --- 0.27, M(4) -_-0.18. Thus, from (3.17), the lifetime expected values of the network in particular states are"

Chapter 5

95

M(1) _= 0.09, M ( 2 ) --- 0.08, M ( 3 ) = 0.09, M ( 4 ) -_- 0.18. If the critical state is r = 2, then from (3.18), the network risk function is given by

r(t) = 1 - exp[-6.25fl] for t >_ 0. The m o m e n t w h e n the network risk exceeds an admissible level 6 = 0.05, from (3.19), is r = r-l(6) -- [-log(1 - 6)/6.25] 1/1 = 0.09 years.

Corollary 5.3 If c o m p o n e n t s o f the h o m o g e n e o u s multi-state series system have reliability functions

R(t,.) = [1,R(t,1),...,R(t,z)], t e (-oo,oo), where

R(t,u) = 1 for t < 0, R(t,u) = r1 exp[ "21 (u) t]+ r 2 exp[--'~'2 (U) t] for t >_ 0, u = 1,2,...,z,

,~,I(U) > O, 22(U)>0 , O___r1 < l , r 2 = 1 - r l ,

and an(u) =

[r 121 (U)+ ?'2~2 (u)]n

, bn(u)= O, u = 1,2,...,z,

then .qi'2 (t,.) = [ 1, .012 (t,1) ,..., .qi'2 (t, z) ], t ~ (-oo,oo), where m

~72 (t, u) = 1 for t < O, ~2 (t,.) = e x p [ - t ] for t > O, u = 1,2, .... z, is its limit reliability function.

Motivation: Since for each fixed u, we have an(U)t + bn(u) < 0 for t < 0 and

an(u)t + bn(u) > 0 for t > 0, then

Reliability of Large Multi-State Systems

96

F(a.(u)t + b.(u),u) = 0 for t < 0 and

F(a.(u)t + b.(u),u) = 1 - [rlexp[-21 (u) (a.(u)t) + r2exp[-22 (u) (a.(u)t)] = [rt d.l (u) + r2 22 (u) ][a.(u)t- o(a.(u))]

= t / n - o(1/n) for t > O. Hence V (t, u) = lim nF(a~(u)t + b~(u),u) = 0 for t < 0 n--.~oo

and V (t, u) = lim nF(a~(u)t + b.(u),u) = lim n[ t / n - o(1/n)] = t for t > O, n----~oo

/I--~oo

which from Lemma 5.1 completes the proof.

Example 5.3 (a bus transportation system) The city transportation system is composed of n, n > 1, buses necessary to perform its communication tasks. We assume that the bus lifetimes are independent random variables and that the system is operating in successive cycles (days) c = 1,2, .... In each of the cycles the following three operating phases of all components are distinguished: j~-components waiting for inclusion in the operation process, lasting from the moment to up to the moment tl, j ~ - components' activation for the operation process, lasting from t l up to t2, j ~ - components operating, lasting from t2 up to t3 = to. Each of the system components during the waiting phase may be damaged because of the circumstances at the stoppage place. We assume that the probability that at the end moment t~ of the first phase the ith component is not failed is equal to p~l), where 0 _< p}l) 0, u = 1,2,

is the reliability function of the ith component performing two tasks. Thus the considered transportation system is a homogeneous three-state series system and according to Corollary 5.3, assuming the normalising constants

an(u ) = 1/[[r 1 / ( 1 5 - 5 u ) + r 2/(lO-2u)]n], bn(u ) = 0 for u = 1,2, we conclude that its limit reliability function is

Reliability of Large Multi-State Systems

98 m

~2 (t,.)

= [1,exp[-t], exp[-t]] for t > 0.

Hence, from (3.49), we get the following approximate formula

R n (t, .) = .ql2 ( ( t - bn(u))/an(u),.) = [ 1 , e x p [ - ( q / 1 0 + r 2 / 8 ) n t ],exp[- ( q / 5 + r 2 / 6 ) n t ]] for t > 0. The mean values of the system lifetimes T(u) in the state subsets, according to (3.13), are:

M(u) = E[T(u)] = l / [ [ q / ( 1 5 - 5u) + r 2 ( 1 0 - 2u)ln], u = 1,2. If we assume that n = 30, rl = 0.8,

r2 =

0.2,

then R3o (t, .) ~ [ 1,exp[-3.15t],exp[-5.80t]] for t > 0

(5.8)

and M(1) = 0.32, M(2) ~ 0.17. Thus, considering (3.17), the expected values of the sojourn times in the particular states are: m

m

M(1) = 0.15, M(2) ~ 0.17. If a critical state is r = 1, then according to (3.18), the system risk function is given by

r(t) =_ 1 - exp[-3.15t] for t > 0. The moment when the system risk exceeds a permitted level 6 = 0.05, according to (3.19), is r = r-l(8) --__-log(1 - 8)/3.15 = 0.016 years =_ 6 days. At the end moment of the system activation phase, which is simultaneously the starting moment of the system operating phase t2 the system is able to perform its tasks with the probability p 0'2) defined by (5.7). Therefore, after applying the formula (5.8), we conclude that the system reliability in c cycles, c = 1,2 .... , is given by the following formula

Chapter 5

99

G(c,.) = [1, p(1,2) exp[-3.15ct4], p(l,2) exp[-5.80ct4]], where t4 = t3 for instance

-

t2 is the time duration of the system operating phase j~. Further, assuming

p(1.2) = p(1)p(2) = 0.99.0.99 = 0.98, t4 = 18 hours = 0.002055 years for the number of cycles c = 7 days = 1 week, we get G(7,-) _=_[1, 0.966, 0.902]. This result means that during 7 days the considered transportation system will be able to perform its tasks in state not worse than the first state with probability 0.966, whereas it will be able to perform its tasks in the second state with probability 0.902. In finding the class of limit reliability functions for non-homogeneous multi-state series systems we use an obvious extension of Lemma 4.3 formulated as follows ([74]). Lemma 5.2 If m

(i) .0i"(t, u) = exp[ - V'(t, u) ], u = 1,2,...,z, is a non-degenerate reliability function, (ii) R n (t,.) = [1,R n (t,1),...,R n (t,z)], t ~ (-oo,oo), is the reliability function of a non-homogeneous multi-state series system defined by (3.33)-(3.34), (iii) a, (u) > 0, b,(u) e (-oo,oo), u = 1,2,...,z, (iv) F(t,u) for each fixed u, is one of distribution functions F~l)(t,u),F~Z)(t,u),...,F~a)(t,u) defined by (3.32) such that (v)

(vi)

3 N(u) V n > N(u) F(an(u)t + bn(u),u) = 0 for t < to(u), F(a,(u)t + b,(u),u) ~ 0 for t >_to(U), where to(u) ~ t0(u), i = 1,2,...,a, u = 1,2,...,z, n~oo F(a, (u)t + b, (u), u) lim

and moreover there exist non-decreasing functions

Reliability of Large Multi-State Systems

100

(vii)

f

d(t,u) =

0

for t < t o (u)

lim ~qidi(an(u)t+bn(u),u)

for t>to(U ),

(5.9)

n---~oo i = 1

where m

(viii)

di (an(U)t +bn(u),u)=

F (i) (a n (u)t + b n (u), u)

(5.10)

F(a n (u)t + b n (u), u)

then w

m

.9i" (t,.) = [ 1, .9i" (t,1) ,..., .9i" (t, z) ], t ~ (-oo,oo), is the multi-state limit reliability function of this system, i.e. --' n (an(u)t + bn(u),u) = .~i" - - (t,u)for t ~ C~,(u) , u = 1,2,...,z, lim R

(5.11)

n - + o o

if and only if m

lim nF(an(U)t+bn(u),u ) d(t,u) = V'(t,u) fort ~ CF.(u), u = 1,2,...,z.

(5.12)

n ...~ o o

For each fixed u, u = 1,2,...,z, assumptions (i)-(viii) of Lemma 5.2 are identical to assumptions (i)-(viii) of Lemma 4.3, condition (5.11) is identical to condition (4.7), and condition (5.12) is identical to condition (4.8). Moreover, since from Lemma 4.3, conditions (4.7) and (4.8) are equivalent, then Lemma 5.2 is valid. [2

Motivation:

Lemma 5.2 and Theorem 4.2 from Chapter 4 establish the class of limit reliability functions for non-homogeneous multi-state series systems pointed out in the form of the next theorem ([74]). T h e o r e m 5.2

The class of limit non-degenerate reliability functions of the non-homogeneous multistate series system, under the assumptions of Lemma 5.2, is composed of 3z reliability functions of the form .qi" (t,.) = [1, .qi" (t,1) ,..., .9/' (t, z) ], t e (-oo,oo),

(5.13)

where gi"(t, u) ~ { .9i"i (t) , .9i"2 (t) , .qi"3 (t) } , U = 1,2,...,Z,

(5.14)

Chapter 5

101

and 9/' i (t) , i = 1,2,3 , are given by (4.9)-(4.11) with d(t) = d(t,u), u = 1,2, .. .z, where m

d (t, u) are defined by (5.9). Motivation" For each fixed u, u = 1,2,...,z, the co-ordinate .~"(t,u)

of the vector

9i"(t,.) defined by (5.13), from Theorem 4.2 that is the consequence of Lemma 4.3, may be one of the three types of reliability functions given by (4.9)-(4.11) with d (t) = d (t, u), where d (t, u) are defined by (5.9). Therefore, the number of different multi-state limit reliability functions of the considered system is equal to the number of z-term variations of the 3-component set (5.14), i.e. 3 z, and they are of the form (5.13). D

Corollary 5.4 If the non-homogeneous multi-state series system is composed of components having Weibull reliability functions

R(~ .) = [1,R(O(t,1),...,R(i)(t,z)], t ~ (-oo,oo), where

R(O(t,u) = 1 for t < O, R(~

= exp[-fli(u) t ai(u) ] for t > O, ai(u) > O, ill(u) > O,

i = 1,2, .... a, u = 1,2,...,z, and

an(U) = (fl(u)n) -t/~u), bn(u) = 0 for u = 1,2 ..... z, where a ( u ) = m i n { a / ( u ) } , ,8(u) =

max

l O, b,(u) e (-oo,oo), u = 1,2 .....z, then 9~(t ; ) = [1,.91(t,1),...,2R(t,z)], t ~ (-oo,~),

is the multi-state limit reliability function of this system, i.e. lim Rn(an(u)t + b,(u),u) = 9~(t,u)

for t ~ C g t ( , ) , u = 1,2,...,z,

(5.15)

1,2 ..... z.

(5.16)

n-.~oo

if and only if lim n R ( a , ( u ) t + b,(u),u) = V(t,u)

for t ~ Cv(u), u =

n-.>oo

For each fixed u, u = 1,2,...,z, assumptions (i)-(iii) of Lemma 5.3 are identical to assumptions (i)-(iii) of Lemma 4.5, condition (5.15) is identical to condition (4.18) and condition (5.16) is identical to condition (4.19). Since, from Lemma 4.5, condition (4.18) and condition (4.19) are equivalent, then conditions (5.15) and (5.16) are equivalent. [2

Motivation:

Lemma 5.3 and Theorem 4.3 from Chapter 4 determine the class of all non-degenerate limit reliability functions for homogeneous multi-state parallel systems. Namely, their application results in the following theorem ([74]). Theorem

5.3

The class of limit non-degenerate reliability functions of the homogeneous multi-state parallel system is composed of 3z reliability functions of the form 9f(t ,. ) = [ 1,.qf(t, 1),...,.qf(t,z)], t ~ (-oo,oo),

(5.17)

Chapter 5

107

where

.ql(t,u) ~ {.qi'l(t),.qi'2(t),.qi'3(t)}, u = 1,2,...,z,

(5.18)

and .qI~(t), i = 1,2,3, are defined by (4.20)-(4.22). Motivation: For each fixed u, u = 1,2,...,z, the co-ordinate 9~(t,u) of the vector .qI(t ,. ) def'med by (5.17), from Theorem 4.3 that is the consequence of Lemma 4.5, may be one of the three types of reliability functions given by (4.20)-(4.22). It means that the number of different limit multi-state reliability functions of the considered system is equal to the number of z-term variations of the 3-component set (5.18), i.e. 3 z, and they are of the form (5.17). [3 C o r o l l a r y 5.5 If components of the homogeneous multi-state parallel system have Weibull reliability functions

R(t,.) = [1,R(t,1) ..... R(t,z)], t ~ (-oo,oo), where

R(t,u) = 1 for t < 0, R(t,u) = exp[-fl(u)t a(u) ] for t > 0, a(u) > 0, fl(u) > 0, u = 1,2,...,z, and

an(U) = bn(u)/( a(u)log n), bn(u) = (log n/fl(u)) va~u), u = 1,2,...,z, then ~3(t,.) = [1,.qi'3(t,1),...,.qi'3(t~z)], t e (-oo,oo), where .~'3(t,u) = 1 - exp[-exp[-t]] for t e ( - ~ , ~ ) , u = 1,2,...,z, is its limit reliability function. Motivation: Since for each fixed u, sufficiently large n and all t e (-0%00), we have

a,(u)t + b,(u) = b,(u)(t/(a(u)log n) +1) > O, then

R(a,(u)t + b,(u),u) = exp[-fl(u)(a,(u)t + b,(u)) ~")] for t e (-~,oo). Hence

108

Reliability of Large Multi-State Systems

nR(a,(u)t + b,(u),u) = n exp[-fl(u)(a,(u)t + bn(u)) ~u)] = n exp[-fl(u)(b,(u))~u)(t/(a(u)log n) + 1))~ O,

then for sufficiently large n

R(a.(u)t + b.(u),u) = e x p [ - 2 ( u ) (a.(u)t + b.(u))] for t ~ (-oo,oo). Hence

V(t, u ) = lira n R(a.(u)t + b.(u),u) n - ~ oo

= lim n exp[-2(u)(a.(u)t + b.(u))] n.-~oo

= lim n exp[- t - log n ] n.-~oo

= exp[-t] for t e (-oo,oo), which from Lemma 5.5 completes the proof.

Example 5.7 (a steel rope, durability) Let us consider a steel rope composed of n = 36 four-state, i.e. z = 3, identical wires having exponential reliability functions with transitions rates between the state subsets 2(u) = 0.2u/year, u = 1,2,3. Assuming that the rope is in the state subset {u, u + 1,..., z} if at least m - 10 of its wires are in this state subset, according to Definition 3.9, we conclude the rope is a homogeneous four-state "10 out of 36" system. Thus, according to (3.24)-(3.25), its reliability function is given by

R ~6~ (t,.) = [1,R ~16~(t,1) ,R ~6~ (t,2) ,R ~6~ (t,3) ], where R ~6~ (t, u) = 1 for t < 0,

R ~16~(t, u) = 1 - ~,

9 exp[-iO.2ut][1 - exp[-O.2ut]] 36-i , u = 1,2,3.

i-0

Applying Corollary 5.7 with normalising constants

Chapter 5

123

a n ( u ) = _5, b n ( u ) = _5 l o g 3 6 , u = 1,2,3, u

u

we conclude that the rope limit reliability function is given by

.9/~O)(t,.)

=

[1,.w~O)(t,

....(0) .--(0> 1),'.hr 3 (t,2),'dt 3 (t,3)], t ~ (-oo,oo),

where 9

exp[-it]

i=o

i!

81~~ ( t , u ) = 1 - ~

exp[-exp[-t]].

Hence, considering (3.49), since a n (1) - 5.00, b n (1) = 17.92, a n (2) = 2.5, b, (2) = 8.96, a n (3) = 1.67, b n (3) = 5.97,

then the approximate formula for the rope reliability function takes the form t-bn(u)

,.)

an(U)

9 exp[-i(O.2t-3.58)] exp[-exp[-O.2t + 3.58]], = [1,1 - 5_.. i=o i! 9 exp[-i(0.4t 1- E i=o i!

3.58)]

1 - ~ exp[-i(0.6t-3.58)] i=0

exp[-exp[- 0.4t + 3.58]],

e x p [ - e x p [ - 0 . 6 t + 3.581] ], t ~ (-0%00).

i!

The approximate mean values of the rope lifetimes T(u) in the state subsets and their standard deviations in years, by (3.13), are: M(1) ~ 6.66, M(2) _=_3.33, M(3) ~ 2.22, o(1) ~_ 1.62, 0(2) _=0.81, 0(3) ~ 0.54, whereas, from (3.17), the approximate mean values of the rope sojourn times in the particular reliability states are:

Reliability of Large Multi-State Systems

124 u

m

M(1) ~3.33, M(2) -= 1.11, M(3) -_-2.22. If the critical state is r = 2, then from (3.18) the rope risk function is approximately given by

r(t)= ~" exp[-i(0.4t-3.58)] i=O i!

exp[-exp[- 0.4t + 3.58]], t ~ (-oo,~).

The moment when the risk exceeds an admissible level 6 = 0.05, after applying (3.19), is r --- 2.0738 years. The behaviour of the rope system reliability function and its risk function are illustrated in Table 5.2 and Figure 5.6. Table 5.2. The values of the still rope multi-state reliability function and risk function

t

,,,

0.2 0.6 1.0 1.4 1.8 2.2 2.6 3.0 3.4 3.8 4.2 4.6 5.0 5.4 5.8 6.2 6.6 7.0 7.4 7.8

9i'~~

!

( t _-bn(1) _~ a,(1) 1.000000 0.999998 0.999990 0.999950 0.999795 0.999283 0.997831 0.994249 0.986488 0.97156'9 0.945898 0.906025 0.849686 0'776749 0.689654 0.593139 0.493317 0.396455 0.307841 0.231067

,1)

I

-qi'~~ ( t-bn(2 ~,2) ) an(2 ) 0.999999 0.999977 0.999609 0.996412 0.980140 0.927919 0.815200 0.642210 0.444152 0.267825 0.141305 0.065843 0.027422 0.010338 0.003571 0.001143 0.000342 0.000097 0.000026 0.000007

~ o ) ( t-bn(3) ~---,3) an(3 ) 0.999998 0.999795 0.994249 0.945898 0.776749 I 0.493317 i l 0.231067 0.080576 0.021678 0.004693 O.OO0851 0.000134 0.000019 O.00OOO2 0.000000 I

o.oooooo o.oo0ooo o.oooooo 0.000000 0.000000

r(t) o.oooooo

I

I

i

0.000023 0.000391 0.003588 0.019860 0.072081 0.184800 0.357790 O.555848 0.732175 0.858695 0.934157 '0.972578 0.989662 0.996429 0.998857 ' 0.999658 0.999903 0.999974 0.999993

Chapter 5

125

Fig. 5.6. The graphs of the still rope multi-state reliability function and risk function Unfortunately, similarly to the case of two-state systems, there are no extensions of Lemmas 4.5-4.7 to non-homogeneous multi-state "m out of n" systems.

5.4. Reliability evaluation of multi-state series-parallel systems In proving facts on limit reliability functions for homogeneous regular multi-state series-parallel systems the following extensions of Lemmas 4.12--4.13 are used ([74]).

Lemma 5.8 If (i) k. ~ oo,

(ii) 9f(t,u) = 1 - e x p [ - V ( t , u ) ] , u = 1,2 .... ,z, is a non-degenerate reliability function, (iii) R kn,tn (t,") = [1,R kn,l, (t,1),...,R k~,tn (t,z)], t ~ (-o0,~), is the reliability function of a homogeneous regular multi-state series-parallel system defined by (3.28)(3.29), (iv) a, (u) > O, bn(u) ~ (-oo,oo), u = 1,2,...,z,

Reliability of Large Multi-State Systems

126 then

.O~(t,. ) = [ 1,.Oft(t,1),...,g~(t,z)], t ~ (-oo,oo), is the limit multi-state reliability function of this system, i.e. lim R k,.t, (a,(u)t + b,(u),u) = .gl(t,u) for t e Cgt(u) , u = 1,2 .....z, n--).oo

(5.36)

if and only if lira kn[R(an(U)t + b , ( u ) , u ) ] t" = V(t,u) f o r t e Cv(u), u = 1,2,...,z. n--~oo

(5.37)

For each fixed u, u = 1,2,...,z, assumptions (i)-(iv) of Lemma 5.8 are identical to assumptions (i)-(iv) of Lemma 4.12, condition (5.36) is identical to condition (4.47) and condition (5.37) is identical to condition (4.48). Moreover, since from Lemma 4.12, conditions (4.47) and (4.48) are equivalent, then conditions (5.36) and (5.37) are equivalent.

Motivation:

Lemma

5.9

If (i) k,, --->k, k > O, l, --->oo, (ii) .q~(t,u), u = 1,2,...,z, is a non-degenerate reliability function, (iii) R k.,t. (t,') = [1,R kn,t. (t, 1) .... ,R k.,t. (t,z)], t e (-oo,oo), is the reliability function of a homogeneous regular multi-state series-parallel system defined by (3.28)(3.29), (iv) a. (u) > 0, bn(u) ~ (-oo,oo), u = 1,2,...,z, then .~(t,. ) = [1,.87(t,1) .... ,.~(t,z)], t ~ (-oo,oo), is its multi-state limit reliability function, i.e. lim R k,.t, (a,(u)t + b,(u),u) = .gl(t,u) for t ~ C ~ ( u ) , u = 1,2,...,z, n-r

(5.38)

if and only if

lim[R(an(u)t + b n ( u ) , u ) ] t"= .qlo(t,u) for t ~ Cgto(u) , u = 1,2,...,z, rt - ~ oo

(5.39)

Chapter 5

127

where 9?o(t,u) is a non-degenerate reliability function and moreover .qi[t,u) = 1 - [ 1 - .9?o(t,u)]k for t ~ (-oo,oo), u = 1,2,...,z.

(5.40)

Motivation: For each fixed u, u = 1,2,...,z, assumptions (i)-(iv) of Lemma 5.9 are identical to assumptions (i)-(iv) of Lemma 4.13. Moreover, condition (5.38) is equivalent to condition (4.49) and condition (5.39) is equivalent to condition (4.50). And moreover, since from Lemma 4.13, condition (4.49) and condition (4.50) are equivalent, then condition (5.38) and condition (5.39) are equivalent and moreover, considering (4.51), the equality (5.40) holds. Lemmas 5.8-5.9 and Theorem 4.6 from Chapter 4 are the basis for formulating the next theorem ([74]). T h e o r e m 5.6 The class of limit non-degenerate reliability functions of the homogeneous regular multi-state series-parallel system is composed of 3 z + 4 z + 3 z reliability functions of the form

9/(t,. ) = [1,9/[t,1),...,9/[t,z)], t ~ (-oo,oo),

(5.41)

where C a s e 1. kn = n, ]In - c log n I>> s, s > 0, c > 0 (under Assumption 4.1). #~(t,u) ~ {~l(t),gi'2(t),gi'3(O}, u = 1,2,...,z,

(5.42)

and 91~(t), i = 1,2,3, are defined by (4.52)-(4.54), C a s e 2. kn = n, In - c log n ~ s, s ~ (-~,oo), c > 0. .ql(t,u) ~ {.qi'4(t),~5(t),.q/6(t),.~i'7(t)}, u = 1,2,...,z,

(5.43)

and 9//(0, i = 4,5,6,7, are defined by (4.55)-(4.58), Case3.

k,---> k, k > O , l~--> oo.

.qi(t,u) ~ { g t s ( 0 , g t 9 ( t ) , g t ~ 0 ( t ) } ,

u = 1,2,...,z,

(5.44)

and .q/~(t), i = 8,9,10, are defined by ( 4 . 5 9 ) - ( 4 . 6 1 ) . Motivation: For each fixed u, u = 1,2,...,z, co-ordinate 9t(t,u) of the vector 9/(t,. ) defined by (5.41), from Theorem 4.6 that is the consequence of Lemma 4.12 and Lemma 4.13, can be a reliability function that is one of three types defined by (4.52)(4.54), or one of four types defined by (4.55)-(4.58), or one of three types defined by (4.59)-(4.61). Thus the number of different multi-state reliability functions of the

Reliability of Large Multi-State Systems

128

considered system is equal to the sum of the number of z-term variations of the 3component sets defined by (5.42) and (5.44) and the 4-component set defined by (5.43), i.e. 3 z + 4 z + 3 z, and they are of the form (5.41).

Corollary 5.8 If components of the homogeneous regular multi-state series-parallel system have Weibull reliability functions R(t,.) = [1,R(t,1) ..... R(t,z)], t ~ (-oo,oo), where

R(t,u) = 1 for t < O, R(t,u) = exp[-fl(u) t a(u) ] for t > O, a(u) > O, fl(u) > O, u = 1,2,...,z, and

kn=n,

/n>O,

a.(u) = bn(u)/(a(u)log n), b.(u) = (log n/(fl(u)l.))l/a"), u = 1,2,...,z, then .q/3(t,') = [1,.qi'3(t,1),...,.qi'3(t,z)], t e (-oo,oo), where .qi'3(t,u) = 1 - e x p [ - e x p [ - t ] ] for t e (-oo,oo), u = 1,2,...,z, is its limit reliability function.

Motivation: Since for each fixed u, sufficiently large n and all t r (-o%00), we have a.(u)t + bn(u) = bn(u)(t/(a(u)log n) + 1) > O, then for t e(-oo,oo) we get

R(a,(u)t + bn(u),u) = exp[-~u)(a,(u)t + b,(u)) au)] = exp[-(log n)ll~.(t/(a(u)log n) + 1) a")] = exp[-(log n)/ln - t/l. - o(1/ln)]. Further, for all t e (-oo,oo), we have

V(t,u)- lim kn[R(an(u)t + bn(u),u) ]t. n -.r

Chapter 5

129

= lim n e x p [ - t - log n + lno(1/ln)] = exp[-t], n----~oo

which from Lemma 5.8 completes the proof.

Example 5.8 (a model series-parallel system) If the homogeneous regular multi-state series-parallel system is such that

kn=30, ln=lO, z=5, and its components have Weibull reliability functions with parameters fl(u) = 10 -5, a(u) = (11 + u)/4, u = 1,2,3,4,5, then, according to Corollary 5.8, assuming normalising constants

an(u) = 4bn(u)/((11 + u).log30), bn(u) = (104-1og30) 4/(ll § u), u = 1,2,3,4,5, considering (3.49), we get the following approximate formula [ 1,R30,10(t, 1),R3o,lo(t,2),R3o, lo(t,3),R3o,lo(t,4),R3o,lo(t,5)] _-- [1,1 - exp[-exp[-0.315t + 10.204]],1 - exp[-exp[-O.446t + 11.054]], 1 - exp[-exp[-0.604t + 11.904]],1 - exp[-exp[-0.789t + 12.755]], 1 - exp[-exp[-1.002t + 13.605]]] for t e (-oo,oo). According to (3.4), the expected values of the system components' sojourn times Tii(u) in the state subsets are given by the formula ([71 ]) M/j(u) = E[T/j(u)] = (fl(u))-l/atU~l((a(u) + 1)/a(u)) = 102~

+ u)/((15 + u)/(11 + u)), u = 1,2,...,5.

Hence, in particular, we have

Mo.(1)=41.46, Mo.(2)=_31.O1, Mo{3)=24.19, 34o.(4)--- 19.40, Mo.(5) = 16.12, and according to (3.8) the expected values of the system components' lifetimes in particular states are" MOO ) = 10.45, M0.(2 ) -_-6.82, M0(3 ) _--_4.79, Mij(4 ) = 3.28, Mij(5 ) =_16.12. The mean values of the system sojourn times T(u) in the state subsets after applying the formula (3.13), are given by ([13], [71])

Reliability of Large Multi-State Systems

130

M(u) = E[T(u)] --- 0.5772 a,(u) + b,(u), u = 1,2,...,5, i.e. M(1) --- 34.23, M(2) _=26.09, M(3) --- 20.67, M(4) -_- 16.89, M(5) -_- 14.16. Hence, from (3.17), the mean values of the system lifetimes in particular states are: M(1) ~_ 8.14, M ( 2 ) ~_ 5.42, M(3) ---3.78, M(4) --2.73, M(5) _=_14.16. If the critical state is r = 2, then from (3.18) the system risk function is r(t) _=exp[-exp[-O.446t + 11.054]]. The moment when the system risk exceeds an admissible level 6 = 0.05, from (3.19), is r = r-1(6) --- [ 1 1 . 0 5 4 - log[-log 6]]/0.446 = 22.32. C o r o l l a r y 5.9 If components of the homogeneous regular multi-state series-parallel system have Weibull reliability functions R(t,.) = [1,R(t,1),...,R(t,z)], t ~ (-0%00), where R(t,u) = 1 for t < O, R(t,u) = exp[-fl(u) t a(u) ] for t > O, a(u) > O, fl(u) > O, u = 1,2 .... ,z, and

a.(u) = (]~u)l.) -va~u), b.(u) = O, then .qi'9(t,.) = [1,~9(t,1),...,.q/9(t,z)] where .qi'9(t,u) = 1 for t < O, ~i'9(t,u) = 1 - [1 - e x p [ - t "(~) ]]3 for t >__O, is its limit reliability function.

(5.45)

Chapter 5

131

Motivation: Corollary 5.9 is a particular case of more general Corollary 5.10, which will be proved in a later part of this chapter. Therefore, we omit its proof. Example 5.9 (a pipeline system) Let us consider the pipeline system composed of k, = 3 lines of pipe segments linked in parallel, each of them composed of 1, = 100 five-state identical segments linked in series. Considering pipe segments as basic components of the pipeline system, according to Definitions 3.12-3.13, we conclude that it is a homogeneous regular fivestate series-parallel system. Therefore, from (3.28)-(3.29), the pipeline system reliability function is given by R 3,100

(t,.) = [ 1,R 3.1ooo(t,1) ,R 3,100 (t,2) ,R 3,100 (t,3) ,R 3,100 (t,4) ],

(5.46)

where

R3,1oo(t,u ) = 1 - [1 - [R( t,u) ]10013 , t

E

(--0%00), U = 1,2,3,4.

(5.47)

Taking into account pipe segment reliability data given in theft technical certificates and expert opinions we assume that they have Weibull reliability functions

R(t,u) = 1 for t < O, R(t,u) = e x p [ - fl(u)t aCu) ] for t > O, u = 1,2,3, 4, with the following parameters: a(1) = 3, /3(1) = 0.00001, a(2) = 2.5, /5'(2) = 0.0001, a(3) = 2, fl(3) = 0.0016, a(4) = 1, fl(4) = 0.05. Hence and from (5.46)-(5.47) it follows that the pipeline system exact reliability function is given by

Rs, loo(t,') = [ 1,1 - [ 1 - exp[-O.O017]]3,1 - [1 - exp[-O.Olt5/2]] 3, 1 - [1 - exp[-0.16t2]]3,1 - [1 - exp[-5t]] 3] for t > 0. From (3.4), the mean values Mo.(u ), u = 1,2,3,4, of the pipe segments in the state subsets in years are: M o O ) =/-'(4/3)(0.00001)-

1/

3 ---41.45, M~j(2) =/-'(7/5)(0.0001) -z/5 =_35.32,

M0(3) = F(3/2)(0.0016) -1/2 -- 22.16, M O O ) = F(2)(0.5) -2 -_-20.00,

132

Reliability of Large Multi-State Systems

while from (3.8), the mean values M o.(u ) , u = 1,2,3,4, of the pipe segments in particular states are: M~/(1) ~ 6.13, MO.(2 ) ~ 13.16, M/j(3) ~ 2 . 1 6 , M0.(4 ) ~ 2 0 . 0 0 . Assuming, according to (5.45), normalising constants an(u) = ( f l ( u ) t n ) - l / a ( u ) ,

bn(u) = 0, u = 1,2,3,4,

and applying Corollary 5.9, we conclude that the limit reliability function of the pipeline system is gl9(t,.) = [1,gi'a(t,1),Bl9(t,2),gi'9 (t,3),g?9 (t,4)], t ~ (-oo,oo), where gi'9(t, 1) : 1 - [1 - exp[-t3]] 3, .gi'9(t,2) = 1 - [1 - exp[-ts/2]] s, .gi'9(t,3) = 1 - [1 - exp[-t2]] 3, .gi'9(t,4) = 1 - [1 - exp[-t]] 3 for t >_ 0. Since, in particular from (5.45), we have a n (1) = 10, a n (2) = 6.31, a n (3) = 2.5, a n (4) = 0.2, b n (u) = 0, u = 1,2,3,4.

then applying the approximate formula (3.49) for t > 0, we get Ra.loo(t,') -= .gi'9((t- b , ( u ) ) / a , ( u ) , . ) = [1,1 - [1 - exp[-0.001t3]]3,1 - [1 - exp[-0.01ts/2]] 3, 1 - [1 - exp[-0.16t2]]3,1 - [1 - exp[-5t]]3].

(5.48)

The expected values M(u), u = 1,2,3,4, of the system sojourn times in the state subsets in years, calculated on the basis of the approximate formula (5.48), according to (3.13), are: M(1) = F(4/3)[3(0.001) - 1 / 3 - 3(0.002) -1/3 + (0.003) -1/3 ] ~ 11.72, M(2) = r ( 7 / 5)[3(0.01) -2/5 - 3(0.02) -2/5 + (0.03) -2/5 ] = 7.67, M(3) = F(3 / 2)[3(0.16) -1/2 - 3(0.32) -1/2 + (0.48) -1/2] ~ 3.23, M ( 4 ) = F(2)[3(5) - ~ - 3(10)-1 + (15) -1] -_-0.37.

Chapter 5

133

Hence, from (3.17), the system mean lifetimes M (u) in particular states are: M(1) -= 4.05, M(2) ~_4.44, M(3) ~ 2.86, M(4) ~ 0.37. If the critical state is r = 2, then the system risk function, according (3.18), is given by

r(t) =

[ 1 - exp[-0.0 lt5/2]]3.

The moment when the system risk exceeds an admissible level 6 = 0.05, from (3.19), is r = r-l(o") = [-lO01og(1- 3.~~')]2/s _--4.62. The behaviour of the exact and approximate multi-state system reliability function coordinate u = 2 and the risk function are presented in Tables 5.3-5.4 and Figures 5.7-5.8.

Table 5.3. The values of the component u = 2 of the exact and approximate piping system reliability function t 0.6 1.5 .......... 3.0 ' 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 .

.

.

bn(2))/an(2))

R 3,1oo(t,2)= .9i'9((t1.000 1.000 0.997 0.957 0.799 0.515 0.242 0.082 0.020 0.004 0.000

.

.

.

.

.

.

.

.

.

.

.

.

.

Table 5.4. The values of the piping system risk function

r(O

,,

.

.

0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0

.

.

J

,

.

.

.

0.000 0.000 0.003 0.043 0.201 0.485 0.758 0.918 0.980 0.996 1.000

Reliability of Large Multi-State Systems

134

I R3,1oo(t,2)

1.0 0.8 0.6 0.4 0.2 0.0

0.0

2.5

5.0

7.5

10.0

12.5

15.0

t

Fig. 5.7. The graph of the component u - 2 of the exact and approximate piping system reliability function

r(0 .0 8

6 4 2 t.0 0.0

I

[

I

2.5

5.0

7.5

I

10.0

12.5

Fig. 5.8. The graph of the piping system risk function

15.0

t

Chapter 5

135

The following extensions of Lemmas 4.15-4.16 are necessary tools in determination of limit reliability functions of non-homogeneous regular multi-state series-parallel systems ([74]). Lemma 5.10

If (i) k. --* ~, (ii) 9i"(t, u) = 1 - e x p [ - V ' (t,u)}, u = 1,2,...,z, is a non-degenerate reliability function, (iii) R'k.,t" (t,.) = [1, R'k.,t. (t,1) ,..., R'k.,t. (t, z) ], t e (-oo,oo), is the reliability function of a non-homogeneous regular multi-state series-parallel system defined by (3.43)-(3.45), (iv)

a. (u) > 0, b.(u) ~ (-o%~), u = 1,2, ....z,

(v) R(t,u) for each fixed u, is one of reliability functions R N(u) R(a.(u)t + b.(u),u) r 0 for t < to(U), R(an(U)t + b.(u),u) = 0 for t >__to(u), where to(u) ~ (-oo,oo>,

(vii)

lim n~

R (i) (a n (u)t + b n (u), u)

< 1 for t < t0(u), i = 1,2,...,a, u = 1,2,...,z,

R(a n (u)t + b n (u), u)

and moreover there exist non-increasing functions lim ~ qidi(an(U)t+bn(u),u ) f o r t t o (u),

where

di(a n (u)t + b n (u), u) = [

R (i) (a n (u)t + b n (u), u) in R(a n (u)t + b n (u), u) ]

then 9i" (t,.) = [ 1, .9i" (t,1) ,..., gi" (t, z) ], t ~ ( - ~ , ~ ) , is its multi-state limit reliability function, i.e.

(5.49)

Reliability of Large Multi-State Systems

136 lim R'

k n ,In

(an(U)t + b , ( u ) , u ) = .ql' (t, u) for t ~ C~.(u ) ~ u = 1,2,..,z, .

(5.50)

if and only if lim kn[R(an(U)t +bn(u),u)] t" d(t,u)=

V'(t,u) for t ~ Cz.(u), u = 1,2,...,z. (5.51)

n - ~ oo

Motivation: Since for each fixed u, u = 1,2,...,z, assumptions (i)-(viii) of Lemma 5.10 are identical to assumptions (i)-(viii) of Lemma 4.15, condition (5.50) is identical to condition (4.66) and condition (5.51) is identical to condition (4.67), which from Lemma 4.15 are equivalent, then Lemma 5.10 is valid. Lemma 5.11 If (i) k, --->k, k > O, l, --->oo, (ii) .qi"( t , u ) , u = 1,2,...,z, is a non-degenerate reliability function, (iii) R'k,,t" (t,.) = [1 ' R' k n ,In (t,1)

,'",

R' k n ,In (t, z) ], t ~ (-oo,oo), is the reliability

function of a non-homogeneous regular multi-state series-parallel system defined by (3.43)-(3.45), (iv) a, (u) > 0, b,(u) ~ (-oo,oo), u = 1,2,...,z, (v) R(t,u) for each fixed u, is one of reliability functions Rt~ defined by (3.45) such that

R(2)(t,u) ..... R(~)(t,u)

(vi) 3 N(u) V n > N(u) R(a,(u)t + bn(u),u) ~ 0 for t < to(U), R(a,(u)t + b,(u),u) = 0 for t > to(u), where to(U) ~ (-oo,oo>,

(vii)

lim n--~oo

R (i) (a n (u)t + b n (u), u) < 1 for t < t0(u), i = 1,2,...,a, u = 1,2,...,z, R(a n (u)t + b n (u), u)

and moreover there exist non-increasing functions

[lim di(an(U)t +bn(u),u ) for t _t o (u), where

Chapter 5 d i (a. (u)t + b,, (u), u)

=

[ 'R(i) (an (u)t + b n (u), u)]l n R(a. (u)t + b. (u), u) '

137

(5.53)

then .qi"(t,.) = [1,.gi"(t,1),...,.gl'(t,z) ], t ~ (-oo,oo), is its multi-state limit reliability function, i.e. lim R'k,.t, (a,(u)t + b,(u),u) = ~ ' (t, u) for t ~ C~..(,) , u = 1,2,...,z, n--~oo

(5.54)

if and only if lim[R(an(u)t + b , ( u ) , u ) ] l" = .9lo(t,u) for t ~ Cgto(u) , u = 1,2,...,z, n ..-) oo

(5.55)

where ~0(t,u) is a non-degenerate reliability function and moreover .qi" (t, u) = 1 - fi [1 - d i (t, u) ~o(t,u) ]qik,

t

~(-~,oo), u = 1,2 .... ,z.

(5.56)

i=1

For each fixed u, u = 1,2,...,z, assumptions (i)-(viii) of Lemma 5.11 and assumptions (i)-(viii) of Lemma 4.16 are equivalent. Condition (5.54) is identical to condition (4.69) and condition (5.55) is identical to condition (4.70). Moreover, from Lemma 4.16 conditions (4.69) and (4.70) are equivalent, which means that conditions (5.54) and (5.55) are also equivalent and from (4.71) the equality (5.56) holds.

Motivation:

Lemma 5.10, Lemma 5.11 and Theorem 4.7 from Chapter 4 establish the class of limit reliability functions for non-homogeneous regular multi-state series-parallel systems given in the following theorem ([74]). Theorem

5.7

The class of limit non-degenerate reliability functions of the non-homogeneous regular multi-state series-parallel system is composed of 3" + 4" + 3 z reliability functions of the form .qi" (t,.) = [1, .qi" (t,1) ,..., ~ ' (t, z) ], t ~ (-r

(5.57)

where Case I. k, = n, I1, - c log n I>> s, s > 0, c > 0 (under Assumption 4.1 and the assumptions of Lemma 5.10). W' (t, u) ~ { .qi"l (t), .qi"2 (t), ~'3 (t) }, u = 1,2,...,z,

(5.58)

Reliability of Large Multi-State Systems

138

and "~'i (t), i'- 1,2,3, are defined by (4.72)-(4.74) with d(t) = d(t,u), u = 1,2,...z, where d(t,u) are defined by (5.49),

Case 2. k, = n, 1~ - c log n ~ s, s ~ (-oo,oo), c > 0 (under the assumptions of Lemma 5.10).

.~?' (t, U) E { *~'4 (t), ~?'5 (t), ~?'6 (t), ~i?' 7 (t) }, u = 1,2,...,z,

(5.59)

and .qi"i (t), i = 4,5,6,7, are defined by (4.75)-(4.78) with d(t) = d(t,u), u = 1,2,...z, where d(t,u) are defined by (5.49),

Case 3. k, ~ k, k > 0 , l, ~ oo (under the assumptions of Lemma 5.11). 9l' ( t , u ) ~

{.qi" s (t), ~ ' 9 (t),.~'lO (t) }, u = 1,2 .... ,z,

(5.60)

and "~'i (t), i = 8,9,10, are defined by (4.79)-(4.81) with di(t) = di(t,u), u = 1,2,...z, where di(t,u) are defined by (5.52). Motivation: For each fixed u, u = 1,2,...,z, co-ordinate gt"(t,u)

of the vector 9i"(t,.)

defined by (5.57), according to Theorem 4.7, that is a consequence of Lemma 4.15 and Lemma 4.16, can be one of the three types of reliability functions given by (4.72)(4.74), or one of the four types of reliability function given by (4.75)-(4.78) with d(t) = d(t,u), where d(t,u) are defined by (5.49), or one of the three types of reliability functions given by (4.79)-(4.81) with di(t) = dj(t,u), where di(t,u) are defined by (5.52). Thus the number of different limit reliability functions of the considered system is equal to the sum of the number of z-term variations of the 3-element sets defined by (5.58) and (5.60) and the 4-element set defined by (5.59), i.e. 3 z + 4 z + 3 ~, and they are of the form (5.57). C o r o l l a r y 5.10 If components of the non-homogeneous regular multi-state series-parallel system have Weibull reliability functions

R (i'j) (t,.) = [ 1, R (i'j) (t,1) ,..., R (i'j) (t, z) ], t G (-oo,oo), where

ROJ)(t,u) = 1 for t < 0, ROJ)(t,u) = exp[-/~j(u) t a~ u = 1,2,...,z, i = 1,2,...,a,j = 1,2,...,ei, and

] for t > O, ao(u ) > O, flit(u) > O,

Chapter 5 a.(u) = (~u)l.) -l/a~"), bn(u) = O,

139

(5.61)

where

a i ( u ) = mill {aO. (u)} , P i ( U )

po.,Ou(u),

(5.62)

fl(u) = min{fli(u) : ai(u ) = a ( u ) } ,

(5.63)

loo

Chapter 5

141

which from Lemma 5.11 completes the proof.

Example 5.10 (a p i p i n g system) The piping system is composed of kn = 3 pipeline lines linked in parallel, each of them composed of In = 100 five-state pipe segments. In two of the pipelines there are 40 pipe segments with exponential reliability functions R 0, and 60 pipe segments with Weibull reliability functions R~l'2)(t, 1) = exp[-0.0015t2], Rr

= exp[-0.0016t2],

RCl'2)(t,3) = exp[-0.002t2], RCl'2)(t,4) = exp[-0.002512] for t >_ 0. The third pipeline is composed of 50 pipe segments with Weibull reliability functions Rt2'l)(t,1) = exp[-0.0007t3], R(2'I)(t,2) = exp[-0.0008t3], R 0, and 50 pipe segments with Weibull reliability functions R(2'2)(t,1) = exp[-0.15 xft ], R(2'2)(t,2) = exp[-0.16 ~ ], R 0. Thus the piping system is a non-homogeneous regular multi-state series-paraUel system in which, according to Definition 3.20, we have kn = k = 3, In = 100, a = 2, q l = 2/3, q2

=

1/3.

Therefore, from (3.44), we get R'

2

3,100

(t,u) = 1 - l-I[1-(R (i)(t,u))100] qi3 i=l

= 1 - [ 1 - (R~l)(t,u))10012 [ 1 - (R~2)(t,u))l~176

where substituting el = 2, Pll = 0.4, Pl2 = 0.6,

Reliability of Large Multi-State Systems

142

a c c o r d i n g to (3.45), we get el

R(t)(t,u) = l-I (R O'j) (t, u)) plj = (R(t,t)(t,u))O.4(R(t,e)(t,u))O.6 j=l

and substituting e2 = 2, P21 = 0 . 5 , P22 = 0 . 5 ,

we get e2

R(2)(t,u) = 1-I (R(2'j) (t, u))P2j = (R(2,t)(t,u))O.S(R(2,2)(t,u))O.5. j=l

H e n c e , f r o m (3.43), the exact reliability function o f the piping s y s t e m takes the f o r m

R!

3,100(t,.)

__

[1,1

.

[1

.

e x p [ -. t

0.097]]2.[1 .

exp[-O.O35t3-7.5x~],

1 - [1 - e x p [ - 1 . 0 4 t -

0.0967]]2.[1 - e x p [ - 0 . 0 4 t 3 - 8 x]~ ],

1 - [1 - e x p [ - 1 . 1 2 t -

0.127]]2.[1 - e x p [ - 0 . 0 5 t 3 - 10x/t ],

1 - [1 - e x p [ - 1 . 2 t -

0.15t2]]2.[1 - e x p [ - 0 . 0 8 t 3 - 1 0 x / t ]] for t > 0.

Further, a p p l y i n g C o r o l l a r y 5.10 and considering (5.62), (5.63) and (5.61), we have

at(u) = min{ 1,2} = 1 for u = 1,2,3,4, i l l ( l ) = 0 . 4 . 0 . 0 2 5 = 0.01, fit(2) = 0 . 4 . 0 . 0 2 6 = 0.0104, fl~(3) = 0 . 4 - 0 . 0 2 8 = 0.0112,131(4) = 0 . 4 . 0 . 0 3 = 0.012,

a:(u) = m i n { 3, 0.5 } = 0.5 for u = 1,2,3,4, f12(1) - 0 . 5 . 0 . 1 5 - 0.075,/32(2) --- 0 . 5 . 0 . 1 6 - 0.08, ,82(3) = 0 . 5 . 0 . 1 8 = 0.09,/32(1) = 0 . 5 . 0 . 2 = 0.1, a ( u ) = m a x { 1,0.5 } = 1 for u = 1,2,3,4, /3(1) = m i n {0.01 } = 0.01,/3(2) = m i n {0.0104} = 0.0104, ~ 3 ) = m i n {0.0112} = 0.0112, fl(1) = m i n {0.012} = 0.012,

Chapter 5

143

a,(1) = 1/(0.01.100)= 1, a,(2) = 1/(0.0104.100)=0.962, an(3) = 1/(0.0112-100) = 0.893, a,(4) = 1/(0.012.100) = 0.833,

b,,(u) = 0

for u = 1,2,3,4,

and we conclude that the system limit reliability function is .9i"9 (t,.) = [ 1,1 - [ 1 - exp[-t]] 2,1 - [ 1 - exp[-t]] 2,1 - [ 1 - exp[-t]] 2, 1 - [1 - exp[-t]] 2] for t >_O. Thus, from (3.49), the approximate formula for the piping system reliability function takes the form

R'3,1o 0 (t,.) = gi" 9 ( ( t -

b,,(u))/a,,(u),.)

= [1,1 - [1 - exp[-t]]2,1 - [1 - exp[-1.04t]]2,1 - [1 - exp[-1.12t]] 2, 1 - [1 - exp[-1.2t]] 2] for t > 0. For instance, the expected values of the first type pipe segments' lifetimes in the state subsets, according to (3.4), are: Mll(1) = 1/0.025 = 40, Mll(2) = 1/0.026 --- 38.46, Mll(3) = 1/0.028 ----35.71, M11(4) = 1/0.030 -__-33.33, and their lifetimes in particular states, according to (3.8), are: M l l (1) --__1.54, MI1 (2) ~ 2.75, Mlt (3) --__2.38,

Mll (4)

~ 33.33.

The approximate mean values of the piping system sojourn times in the state subsets, according to (3.13), are: M(1) -_- 1.5, M(2) --- 1.44, M(3) --- 1.34, M(4) _=_1.25. Hence, from (3.17), the mean values of the piping system lifetime in particular states are: M(1) -_--0.06, M ( 2 ) -_-0.10, M(3) ~ 0.09, M ( 4 ) ~ 1.25. If the critical reliability state of the system is r = 2, then according to (3.18) its risk function is given by r(t) --[1 - e x p [ - 1 . 0 4 t ] ] 2.

Reliability of Large Multi-State Systems

144

Further, from (3.19), the moment when the system risk exceeds an admissible level 6 = 0 . 0 5 is

~'= r-l(0-) =--(1/1.04)log(1 -- ~ ) ----___0.24 years.

5.5. Reliability evaluation of multi-state parallel-series systems In proving facts on limit reliability functions for homogeneous regular multi-state parallel-series systems the following slight extensions of Lemmas 4.18-4.19 are used ([74]). L e m m a 5.12 If (i)

kn ~

0%

(ii)

.gi'(t,u) = e x p [ - V (t,u) ], u = 1,2,...,z, is a non-degenerate reliability function, m

m

(iii) Rk.,z" (t,.) = [1, Rk.,l n (t,1), .... Rkn,ln (t,z))], t e (-oo,~), is the reliability function of a homogeneous regular multi-state parallel-series system defined by (3.30)(3.31), (iv)

a. (u) > 0, b.(u) r ( - ~ , ~ ) , u = 1,2,...,z,

then .qi' (t,.) = [ 1, .qi'(t,1) ,..., 9i' (t, z) ], t e (-oo,oo), is its limit multi-state reliability function, i.e. m

lim Rk.,t" (a.(u)t + b.(u),u)= ~ ( t , u ) fort ~ C~(u), u = 1,2,...,z,

(5.64)

n ---->oo

if and only if lim kn[F(an(u)t + b n ( u ) , u ) ] t" = V ( t , u ) for t ~ CV(u) , u = 1,2,...,z.

(5.65)

n ...~ oo

For each fixed u, u = 1,2,...,z, assumptions (i)-(iv) of Lemma 5.12 are identical to assumptions (i)-(iv) of Lemma 4.18. Moreover, condition (5.64) is identical to condition (4.88) and condition (5.65) is identical to condition (4.89). Therefore, since

Motivation:

Chapter 5

145

from Lemma 4.18, condition (4.88) and condition (4.89) are equivalent, then conditions (5.64) and (5.65) are also equivalent. Lemma

5.13

If (i)

kn--->k, k>0,/,---~oo,

(ii)

9/(t, u), u = 1,2,...,z, is a non-degenerate reliability function,

(iii)

Rk,,l n (t,.) = [1, Rk,,l n (t,1),..., Rk,,l" (t, z) ], t e (-~,oo), is the reliability function

m

of a homogeneous regular multi-state parallel-series system defined by (3.30)(3.31), (iv)

a, (u) > 0, b,(u) e (-~,oo), u = 1,2,...,z,

then .gi'(t,.) = [1,.~i'(t,1) .....

.ql (t, z) ], t e (-o%~),

is its limit multi-state reliability function, i.e. m

m

lim Rkn,l n (an(U)t + bn(u),u) = .~(t, u) for t e C~(u) , u = 1,2,...,z,

(5.66)

n-,~

if and only if l i m [ F ( a n (U)t +bn (u),u)] l" = Jo(t,u) for t e C % ( u ) , u = 1,2,...,z,

(5.67)

n-.9,oo

where ~0(t,u) is a non-degenerate distribution function and moreover .~(t, u) = [1 - J0(t,u)] k for t e (-oo,oo), u = 1,2,...,z.

(5.68)

Motivation: For each fixed u, u = 1,2 .... ,,z, assumptions (i)-(iv) of Lemma 5.13 are identical to assumptions (i)-(iv) of Lemma 4.19. Moreover, condition (5.66) is identical to condition (4.90) and condition (5.67) is identical to condition (4.91). Since from Lemma 4.19, condition (4.90) and condition (4.91) are equivalent, then condition (5.66) and condition (5.67) are also equivalent and moreover due to (4.92) the equality (5.68) holds. Lemma 5.12, Lemma 5.13 and Theorem 4.8 from Chapter 4 yield the next theorem ([74]).

146

Reliability o f Large Multi-State Systems

Theorem 5.8

The class of limit non-degenerate reliability functions of the homogeneous regular multi-state parallel-series system is composed of 3 ~ + 4 z + 3 z reliability functions of the form m

m

.gl(t,.) = [ 1 , g l ( t , 1 ) , . . . , 9 1 ( t , z )

], t ~ (-oo,oo),

(5.69)

where Case 1. k, = n, l ln - c log n I>> s, s > 0, c > 0 (under Assumption 4.1).

.9i' (t, u) ~ { .9i'1(t), .q/2 (t), .9/3 (t) }, u = 1,2,...,z, and

*~i (t),

(5.70)

i = 1,2,3, are defined by (4.93)-(4.95),

Case 2. k, = n, 1,, - c log n ~ s, s ~ (-o%~), c > 0.

(t, u) ~ { .9/4 (t), .9i'5 (t), ~6 (t), .9?7 (t) }, u = 1,2,...,z, and

"~i (t),

(5.71)

i = 4,5,6,7, are defined by (4.96)-(4.99),

Case 3. k, --~ k, k > 0, l, ~ ~.

.9? (t, u) e { "~'s (t), .9i'9 (t), .gi'10(t) }, u = 1,2,...,z,

(5.72)

and g l i ( t ) , i = 8,9,10, are defined by (4.100)-(4.102). Motivation: For each fixed u, u = 1,2,...,z, co-ordinate g l ( t , u ) of the vector .gi'(t,.) defined by (5.69), according to Theorem 4.8 that is the consequence of Lemma 4.18 and Lemma 4.19, can be one of the three types of reliability function given by (4.93)-(4.95), or one of the four types of reliability function given by (4.96)-(4.99), or one of the three types of reliability function given by (4.100)-(4.102). Thus the number of different limit multi-state reliability functions of the considered system is equal to the sum of the numbers of z-term variations of the 3-component sets defined by (5.70) and (5.72) and the number of the 4-component set defined by (5.71). It means that this number is equal to 3 z + 4 z + 3 z and they are of the form (5.69). Corollary 5.11 If components of the homogeneous regular multi-state parallel-series system have Weibull reliability functions R(t,.) = [1, R(t,1),..., R(t, z) ], t ~ (-oo,oo),

Chapter 5

147

where

R(t,u) = 1 for t < O, R(t,u) = exp[-fl(u)t a(u) ] for t > O, a(u) > O, fl(u) > O,

(5.73)

u = 1,2,...,z,

and

kn---~ k, k > O , l,---~ oo,

(5.74)

an(U) = bn(u)/(a(u)fl(u)(bn(u))a~u)), bn(u) = [(log l,,)/fl(u)] l/a(u), u = 1,2,...,z,

(5.75)

then

"~10 (t,-) = [ 1, "~1o (t,1), ..., *~1o (t, z) ], t ~ (-oo,oo), where

.qi"--lo(t,u) = [1 - e x p [ - e x p [ - t ] ] * for t ~ (-oo,oo), u = 1,2,...,z, is its limit reliability function. Since for each fixed u, sufficiently large n and all t ~ (-oo,oo), according to (5.74) and (5.75), we have

Motivation:

an(U)t + bn(u) > O, and from (5.73)

F(an(u)t + bn(u),u) = 1 - exp[-p(u)(an(U)t + bn(u)) a~u)] for t ~ (-oo,oo), and further from (5.67)

.~o(t,u) = lim [F(a.(u)t + b.(u),u)] t. n .-~ oo

= lim [ 1 - exp[-.8(u)(bn(u))a~u)( (1 + ((an(u)/bn(u))t)a~u)]] t. tl.--~oo

= l i m [ 1 - e x p [ - (log l.)(1 + t / ( a ( u ) l o g / . ) ) ~ u ) ] ] tl---~oo

= lim

[ 1 - (1/l.)exp[-t

+ o ( 1 ) ] ] 1.

n.-.~oo

= exp[-exp[-t]] f o r t ~ (-oo,oo).

t.

Reliability of Large Multi-State Systems

148

Thus, after considering (5.68), from Lemma 5.13, .qi'10(t,.) is the limit reliability function of the considered system.

Example 5.11 (an electrical energy distribution system) Let us consider a model energetic network stretched between two poles and composed of three energetic cables, six insulators and two bearers and analyse the reliability of all cables only. Each cable consists of 36 identical wires. Assuming that the cable is able to conduct the current if at least one of its wires is not failed we conclude that it is a homogeneous parallel-series system composed of k, = 3 parallel subsystems linked in series, each of them consisting of In = 36 basic components. Further, assuming that the wires are four-state components, i.e. z = 3, having Weibull reliability functions with parameters a(u) = 2, fl(u) = (7.07) 2u- 8, u = 1,2,3, according to Corollary 5.11, assuming normalising constants

an(u) = (7.07) 4- u/(2 ~/log 36 ), bn(u) = (7.07) 4-u x/l~ 36, u = 1,2,3, and applying (3.49) we obtain the following approximate form of the system multi-state reliability function

R3,36(t,.)

= [ 1, R3,36(t,1),

R3,36(t,2), R3,36(t,3) ]

[1,[1 - exp[-exp[-0.01071t + 7.167]]] 3, [1 - exp[-exp[-0.07572t + 7.167]]] 3, [1 - exp[-exp[-0.53543t + 7.167]]] 3] for t e (-oo,oo). The values of the system sojourn times T(u) in the state subsystems in months, after applying (3.13), are given by oo

E[T(u)] = ~[1- exp[- exp[-(7.07) u-4 2~/log 36t + 2 log 36]]] 3 dt, u = 1,2,3, 0

and particularly M(1) _=_650, M(2) ~ 100, M(3) ~ 15. Hence, from (3.17), the system mean lifetimes in particular states are: M(1) ~ 550, M(2) ~ 85, M(3) ~ 15.

Chapter 5

149

If the critical reliability state of the system is r = 2, then its risk function, according to (3.18), is given by r(t) =_ 1- [1 - exp[-exp[-0.07572t + 7.167]]] 3. The moment when the system risk exceeds an admissible level 6 = 0.05, calculated due to (3.19), is r = r-l(6) __--[7.167 - log[-log[1 - (1

-

0") 1/3

]]]/0.07572 = 76 months.

The extensions of Lemmas 4.22-4.23 are essential tools for finding limit reliability functions for non-homogeneous regular multi-state parallel-series systems. They may be formulated as follows ([74]). Lemma

5.14

If (i) k. --~ ~, m

(ii)

gi" (t, u) = exp[ - V'(t, u) ], u = 1,2,...,z, is a non-degenerate reliability function,

(iii) R k.,t. (t,.) = [1 , R kn,t. (t,1)

,...,

R'

k,,l,,

(t, z) ], t ~ (-oo,~), is the reliability

function of a non-homogeneous regular multi-state parallel-series system defined by (3.46)-(3.48), (iv) a. (u) > 0, b.(u) ~ (-oo,oo), u = 1,2,...,z, (v) F(t,u) for each fixed u, is one of the distribution functions F(1)(t,u),F(Z)(t,u),..., F(a)(t,u) defined by (3.48) such that (vi) 3 N(u) V n > N(u) F(an(U)t + bn(u),u) = 0 for t < to(u), F(an(U)t + bn(u),u) r 0 for t > to(U), where to(u) ~ t0(u), i = 1,2 .... ,a, u = 1,2, .... z,

and moreover there exist non-decreasing functions

_

[

(viii) d ( t , u ) = 1

0 a _ lim ~,qidi(an(U)t+bn(u),u)

I, n''~~176i

where

for t < t o (u) for t>_.to(U),

(5.76)

150

Reliability o f Large Multi-State Systems F (i) (a n (u)t + b n (u), u) ]l n

-

d i (a n (u)t + b n (u), u) = [ -F~a-n-(~tt +.-b-n i~,-u- ~

,

then m

m

m

.qi" (t,.) = [ 1, ~ ' ( t , 1 ) ,..., ~ ' (t, z) ], t e (-0%00), is its limit multi-state reliability function, i.e. lim R'k,,t, (a,(u)t + b,(u),u) = ~ ' ( t , u )

n --~oo

for t ~ C~.~u ) , u = 1,2,...,z,

(5.77)

if and only if lim k n ( F ( a n ( U ) t + b , ( u ) , u ) )

t. d ( t , u ) = V ' ( t , u ) fort ~ CV,tu), u = 1,2,...,z. (5.78)

n----~oo

Motivation: Since for each fixed u, u = 1,2,...,z, assumptions (i)-(viii) of Lemma 5.14 are identical with assumptions (i)-(viii) of Lemma 4.22, condition (5.77) is identical to condition (4.105) and condition (5.78) is identical to condition (4.106), which from Lemma 4.22 are equivalent, then Lemma 5.14 is valid. Lemma 5.! 5 If

(i) kn -+ k, k > O , l, ---~ oo, !

(ii)

.gi"(t,.), u = 1,2,...,z, is a non-degenerate reliability function,

(iii) R k,.1, (t,.) = [1 R' k n,ln (t,1)

,'",

R' k n ,In (t, z) ], t e (-oo,oo), is the reliability

function of a non-homogeneous regular multi-state parallel-series system defined by (3.46)-(3.48), (iv) an (u) > O, bn(u) e (-oo,oo), u = 1,2,...,z, (v) F!t,u ) for each fixed u, is one of the distribution functions F(l)(t,u),F(2)(t,u), .... F~a)(t,u) defined by (3.48) such that (vi) 3 N(u) V n > N(u) F(an(U)t + bn(u),u) = 0 for t < to(u), F(an(u)t + bn(u),u) , 0 for t > to(U), where to(U) e 0 (under the assumptions of Lemma 5.14).

(5.85)

gi"(t, u) ~ { .gi' 4 (t) , .9i"5 (t) , .gi" 6 (t) , gi' '7 (t) } , u = 1,2, " .,z, ~

!

m

and ~

i (t),

i "-"

D

4,5,6,7, are given by (4.114)-(4.117) with d (t) = d (t, u), u = 1,2 .... z,

m

where d (t, u) are defined by (5.76),

Case 3. k, --~ k, k > 0, l, ---) oo (under the assumptions of Lemma 5.15). (5.86)

9 t ' ( t , u ) ~ { ~ s (t), 9i" 9 (t), .O/'lO (t) }, u = 1,2,...,z, M

!

m

m

and .gi' i (t), i = 8,9,10, are given by (4.118)-(4.120) with di(t ) = d i ( t , u ) , u = 1,2,...z, where d i (t, u) are defmed by (5.79). Motivation: For each fixed u, u = 1,2,...,z, co-ordinate .gi" (t,u) of the vector .gi" (t,.)

defined by (5.83), from Theorem 4.9 being the consequence of Lemma 4.22 and Lemma 4.23, may be one of the three types of reliability function defined by (4.111)-(4.113), or one of the four types of reliability function defined by (4.114)-(4.117) with d (t) = m

d(t,u),

where d ( t , u )

are given by (5.76), or one of the three types of reliability m

_

B

function given by (4.118)-(4.120) with di(t ) = d i ( t , u ) , where d i ( t , u ) are defined by (5.79). Thus the number of different limit multi-state reliability functions of the system is equal to the sum of the number of z-term variations of the 3-element sets given by (5.84) and (5.86) and the number of the 4-element set given by (5.85). It means this number is equal to 3 z + 4 z + 3 z and they are of the form (5.83).

CHAPTER 6

RELIABILITY EVALUATION OF PORT AND SHIPYARD TRANSPORTATION SYSTEMS The multi-state asymptotic approach is applied to the reliability and rb~k characteristics evaluation of selected large transportation systems used in ports and shipyards. Reliability analysis of multi-state series, series-parallel and parallel-series transportation systems is based on corollaries formulated and proved in this chapter and corollaries given in the previous chapter. Corollaries are used to evaluate reliability characteristics of three transportation systems used at the Port of Gdynia and one operating at the Naval Shipyard of Gdynia. The port grain transportation system built of three-state non-homogeneous series-parallel subsystems, the port oil piping transportation system composed of three-state non-homogeneous series-parallel subsystems, the port bulk transportation system built o f f our-state non-homogeneous series-parallel and series subsystems and the shipyard rope transportation system that is a four-state homogeneous parallel-series system are considered. Multi-state reliability functions, mean values of sojourn times in the state subsets and their standard deviations, mean values of lifetimes in the particular states, risk functions, and exceeding moments of a permitted risk level are determined for these systems. The accuracy of the asymptotic approach to the reliability evaluation of these systems is also illustrated. System components reliability data and system operation processes data come from experts operating these systems, component technical certificates and obligatory norms. Reliability data by necessity are approximate and concerned only with the mean values of the system components'sojourn times in the state subsets and hypothetical distributions of these lifetimes.

6.1. Auxiliary results Assuming in Corollary 5.4 an(U) = [d(t,u)/3(u)n] -1/~(") = [n

Y~qifli(u)] -l/a(") (i:ai(u)=a(u))

Port and Shipyard Transportation Systems

154

bn(u) = 0 for u = 1,2 ..... z, after considering the justification given in Chapter 1, prior to Definition 1.6, we obtain the following result. C o r o l l a r y 6.1 If components of the non-homogeneous reliability functions

multi-state series system have Weibull

R (i) (t,.) = [1, R (i) (t,1) ,..., R (i) (t, z) ], t e (-oo,oo), where

R(i)(t,u) = 1 for t < O, R(~

= exp[-fl/(u) t a'(u) ] for t > O, ai(u) > O, fli(u) > O, (6.1)

i = 1,2,...,a, u = 1,2,...,z, and

an(U) = Lt3(u)n] -va~u), bn(u) = 0 for u = 1,2,...,z,

(6.2)

where a ( u ) = m i n { a i ( u ) } , fl(u) = l 0, two drums with reliability functions RCl'2)(t,1) = exp[-0.0015t2], RCl'2)(t,2) = exp[-0.0018t 2] for t > 0,

Port and Shipyard Transportation Systems

158

117 channelled rollers with reliability functions R(l'3)(t, 1) = exp[-0.005ta], R(l'3)(t,2) = exp[-0.0075t 2] for t > 0, and nine supporting rollers with reliability functions R(l'a)(t,1) = exp[-0.004t2], R(l'4)(t,2) = exp[-0.005t 2] for t > 0. Thus it is a non-homogeneous regular multi-state series-parallel according to Definition 3.20, we have

system where,

k,,=k=2,1n = 129, a = 1, ql = 1, el = 4, Pll

=

1/129, Pl2

=

2/129, P13

=

117/129, PI4

=

9/129

and according to (3.43)-(3.45) its exact reliability function is R!

2,129 (t,-) = [1,1

.

[1

exp[-0.6365t2]]2,1 . .

.

[1

exp[-0.9481ta]] 2] for t > 0.

Further, applying Corollary 5.10, according to (5.62), we have al(1) = rain{2, 2, 2, 2} = 2, ill(l) =

2 117 9 1 0.0125 + 0.0015 + 0.005 + 0.004 = 0.004934108, 129 129 129 129

al(2) = min{2, 2, 2, 2} = 2, ill(2) =

2 117 9 1 0.022 + .... 0.0018 + 0.0075 + 0.005 = 0.007349612, 129 129 129 129

and according to (5.63) a(1) = max{2} = 2, ,O(1)= rain{0.004934108} =0.004934108, a(2) = max{2} = 2, ,8(2) = min{O.O07349612} = 0.007349612, and according to (5.61) a,(1) = (0.004934108.129) -~/2 = 1.253432119, b,(1) = 0, a,(2) = (0.007349612.129) -1/2 = 1.027005872, b,(2) = 0, and we conclude that the limit reliability function of subsystem Sl is

Chapter 6

159

.~'9 (t,-) = [ 1,1 - [ 1 - exp[-tz]] 2,1 - [ 1 - exp[-ta]] 2] for t > 0. Hence, from (3.49), we get the approximate subsystem reliability function (the formula is exact in this case) given by 1~'2,12 9 9 (t,.) =__ ~?'9 ( ( t -

b,(u))/a,(u),. )

= [1,1 - [1 - exp[-0.6365ta]]2,1 - [1 - exp[-0.9481ta]] 2] for t >_0. The expected values of the subsystem lifetimes in the state subsets and their standard deviations, from (3.13)-(3.15), are: M(1) -_- 1.44 years, M(2) -_- 1.18 years, o(1) _=_0.53 years, o(2) _-_-0.44 years. Hence and from (3.17), the subsystem mean lifetimes in particular states are: u

M(1) _--_0.26 years, M ( 2 ) =__1.18 years. If the critical reliability state of the subsystem is r = 2, then according to (3.18) its risk function is given by

r(t) --- [ 1 - exp[-0.9481 ta]] 2. Hence and from (3.19), the moment when the subsystem risk function exceeds an admissible level 8 = 0.05 is r = r-l(b') = [-(1/0.9481)1og(1 - ~f6 )]in ___0.52 years. Subsystem Sz is composed of three identical bucket elevators ([ 103]) each composed of a buttress belt, a drum driving the belt, a reversible drum and 740 buckets. Thus subsystem Sz consists of kn = 3 elevators, each composed of 1, = 743 components. In each elevator there is one belt with reliability functions R(~'t)(t, 1) = exp[-0.025t2], R(~'l)(t,2) = exp[-0.026t 2] for t > 0, two drums with reliability functions R(t'2)(t, 1) = exp[-0.0015t2], R(l'2)(t,2) = exp[-0.0018t z] for t >__0, 740 buckets with reliability functions R(l'3)(t,1) = exp[-0.03ta], R(l'3)(t,2) = exp[-0.06t 2] for t > 0.

Port and Shipyard Transportation Systems

160

Thus it is a non-homogeneous regular multi-state series-parallel according to Definition 3.20, we have

system where,

k,=k =3, 1 , = 7 4 3 , a = 1, ql = 1, el = 3, Pll = 1/743, P12 = 2/743, PI3 = 7 4 0 / 7 4 3

and according to (3.43)-(3.45) its exact reliability function is given by R!

3,743(t,.)

~_

[1,1

.

[1

.

exp[-22.228t2]]3,1 . .

[1

exp[-44.4296t2]]3] for t >_ 0.

Further, applying Corollary 5.10, according to (5.62), we have a,(1) = min{2, 2, 2} = 2, p,(1) =

2 740 1 0.025 + 0.0015 + 0.03 = 0.029916554, 743 743 743

a1(2) = min{2, 2, 2} = 2,

p,(2) =

1

743

0.026 +

2 743

0.0018 +

740 743

0.06 = 0.059797577,

and according to (5.63) a(1) = max{2} = 2, fl(1) = min{O.029916554} = 0.029916554, a(2) = max{2} = 2, ,6(2) = min{0.059797577} = 0.059797577, and according to (5.61) a,(1) = (0.029916554.743) -1/2 = 0.212104464, b,(1) = 0, a,(2) = (0.059797577.743) -1/2 = 0.0150025056, b,(2) = 0, and we conclude that the limit reliability function of subsystem $2 is ~ ' 9 (t,.)

-

-

[1,1 - [1 - exp[-t2]]3,1 - [1 - exp[-t2]] 3] for t > 0.

Thus, from (3.49), the approximate formula (it is exact in this case) takes the form R'3,743 (t,')

--- 9i" 9 ( ( t - b,(u))/an(u),.) - [1,1 - [1 - exp[-22.228t2]] 3,1 - [1 - exp[-44.4296t2]] 3] for t > O.

Chapter 6

161

The expected values and standard deviations of the subsystem lifetimes in the state subsets, according to (3.13)-(3.15), are: M(1) _=0.27 years, M(2) --- 0.19 years, o(1) ~ 0.10 years, o(2) ~ 0.07 years. Hence and from (3.17), the subsystem mean lifetimes in particular states are" m

M(1) ~ 0.08 years, M(2) ~ 0.19 years. If the critical reliability state of the subsystem is r = 2, then from (3.18) its risk function has the form r(t) ~ [1 - exp[-44.2296t2]] 3.

Hence, and from (3.19), the moment when the subsystem risk function exceeds an admissible level 8 = 0.05 is r = r-t(b) = [-(1/44.2296)1og(1 - 3 ~ )]re ~ 0.10 years. Subsystem $3 is composed of two identical belt conveyors of the second type each composed of a buttress belt, a drum driving the belt, a reversible drum, 117 channelled rollers and 19 rollers supporting the belt. Thus subsystem $3 consists of k, = 2 conveyors, each composed of l~ = 139 components. In each conveyor there is one belt with reliability functions R(i'l)(t,1) = exp[-0.0125t2], R(t't)(t,2) = exp[-0.022t 2] for t > 0, two drums with reliability functions RO'2)(t, 1) = exp[-0.0015t2], R(l'2)(t,2) = exp[-0.0018t 2] for t > 0, 117 channelled rollers with reliability functions R~

= exp[-0.005t2], R(t'3)(t,2) = exp[-0.0075t 2] for t > 0,

and 19 supporting rollers with reliability functions R(l'4)(t,1) = exp[-0.004t2], R(l'4)(t,2) = exp[-0.005t 2] for t >_0. Thus it is a non-homogeneous regular multi-state series-parallel system where, according to Definition 3.20, we have k,=k=2,1,

= 139, a = 1, ql = 1,

Port and Shipyard Transportation Systems

162

el = 4, pll = 1/139, p12 = 2/139, p13 = 117/139, p14 = 19/139 and according to (3.43)-(3.45) its exact reliability function is R!

2,139 (t,')

__

[1,1

.

[1

.

exp[-0.6765t2]]2,1 . .

[1

exp[-0.9981t2]] 2] for t > 0.

Further, applying Corollary 5.10, according to (5.62), we have a~(1) = min{2, 2, 2, 2} = 2 , ill(l) =

2 117 19 0.004 = 0.004866906, 1 0.0125 + 0.0015 + 0.005 + 139 139 139 139

al(2) = rain{2, 2, 2, 2} = 2, fl1(2) =

2 117 19 1 0.022 + 0.0018 + 0.0075 + 0.005 = 0.007180575, 139 139 139 139

and according to (5.63) a,(1) = max{2} = 2, g l ) =

min{O.O04866906} =0.004866906,

a(2) = max {2 } = 2,/3(2) = min {0.007180575} = 0.007180575, and according to (5.61) an(l) = (0.004866906.139) -1/2 = 1.215811147, b,(1) = 0, an(2) = (0.007180575.139) -1/2 = 1.000951394, bn(2) = 0, and we conclude that the limit reliability function of subsystem $3 is 9(t,')

[1 1

[1

exp[-t2]] z,1

[1

exp[-t2]] 2] for t >__0.

Thus, from (3.49), the approximate formula (it is exact in this case) for the subsystem reliability function takes the form

R'2,139 (t,.) =__9?' 9 ( ( t - b,,(u))/an(U),') = [1,1 - [1 -exp[-0.6765t2]]2,1 - [1 - exp[-0.9981t2]] 2] for t >_0. The expected values and standard deviations of the subsystem sojourn times in the state subsets, according to (3.13)-(3.15), are:

Chapter 6

163

M(1) ~ 1.39 years, M(2) ~ 1.15 years, o(1) ~ 0.53 years, o(2) _=0.42 years. Hence and from (3.17), the mean lifetimes of the subsystem in particular states are: M(1) ~0.24 years, M(2) ~ 1.15 years. If the critical reliability state of the subsystem is r = 2, then according to (3.18) its risk function takes the form

r(t) --- [1 - exp[-0.9981t2]] 2. Hence, and from (3.19), the moment when the subsystem risk exceeds an admissible level 8 = 0.05 is r = r-l(b") = [-(1/0.9981)1og(1 - ~

)]v2 ~ 0.50 year.

Subsystem $4 is composed of three chain conveyors, each composed of a wheel driving the belt, a reversible wheel and 160, 160 and 240 links respectively. The subsystem consists of three conveyors. Two of these are composed of 162 components and the remaining one is composed of 242 components. Thus it is a non-homogeneous nonregular multi-state series-parallel system. In order to make it a regular system we conventionally complete two first conveyors having 162 components with 80 components that do not fail. After this supplement subsystem $4 consists of kn = 3 conveyors, each composed of In = 242 components. In two of them there are two driving wheels with reliability functions R(t'~)(t,1) = exp[-0.005t2], R(l'~)(t,2) = exp[-0.008t 2] for t > 0, 160 links with reliability functions R(l'2)(t,1) = exp[-0.012t2], R(l'2)(t,2) = exp[-0.018t 2] for t >_0, and 80 components with "reliability functions" R(l'3)(t,1) = exp[-ill t2], R(l'3)(t,2) = exp[-f12 t2] for t >_0, where ,81 =

/~2 = 0 .

The third conveyer is composed of two driving wheels with reliability functions R(2'~)(t,1) = exp[-0.022t 3/2 ], R(2'l)(t,2) = exp[-0.026t 3/2] for t > 0, and 240 links with reliability functions R(2'2)(t,1) = exp[-0.034t3/2], R(2'E)(t,2) = exp[-0.042t 3/2] for t >__0.

Port and Shipyard Transportation Systems

164

Thus the subsystem is a n o n - h o m o g e n e o u s regular multi-state series-parallel system where, according to Definition 3.20, we have k,, = k = 3, l, = 242, a = 2, ql = 2/3, q2 = 1/3, et = 3, Pll = 2/242, PI2 = 160/242, Pl3 = 80/242, e2 = 2, P21 = 2/242, P22 = 240/242, and according to (3.43)-(3.45) its exact reliability function is given by R !

3,242

(t,.)

~

[1,1

m

[1

m

exp[-1.93t2]]2.[1

m

exp[m8.204t3/2],

1 - [1 - exp[-2.896t2]]2.[1 - exp[-10.132t3/2]]] for t > 0. Further, applying Corollary 5.10, from (5.62), we have al(1) = min{2, 2, 2} = 2 ,

fl~(1) =

160 80 2 0.005 + 0.012 + 0 =0.007975206, 242 242 242

a2(1) = min {3/2, 3/2 } = 3/2, /32(1) =

2 0.022 + 240 0.034 = 0.033900826, 242 242

ai(2) = min{2, 2, 2} = 2, 2 160 80 .... 0 = 0.011966942, /31(2) = 242 0.008 + 242 0.018 + 242 a2(2) - min {3/2, 3/2} = 3/2, ,82(2) =

240 2 0.026 + 0.042 = 0.041867768, 242 242

and from (5.63) a(1) = max{2, 3/2} = 2,/3(1) = min{O.O07975206} = 0.007975206, a(2) = m a x {2, 3/2 } = 2, fl(2) = min {0.011966942} = 0.011966942, and from (5.61 )

Chapter 6

165

an(l) = (0.007975206. 242) -~/2 = 0.719815778, bn(1) = 0, an(2) = (0.011966942. 242) -1/2 = 0.587625622, bn(2) = 0, and we conclude that the limit reliability function of subsystem $4 is 9/'9 (t,.) = [1, 1 - [1 - exp[-t]] 2, 1 - [1 - exp[-t]] 2] for t _>0. Thus, from (3.49), we obtain the following approximate formula

R'3,242 (t,.) --- .~i"9 ( ( t - bn(u))/an(U),. ) = [ 1,1 - [ 1 - exp[- 1.937]] 2,1 - [ 1 - exp[-2.896t2]] 2] for t > 0. The approximate mean values and standard deviations of the subsystem lifetimes in the state subsets, according to (3.13)-(3.15), are: M(1) = 0.82 years, M(2) = 0.67 years, o(1) ~ 0.32 years, o(2) ~ 0.26 years. Hence, and from (3.17), the subsystem mean lifetimes in particular states are: m

m

M(1) ~0.15 years, M ( 2 ) --0.67 years. If the critical reliability state of the subsystem is r = 2, then due to (3.18) its risk function is given by

r(t) = [1 - exp[-2.896t2]] 2. Hence, and from (3.19), it follows that the moment when the subsystem risk exceeds an admissible level 8 = 0.05 is r = r-t(b") = [-(1/2.896)1og(1 - ~

)]1/2 ~ 0.30 years.

Since the considered subsystems create a series structure in a reliability sense, then according to Definition 3.17 and formulae (3.33)-(3.34), the reliability function of the whole transportation system is given by R'(t,.) =_ [1, R'(t,1), R'(t,2) ], t __ 0, where R' (t,1) = 24exp[-25.471t 2] - 12exp[-27.401t 2] - 12exp[-26.1475t 2] + 6exp[-28.0775t 2] - 24exp[-47.699t 2] + 12exp[-49.629t 2]

Port and Shipyard Transportation Systems

166

+ 12exp[-48.3755t 2] - 6exp[-50.3055t 2] + 8exp[-69.927t 2] - 4exp[-71.857t 2] - 4exp[-70.6035t 2] + 2exp[-72.5335t 2] - 12exp[-26.1075t 2] + 6exp[-28.0375t 2] + 6exp[-26.784t 2] - 3exp[-28.714t 2] + 12exp[-48.3355t 2] -6exp[-50.2655t 2] - 6exp[-49.012t 2] + 3exp[-50.942t 2] - 4exp[-70.5635t 2] + 2exp[-72.4935t 2] + 2exp[-71.24t 2] -exp[-73.17t2], R'(t,2)

~ 24exp[-49.2718t 2] - 12exp[-52.1678t 2] - 12exp[-50.2699t 2] + 6exp[-53.1659t 2] - 24exp[-93.7014t 2] + 12exp[-96.5974t 2] + 12exp[-94.6995t 2] - 6exp[-97.5955t 2] + 8exp[-138.13 It 2] - 4exp[-141.027t 2] - 4exp[-139.1291 t2] + 2exp[-142.025 It 2] - 12exp[-50.2199t 2] + 6exp[-53.1159t 2] + 6exp[-51.218t 2] - 3exp[-54.114t 2] + 12exp[-94.6495t 2] - 6exp[-97.5455t 2] - 6exp[-95.6476t 2] + 3exp[-98.5436t 2] - 4exp[-139.0791 t2] + 2exp[-141.975 It 2] + 2exp[-140.0772t 2] -exp[-142.9732t2].

The approximate mean values and standard deviations of the system lifetimes in the state subsets, from (3.13)-(3.15), are: M(1) _=_0.27 years, M(2) ~ 0.19 years, o(1) --- 0.09 years, o(2) =_0.07 years. Hence, from (3.17), the mean values of the system lifetime in particular states are" B

M(1) --- 0.08 years, M(2) _--_0.19 years. If a critical reliability state of the system is r = 2, then from (3.18) its risk function takes the form m

r(t) ~ 1 - R'(t,2).

Chapter 6

167

Hence, from (3.19), the moment when the risk exceeds the critical level 8 = 0.05 is r = r-l(o") _=0.10 years. The behaviour of the exact and approximate reliability functions and the risk function of the considered system is illustrated in Table 6.1 and Figure 6.2. Moreover, in Table 6.1 the differences between the exact and approximate values of the components of system multi-state reliability functions marked by d(1) and d(2) are given. These differences show that replacement of the system's exact reliability function by its approximate form does not result in large mistakes in this evaluation.

Table 6.1. The values of the components of multi-state reliability functions and the risk function of the port grain transportation system L

m

R'(t,1)

R'(t,2) |

t

|

i

exact reliability function 1.00000 0.00 0.05 0.99984 0.10 0.99191 0.15 0.93800 0.20 0.79233 0.25 0.57025 0.30 0.34445 0.35 0.17559 0.40 0.07652 0.45 0.02885 0.50' 0.00948 0.55 0.00273 0.60 i 0.00069 9

|

approximate A(1) exact reliability reliability function function 1.00000 I 0.0000 1.ooooo 0.99981 0.0000 i 0.99882 ' 0.99164 'i 0.0003" 0.95345 0.93695 0.0011 0.74554 I 0.79023 0.0021 0.42211 0.56759 0.0027 0.17068 0.34213 0.0023 0.05099 0.17411 0.0015 0.01162 0.07579 0 . 0 0 0 7 0.00205 0.02857 . 0.0003. 0.00028 i 0.00940 0.0001 0.00003 0.00271 0.0000 i 0.00000 ' 0.00069 'lO.O000 I 0.00000 9

,

9

,

|

82

.

approximate reliability function 1.ooooo 0.99877 i 0.95288 0.74389 0.42005 0.16933 0.05046 0.01148 0.00203 . 0.00028 ~ 0.00003 0.00000 i 0.00000 9

.

.

.

.

.

.

.

.

.

.

r(t)

,4(2) |

|

0.0000 0.0000 0.0001 0.0012 '0.0006'0.0471 ' 0.0017 0.2561 0.0021 0.5779 0.0014 0.8307 0.0005 0.9495 0.0001 0.9885 0.0000 0.9980 . 0.0000 i 0.9997 ! 0 . 0 0 0 0 1.0000 0.0000 1.0000 ' 0.0000 1.0000 I

.

i

Port and Shipyard Transportation Systems

168

R ' (t,.)

u=0

1.00

t) 0.80 0.60 exact 0.40 -

.................approximate

1

0.20 0.00 0.( )0 w

I

I

I

I

f

0.60

0.80

t

L

0.20

0.40

r

Fig. 6.2. The graphs of the components of multi-state reliability functions and the risk function of the port grain transportation system

6.3. Reliability of a port oil transportation system Oil Terminal No. 21 in D~bog6rze, presented in Figure 6.3, is set up to receive from ships, store and send by carriages or cars oil products such as petrol, driving oil and fuel oil. Three terminal parts A, B and C fulfil these purposes. They are linked by the piping transportation systems. The unloading of tankers is performed at the pier placed in the Port of Gdynia. The pier is connected to terminal part A through the transportation subsystem $1 built of two piping lines composed of steel pipe segments with diameter of 600 mm. In part A there is a supporting station fortifying tankers' pumps and making possible further transport of oil by means of subsystem $2 to terminal part B. Subsystem $2 is built of two piping lines composed of steel pipe segments of diameter 600 mm. Terminal part B is connected to terminal part C by subsystem $3. Subsystem $3 is built of one piping line composed of steel pipe segments of diameter 500 mm and two piping lines composed of steel pipe segments of diameter 350 mm. Terminal part C is set up for loading the rail cisterns with oil products and for the wagon carrying these to the railway station of the Port of Gdynia and further into the country.

Chapter 6

169

Fig. 6.3. The scheme of the oil transportation system Considering the safety of the operation of the piping in the system, the following three reliability states of the piping components are distinguished: state 2 - the piping operation is fully safe, state 1 - the piping operation is less safe and dangerous because of the possibility of environment pollution, state 0 - the piping is destroyed. Subsystem S~ consists of two identical piping lines, each composed of 176 pipe segments of length 12 m and two valves. It means that it is built of kn = 2 piping lines, each composed of In = 178 components. In each of the lines there are 176 pipe segments with reliability functions R(l'l)(t, 1) = exp[-0.00000000 lt4], R(l'l)(t,2) = exp[-0.000000004t 4] for t >_0, and two valves with reliability functions R(l'2)(t, 1) = exp[-0.000000052t4], R(l'2)(t,2) = exp[-0.000000107t 4] for t _>0. Thus subsystem $1 is a non-homogeneous regular multi-state series-parallel system, where according to Definition 3.20 kn = k = 2, 1,= 178, a = 1, ql = 1, el

=

2, Pll = 176/178, P12 = 2/178.

The exact subsystem reliability function for t > 0, according to (3.43)-(3.45) is given by R!

2,178(t,.)=[11~

.

[1

. . . exp[-0.00000028t4]]2,1

[1

exp[-0.000000918t4]]2].

Port and Shipyard Transportation Systems

170

Applying Corollary 5.10, according to (5.62), we have a,(1) = rain{4, 4} = 4 , ill(l) =

176 178

0.000000001 +

2 . 0.000000052 = 0.000000001573, 178

a~(2) = min{4, 4} = 4, 2 /31(2) = 176 0.000000004 + 0.000000107 = 0.0000000051573, 178 178 and from (5.63) a(1) = max{4} = 4 , fl(1) = min {0.000000001573 } = 0.000000001573, a,(2) = max{4} = 4 , /3(2) = rnin {0.000000005157 } = 0.0000000051573, and from (5.61) an(1) = (0.000000001573.178)-1/4

--.__

43.47208719, b,(1) = 0,

an(2) = (0.000000005157"178) -1/4 = 32.30645681, bn(2) = O, and we conclude that the subsystem limit reliability function is given by .gi"9 (t,.) = [1,1 - [1 - exp[-t4]]2,1 - [1 - exp[-t4]] 2] for t >_ 0. Thus, from (3.49), for t _>0, we have R!

2,178 (t,.) ~ [1,1

.

[1

exp[-0.00000028t4]]2,1 . . .

[1

exp[-0.000000918t4]]2].

The mean values and standard deviations of subsystem S~ sojourn times in the state subsets, from (3.13)-(3.15), are: M(1) ~ 45.67 years, M(2) ~ 33.94 years, o(1) ~ 8.92 years, o(2) _= 6.63 years. Hence, from (3.17), the mean lifetimes of the subsystem in the particular states are:

Chapter 6 m

171

m

M(1) --- 11.73 years, M ( 2 ) =_33.94 years. If a critical state is r = 2, then according to (3.18) the system risk function takes the form r(t) = [1 - exp[-0.000000918t4112

and from (3.19) the moment when the risk exceeds the critical level 8 = 0.05 is r = r-l(o") = [-(1/0.000000918)1og(1 - xf~ )]1/4 ~_ 22.91 years. Subsystem $2 consists of two identical piping lines, each composed of 717 pipe segments of length 12 m and two valves. It means that it is built of kn = 2 piping lines, each composed of In = 719 components. In each of the lines there are 717 pipe segments with reliability functions R(l'l)(t, 1) = exp[-0.00000000 lt4], R(l't)(t,2) = exp[-0.000000004t 4] for t > 0, and two valves with reliability function R(l'2)(t, 1) = exp[-0.000000052t4], R(l'2)(t,2) = exp[-0.000000107t 4] for t >_ 0. Thus subsystem S 2 is a non-homogeneous regular multi-state series-parallel system, where according to Definition 3.20 kn = k = 2, In =719, a = 1, qt = 1, el = 2, Pll = 717/719, PI2 = 2/719. The exact subsystem reliability function for t > 0, according to (3.43)-(3.45), is given by R!

2,719 (t,.) = [1 | ,

.

[1

exp[-0.000000821 t4]]2,1 .

.

.

[1

exp[-0.000003082t4]]z].

Applying Corollary 5.10, according to (5.62), we have a,(1) = min(4, 4} = 4, 2 ill(l) = 717 .... 0.000000001 + 0.000000052 = 0.0000000011418, 719 719 al(2) = min{4, 4} = 4,

Port and Shipyard Transportation Systems

172

fl,(2) =

717 2 0.000000001 + 0.000000052 = 0.0000000042865, 719 719

from (5.63) a,(1) = rnax{4} = 4 , ,B(1) = min {0.0000000011418 } = 0.0000000011418, a(2) = max{4} =4, fl(2) = rnin {0.0000000042865 } = 0.0000000042865, and from (5.61) a,,( 1) = (0.0000000011418.719) -1/4 = 33.22111547, b,,( 1) = O, a,,(2) = (0.0000000042865.719) -t/4 = 23.86667072, bn(2) = 0, and we conclude that the subsystem $2 limit reliability is given by 9(t,') =[1 1

[1

exp[-t4]] 2,1

[1

exp[-t4]] 2 ] f o r t > 0 .

Hence, from (3.49), for t > 0,we get

R'2,719 (t,-)

--- [1,1 . [1

exp[-O.OOOOOO821t4]]2,1 . . .

[1

exp[-O.OOOOO3082t4]] 2].

The mean values and standard deviations of the subsystem sojourn times in the state subsets, according to (3.13)-(3.15), are: M(1) =_ 34.90 years, M(2) =_25.07 years, o(1) --- 6.82 years, o(2) _=_4.92 years, and from (3.17), the mean values of the subsystem lifetimes in the particular states are" M(1) -_-9.83 years, M(2) _=_25.07 years. If the critical state is r = 2, then from (3.18), the subsystem risk function takes the form

r(t) ~ [1 - exp[-O.OOOOO3082t4]] 2. Hence, from (3.19), the moment when the subsystem risk exceeds the critical level 8 = 0 . 0 5 is

Chapter 6

r = r-~(o") = [-(1/0.000003082)1og(1 - ~

173

)]~/4 =__16.93 years.

Subsystem $3 consists of three different piping lines, each composed of 360 pipe segments of either 10 m or 7.5 m length and two valves. It means that it is built of kn - 3 piping lines, each composed of I, - 362 components. In two lines there are 60 pipe segments with reliability functions R(l'l)(t,1) = exp[-0.0000000008t4], R(l'l)(t,2) = exp[-0.000000002t 4] for t > 0 and two valves with reliability functions R(l'2)(t, 1) = exp[-0.000000052t4], R(l'2)(t,2) = exp[-0.000000107:] for t > 0. In the third line there are 360 pipe segments with reliability functions R(2'l)(t, 1) = exp[-0.00000022t3], R(2'l)(t,2) = exp[-0.0000003t 3] for t > 0 and two valves with reliability functions R(2'2)(t, 1) = exp[-0.000000052t4], R(2'2)(t,2) = exp[-0.000000107t 4] for t >_ 0. Thus subsystem $3 is a non-homogeneous regular multi-state series-parallel system, where according to Definition 3.20 k, = k = 3, I, = 362, a = 2, ql = 2/3, q2 = 1/3, el = 2, Ptl = 360/362, Pl2 = 2/362, e2 =

2, P21 = 360/362, P22 = 2/362,

The exact subsystem reliability function for t > 0, according to (3.43)-(3.45), is given by R'3,362 ( t , ' )

= [1,1 - [ 1 - exp[-0.000000392t4112.[1 - exp[-0.0000792t 3 - 0.000000104t4]], 1 - [ 1 - exp[-0.000000934t4]]z.[ 1 - exp[-0.000108t 3 - 0.000001294:]]]. Applying Corollary 5.10, according to (5.62), we have a~(1) = min{4, 4} = 4 , ill(l) = 360 0.0000000008 + 2 0.000000052 = 0.0000000010828, 362 362

Port and Shipyard Transportation Systems

174

a2(1) = min{3, 4} = 3 , f12(1) = 360 0.00000022 + 2 0.000000052 = 0.000000218, 362 362

al(2 )

=

min {4, 4} = 4,

360 2 ,8~(2) = 362 0.000000002 + 362 0.000000107 = 0.0000000025801, a2(2) = min{3, 4} = 3, /32(2) = 360 0.0000003 + 2 0.000000107 = 0.000000298, 362 362 next from (5.63) a(1) = max{4, 3} = 4, fl(1) = min{0.0000000010828} = 0.0000000010828, a(2) = max {4, 3 } = 4,/3(2) = min{O.O000000025801 } = 0.0000000025801, and from (5.61 ) a,(1) =(0.0000000010828-362) -1/4 - 39.96487724, b,(1) = 0, a,(2) =(0.0000000025801.362) -1/4 = 32.1672016, b,(2) = 0, and we conclude that the subsystem $3 limit reliability function is given by .~i"9 (t,.) = [1,1 - [1 - exp[-t4]] 2,1 - [1 - exp[-t4]] 2] for t > 0. Hence, applying (3.49), for t > 0,we get R!

3,362(t,') ~[1,1--[1

m

exp[-0.000000392t4]]2,1

~

[1

u

4

2

exp[-0.000000934t ]] ].

The mean values and standard deviations of the subsystem sojourn times in the state subsets, according to (3.13)-(3.15), are: M(1) =- 41.99 years, M(2) --__33.80 years, o(1) -- 8.17 years, 0(2) _=_6.57 years.

Chapter 6

175

Hence, from (3.17), the mean values of the subsystem lifetimes in the particular states are: m

M(1) =__8.19 years, M(2) --- 33.80 years. If the critical state is r = 2, then from (3.18), the subsystem risk function takes the form r(t) _=_[1 - exp[-0.000000934t4]] 2 and from (3.19) the moment when the risk exceeds the critical level 8 = 0.05 is r = r-t(8) = [-(1/0.000000934)1og(1 - ~

)]1/4 _=_22.82 years.

Since the subsystems create a series reliability structure, then by (3.33)-(3.34) the multi-state reliability function of the whole oil transportation system is given by Q

u

R'(t,.) = [1, R'(t,1),R'(t,2) ],

where R" (t,1) --- 8exp[-0.000001493?] - 4exp[-0.000001885?] - 4exp[-0.000002314t 4] + 2exp[-0.000002706t 4] - 4exp[-0.000001773?] + 2exp[-0.000002165t 4] + 2exp[-0.000002594t 4] - exp[-0.000002986t4], R" (t,2) --- 8exp[-0.000004934t 4] -4exp[-0.000005868t 4] - 4exp[-0.000008016t 4] + 2exp[-0.00000895t 4] -4exp[-0.000005852t 4] + 2exp[-0.000006786t 4] + 2exp[-0.000008934t 4] - exp[-0.000009868t4]. From (3.13)-(3.15), the approximate mean values and standard deviations of the system sojourn times in the state subsets are: M(1) --- 32.60 years, M(2) =_23.98 years, o(1) -_- 5.90 years, o(2) -_-4.43 years. While, according to (3.17), the mean values of the system lifetimes in particular states are; M (1) ~ 8.62 years, M (2) =_23.98 years. If the critical state is r - 2, then from (3.18) the system risk function is given by

176

Port a n d S h i p y a r d Transportation Systems w

r(t) = 1 - R ' (t,2).

Hence, from (3.19), the moment when the risk exceeds the admissible level # = 0.05 is r = r-l(6) --__16.50 years. The behaviour o f the system's exact and approximate reliability functions, the differences between them and its risk fimction are illustrated in Table 6.2 and Figure 6.4.

Table 6.2. The values of the multi-state reliability functions components and the risk function of the port oil transportation system R'(t,1)

. R'(t,2)_.. , l exact r app roximate ' A(2) exact approximate A(1) reliability reliability reliability reliability function function i function . function . 0 1.00000 1.00000 0.0000 9 1.00000 ' 1.00000 ' 0.0000' 4 1.00000 1.00000 0.0000 1.00000 1.00000 0.0000 8 0.99999 0.99998 0.0000 0.99983 0.99981 0.0000 12 0.99967 0.99962 0.0001 0 . 9 9 5 7 4 0.99545 0.0003 |,

9

t

16 20

0.99674 0.98122

.

24 28 32 36 40 44 48 52 56 60

0.92630 0.79042 " 0.55756 0.29212 ! 0.10179 0.02121 0.00239 0.00013 0.00000 0.00000

.

.

0.99628 0.97932

. 0.0005 . 0.96187 , 0.0019 0.82208

0.95986 0.81658

0.92181 0.78421 0.55282 0.29028 0.10146 0.02118 0.00239 0.00013 0.00000 0.00000

0.0045 0.51602 " 0.0062 ' 0 . 1 8 5 0 9 ' 0.0047 0.02933 0.0018 0.00161 0.0003 0.00002 0.0000 ' 0.00000 . 0.0000 , 0.00000 ! 0.0000 0.00000

0.51032 0.18326 0.02919 0.00161 0.00002 0.00000 0.00000 0.00000

, 0.0000. 0.00000. o.oooo, o . o o o o o

0.00000 o.ooooo

.

.

,

r(t)

0.0000 0.0000 0.0002 0.0046 j 0.0020 . 0.0401 i i0.0055 0 . 1 8 3 4 0.0057 0.4897 ' 0 . 0 0 1 8 0.8167 0.0001 0.9708 0.0000 0.9984 0.0000 1.0000 0.0000 1.0000 j 0 . 0 0 0 0 . 1.0000 0.0000 1.0000 .0.0000 i 1.0000 ; . o.oooo 1.oooo

,

,

,

Chapter 6

177

R ' (t,-) u=O

1.00

)

0.80 0.60 0.40 -

exact ................approximate

0.20 0.00

...... 0.00

,

,

20.00

40.00

': 60.00

Fig. 6.4. The grap h s of the multi-state reliability functions and the risk function of the port oil transportation system

The considered reliability structure of the oil transportation system is adequate in the case when each of the oil products may be transported by each of three system pipelines. However in such transportation some parts of different products are mixed and they have to be refined. If mixing of different oil products is not tolerated, the system structure should be considered as a non-homogeneous series system composed of n = 2880 components. In this case the system is composed of 1786 pipe segments with reliability functions R~l)(t, 1) = exp[-0.000000001t4], R~l)(t,2) = exp[-0.000000004t 4] for t > 0, 720 pipe segments with reliability functions R~2)(t, 1) - exp[-0.0000000008t4], R~2)(t,2) = exp[-0.000000002t 4] for t > 0, 360 pipe segments with reliability functions R(3)(t,1) = exp[-0.00000022t3], R(3)(t,2) = exp[-0.0000003t 3] for t > 0, and 14 valves with reliability functions R(4)(t,1) - exp[-0.000000052t4], Rr

= exp[-0.000000107~ 4] for t > 0.

Port and Shipyard Transportation Systems

178

According to Definition 3.17, the considered system is a non-homogeneous multi-state series system with parameters n=2880, a = 4 , qm = 1786/2880, q2 = 720/2880, q3 = 360/2880, q4 = 14/2880, and according to (3.33)-(3.34) its exact reliability function is given by R" 2880

(t,.) = [1,exp[-0.00000309t

4 -

0.0000792?]

exp[-0.000010082t 4 - 0.000108t3]] for t > 0. The expected values of the system sojourn times in the state subset and their standard deviations, according to (3.13)-(3.15), are: M(1) ~ 17.30 years, M(2) _=_14.00 years, o(1) =- 5.60 years, 0(2) _=_4.40 years. Hence, from (3.17), the system mean lifetimes in the states are: M(1) ~ 3.30 years, M(2) ~ 14.00 years. If a critical system reliability state is r = 2, then according to (3.18) its risk function takes the form

r(t) ~ 1 - exp[-0.000010082t 4 - 0.000108fl]. Further, from (3.19), the moment when the system risk exceeds g; = 0.05 is

a

permitted level

r ~ r-t(b-) ~ 6.20 years.

6.4. Reliability of a port bulk transportation system The bulk conveyor system is the part of the Baltic Bulk Terminal of the Port of Gdynia assigned to load ships with bulk cargo from Terminal Storage. Its scheme is given in Figure 6.5. Three self-acting loading machines, the transportation system composed of belt conveyors and the coastal loading system carry out the loading of the ships. In the conveyor loading system we distinguish the following transportation subsystems: S t - the dosage conveyor, $ 2 - the horizontal conveyor, $ 3 - the horizontal conveyor, $ 4 - the sloping conveyors,

Chapter 6

179

Ss - the dosage conveyor with buffer, $ 6 - the loading system. The transporting subsystems have steel covers and they are provided with drives in the form of electrical engines with gears. In their reliability analysis we omit their drives, as they are mechanisms of different types. We also omit their covers as they have a high reliability and, practically, do not fail.

_ 0

$5

$4

[LOADING

//g

......

IWHARFI

[WAREHOUSEI

Fig. 6.5. The scheme of the bulk cargo transportation system

Taking into account the efficiency of the considered transportation system we distinguish the following three reliability states of its components: state 3 - the state ensuring the largest efficiency of the conveyor, state 2 - t h e state ensuring less efficiency of the conveyor caused by throwing material off the belt, state 1 - the state ensuring least efficiency of the conveyor caused by throwing material off the belt and needing human assistance, state 0 - the state involving failure of the conveyer. Subsystem SI is composed of three identical belt conveyers, which consist of a ribbon belt, a drum driving the belt, a reversible drum, 12channelled rollers and three rollers supporting the belt. It means that subsystem Sl consists of k~ = 3 conveyors, each composed of l~ = 18 components. In each conveyer there is one belt with reliability functions R~l'l)(t, 1) = exp[-0.012t2], RO't)(t,2) = exp[-O.022t2], R~l'l)(t,3) = exp[-O.049t 2] for t > 0, two drums driving the belt with reliability functions RCt'2)(t,1) = exp[-0.0019t2], RCl'2)(t,2) = exp[-0.0024t2], RCl'2)(t,3) = exp[-0.0029t 2] for t > 0, 12 channelled rollers with reliability functions R(l'3)(t, 1) = exp[-0.028t2], R(l'3)(t,2) = exp[-0.03t2], R(l'3)(t,3) = exp[-0.032t 2] for t > 0, and three supporting rollers with reliability functions R~l'4)(t,1) = exp[-0.0075t2], R~l'4)(t,2) = exp[-0.0 lt2], R~l'4)(t,3) = exp[-0.02t 2]

Port and Shipyard Transportation Systems

180

for t > 0. Thus, according to Definition 3.20, it is a n o n - h o m o g e n e o u s regular multi-state seriesparallel s y s t e m with parameters:

k,,=k=3,1n = 18, a = 1, ql = 1. el = 4, Ptl = 1/18, Pl2

=

2/18, PI3

=

12/18, Pl4

=

3/18,

and from (3.43)-(3.45) its exact reliability function is given by 3,18(t,')

[1,1-[1

exp[-0.3743t2]]3,1

[1

exp[-0.4168t2]] 3,

1 - [1 - exp[-O.4988t2]] 3] for t > O. Applying Corollary 5.10, according to (5.62), we have a~(1) = min{2, 2, 2, 2} = 2 , 2 12 3 ill(I) = . 1 .0.012 . +. .0 . 0 0 1.9 + . . 0 . 0 2 8 + 0.0075 18 18 18 18 al(2)

=

= 0.020794444,

min{2, 2, 2, 2} = 2,

2 12 3 ill(2) = 1 . 0.022 + 90.0024 + 90.03 + 0.01 18 18 18 18

0.023155555

a1(3) = min{2, 2, 2, 2} = 2, 2 12 3 fl~(3) = . 1 0.049 . +. .90.0029 + 90.032 + n . 0.02 18 18 18 18

= 0.027711111,

and according to (5.63) a(1) = m a x {2} = 2,13(1) = min{O.020794444} - 0.020794444, a(2) = max{2} = 2,/3(2) = min{O.023155555} = 0.023155555, a(3) = max{2} = 2, fl(3) = min{O.027711111} = 0.027711111, and according to (5.61) an(l)

= (0.020794444"18) -1/2 = 1.634519443, b,(1) = 0,

a,(2)

= (0.023155555.18) -1/2 = 1.548945546, b,(2) = 0,

Chapter 6

181

an(3) = (0.027711111.18) -1/2 = 1.415913682, bn(3) = 0, and we conclude that the limit reliability function of the subsystem $1 is given by 9(t,-) =[1,1

[1

exp[-t2]]3,1

[1

exp[-t2]]3,1

[1

exp[-t2]]3]fort>0.

Therefore, from (3.49), the approximate reliability function (the formula is exact in this case) takes the form R'3,18 (t,-) ~ -~"9 ( ( t - bn(u))/an(U)) = [1,1 - [1 - exp[-0.3743t2]]3,1 - [1 - exp[-0.4168t2]] 3, 1 - [1 -exp[-O.4988t2]] 3] for t > O. The expected values of the subsystem lifetimes in the state subsets and their standard deviations, according to (3.13)-(3.15), are: M(1) --- 2.11 years, M(2) -_--2.00 years, M(3) -- 1.83 years, o(1) --- 0.67 years, o(2) ~ 0.63 years, 0(3) =_0.57 years. Hence, from (3.17), the subsystem mean lifetimes in the states are: m

m

M(1) =0.11 years, M(2) ~ 0.17 years, M(3) ~ 1.83 years. If a critical state is r = 2, then from (3.18) the subsystem risk function is given by

r(t) -- [ 1 - exp[-0.4168t2]] 3. Hence, from (3.19), the moment when the subsystem risk exceeds the permitted level 6 = 0.05 is r = r-1(8) = [- (1/0.4168)1og(1 - 3 ~ )]1/2 =_ 1.05 year. Subsystem $2 is composed of one belt conveyor that consists of a ribbon belt, a drum driving the belt, a reversible drum, 125 channelled rollers and 45 rollers supporting the belt. It means that subsystem $2 consists of one conveyor composed of n = 173 components. In the conveyor there is one belt with reliability functions R(l)(t,1) = exp[-0.012t2], R(t)(t,2) = exp[-0.022t2], R(~ two drums driving the belt with reliability functions

= exp[--0.049t 2] for t > 0,

Port and Shipyard Transportation Systems

182

R(2)(t,1) = exp[-0.0019t2], R(2)(t,2) = exp[-0.0024t2], R(2)(t,3) = exp[-0.0029t 2] for t >_ 0, 125 channelled rollers with reliability functions R(3)(t,1) = exp[-0.0074t2], R(3)(t,2) = exp[--0.012t2], R(3)(t,3) = exp[-0.021t 2] for t > 0 and 45 supporting rollers with reliability functions R(4)(t, 1) = exp[-0.002t2], R(4)(t,2) = exp[-0.0025t2], R(4)(t,3) = exp[-0.003t 2] for t >_0. Thus, according to Definition 3.17, it is a non-homogeneous multi-state series system with parameters n = 173, a = 4 , ql = 1/173, q2 = 2/173, q3 = 125/173, q4 =45/173. and according to (3.33)-(3.34) its exact multi-state reliability function is given by

R"173(t,.)

= [1,exp[-1.0308t2],exp[-1.6393t2],exp[-2.81487]]

Next, applying Corollary 5.4, we have a(1) = min{2, 2, 2, 2} = 2 , /3(1) = max{0.012, 0.0019, 0.0074, 0.002} = 0.012, a(2) = min{2, 2, 2, 2} = 2, fl(2) = max{0.022, 0.0024, 0.012, 0.0025} = 0.022, a(3) = min{2, 2, 2, 2} - 2, ,8(3) = max{0.049, 0.0029, 0.021, 0.003} = 0.049, an(l) = (0.012'173) -1/2 = 0.694042915, b,(1) = 0, a,(2) = (0.022.173) -~/2

= 0.512584663, b,(2) = 0,

a,(3) = (0.049.173) -t/2

= 0.343462169, bn(3) = 0 ,

and moreover

for t > 0.

Chapter l)-;"t,'" Cl(

1 ~0.012 + 173 0.012

=

2)-:'t,-" a( 3)-:'t,-" el(

1 0.022 . . . . . 173 0.022

=

.

2 . -0.0019 ---........173 0.012

1 . 0.049 . 173 0.049 .

+

+

+

2. . 0.0024 . . 173 0.022

6

125 . 0.0074 ~ + 173 0.012 +

125 . . .0.012 __...._ 173 0.022

~

2 . 0.0029 125 0.021 ~ + ~ . ~ + 173 0.049 173 0.049 ~

183

.

45. .0.002 . -0.49653175, 173 0.012 45 0.0025 -0.430714636, 173 0.022

+

~

.

45. .0.003 . -0.332051428, 173 0.049

and we conclude that the subsystem limit reliability function is given by ~9' 2 (t,.) = [1,exp[-0.49653175t2],exp[-0.430714636t2],exp[-0.332051428t2]], t >__0. Therefore, from (3.49), the approximate reliability function (the formula is exact in this case) takes the form m

R'17

3

(t,.) ~ .gi"2 ((t-bn(u))/an(U)) = [1,exp[-1.0308t2],exp[-1.6393t2],exp[-2.8148t2]] for t >_0.

The expected values of the subsystem lifetimes in the state subsets and their standard deviations, according to (3.13)--(3.15), are: M(1) =_0.87 years, M(2) ~ 0.69 years, M(2) ~ 0.53 years. o(1) ~ 0.46 years, o(2) --- 0.36 years, o(3) _=_0.28 years. Hence, from (3.17), the subsystem mean lifetimes in the states are" M(1) =_0.18 years, M(2) --- 0.16 years, M(3) =_0.53 years. If a critical state is r = 2, then from (3.18) the subsystem risk function is given by r(t) ~ 1 - exp[-1.6393t2]. Hence, from (3.19), the moment when the subsystem risk exceeds the permitted level 6 = 0.05 is r = r - l ( ~ = [- (1/1.6393)1og(1 - b)] v2 ~ 0.18 years. We will perform the reliability evaluation of the remaining subsystems using Corollary 6.1, which is a simplified modification of Corollary 5.4 and is much easier in use. Subsystem $3 is composed of one belt conveyor that consists of a ribbon belt, a drum driving the belt, a reversible drum, 65 channelled rollers and 20 rollers supporting the

Port and Shipyard Transportation Systems

184

belt. It means that subsystem $3 consists of one conveyor composed of n = 88 components. In the conveyor there is one belt with reliability functions R(O(t,1) = exp[--0.012fl], RO)(t,2) = exp[-0.022t2], R(~

= exp[-0.049t 2] for t > 0,

two drums driving the belt with reliability functions R(2)(t,1) = exp[-0.0019t2], R(2)(t,2) = exp[-O.0024t2], R(2)(t,3) = exp[-0.0029fl] for t > 0, 65 channelled rollers with reliability functions R(a)(t,1) = exp[-O.0074t2], R(3)(t,2) = exp[--0.012t2], R(3)(t,3) = exp[-0.021t 2] for t >_0 and 20 supporting rollers with reliability functions R(4)(t, 1) = exp[-0.002t2], R(4)(t,2) = exp[-0.0025t2], R(4)(t,3) = exp[-0.003t 2] for t > 0. Thus, according to Definition 3.17, it is a non-homogeneous multi-state series system with parameters n = 88, a = 4, ql = 1/88, q2 = 2/88, q3 = 65/88, q4 = 20/88. and according to (3.33)-(3.34) its exact multi-state reliability function is given by R"s8 (t,.) = [1,exp[-0.5368t2],exp[-0.8568t2],exp[-1.4798t2]] for t >_O. Next, applying Corollary 6.1, according to (6.3), we have a(1) = min{2, 29 2, 2} =2, 2 65 20 /3(1) = 1 .0.012+ .0.0019+--.0.0074+ .0.002 88 88 88 88

0.0061

a(2) = rain{2, 2, 2, 2} =2, 2 65 20 fl(2) = __.1 0.022 + - - . 0.0024 + ~ . 0 . 0 1 2 + ~ .0.0025 = 0.009736363, 88 88 88 88 a(3) = min{2, 2, 2, 2} =2, 2 65 20 fl(3) = . 1 0.049+ . . .0.0029+ . .0.021+ .0.003 =0.016815909, 88 88 88 88

Chapter 6

185

and according to (6.2) a.(1) = (0.0061.88) -1/2 = 1.364877726, b.(1) = O, a.(2) = (0.009736363.88) -1/2 = 1.080339575, b.(2) = O, a,(3) = (0.016815909-88) -1/2 = 0.822050484, b,(3) = O, and we conclude that the subsystem limit reliability function is given by "~'2 (t,-) = [1,exp[-t2],exp[-t2],exp[-t2l] for t > O. Therefore, from (3.49), the approximate reliability function (the formula is exact in this case) takes the form w

R' 8s (t,.) = .qi''2 ((t-bn(u))/an(U)) = [ 1,exp[-0.5368t2],exp[-0.8568t2],exp[-1.4798t2]] for t > 0. The expected values of the subsystem lifetimes in the state subsets and their standard deviations, according to (3.13)-(3.15), are: M(1) ~ 1.21 years, M(2) _=_0.96 years, M(2) _=_0.73 years. o(1) ~ 0.63 years, o(2) ~ 0.50 years, o(3) _=0.38 years. Hence, from (3.17), the subsystem mean lifetimes in the states are" M(1) _=_0.25 years, M(2) ~ 0.23 years, M(3) ~ 0.73 years. If a critical state is r = 2, then from (3.18) the subsystem risk function is given by

r(t) ~ 1 - exp[-0.8568t2]. Hence, from (3.19), the moment when the subsystem risk exceeds the permitted level 8 = 0 . 0 5 is r = El(b") = [-(1/0.8568)1og(1 - o0)]1/2 =_0.24 years. Subsystem $4 is composed of one belt conveyor that consists of a ribbon belt, a drum driving the belt, a reversible drum, 12 channelled rollers and three rollers supporting the belt. It means that subsystem $4 consists of one conveyor composed of n = 18 components. In the conveyor there is one belt with reliability functions

Port and Shipyard Transportation Systems

186 Rr

1) = exp[-0.012t2], R~l)(t,2) = exp[-O.022t2], R~l)(t,3) = exp[-0.049t 2] for t > 0,

two drums driving the belt with reliability functions R~2)(t,1) = exp[-O.0019t2], Rt2)(t,2) = exp[-0.0024t2], Rt2)(t,3) = exp[-0.0029t 2] for t > 0, 12 channelled rollers with reliability functions R(3)(t, 1) = exp[-0.028t2], R(3)(t,2) = exp[-0.03t2], R(3)(t,3) = exp[-0.032t 2] for t > 0, and three supporting rollers with reliability functions RC4)(t,1) = exp[-0.0075t2], R~4)(t,2) = exp[-0.01t2], R~4)(t,3) = exp[-0.02t 2] for t >_0. Thus, according to Definition 3.17, it is a non-homogeneous multi-state series system with parameters n=18, a=l, ql = 1/18, q2

=

2/18, q3

=

12/18, q4 = 3/18,

and according to (3.33)-(3.34) its exact multi-state reliability function is given by R"ls (t,.) = [ 1,exp[-0.3743t2],exp[-0.4168t2],exp[-0.4988t2]] for t > 0. Next, applying Corollary 6.1, according to (6.3), we have a(1) = min {2, 2, 2, 2} = 2, 2 12 3 fl(1) = __1 .0.012 + - - . 0 . 0 0 1 9 + m . 0 . 0 2 8 + _ _ . 0 . 0 0 7 5 18 18 18 18

= 0.020794444,

a(2) = min {2, 2, 2, 2} = 2, 2 12 3 /3(2) = 1 . 0 . 0 2 2 + - - . 0 . 0 0 2 4 + .0.03+--.0.01 18 18 18 18

0.023155555

a(3) = min{2, 2, 2, 2} =2, 2 12 3 ~ 3 ) = 1 0.049+ .0.0029+ .0.032+ .0.02 18 18 18 18 and according to (6.2)

0.027711111

Chapter 6

187

an(l) = (0.020794444-18) -1/2 = 1.634519443, bn(1) = 0, a~(2) = (0.023155555-18) -1/2 = 1.548945546, b,(2) = 0, an(3) = (0.027711111.18) -1/2 = 1.415913682, b,(3) = O, and we conclude that the subsystem limit reliability function is given by ~ ' 2 (t,.) = [1,exp[-t2],exp[-t2],exp[-t2]] for t >_0. Therefore, from (3.49), the approximate reliability function (the formula is exact in this case) takes the form R'18 (t,') ~ ,0? 2 ((t-bn(u))/an(U)) = [1,exp[-0.3743t2],exp[-0.4168t2],exp[-0.4988t2]] for t > 0. The expected values of the subsystem lifetimes in the state subsets and their standard deviations, according to (3.13)-(3.15), are" M(1) ~ 1.45 years, )1//(2) -_- 1.37 years, M(3) ~ 1.25 years. o(1) =__0.76 years, o(2) ~ 0.72 years, o(3) _=_0.66 years. Hence, from (3.17), the subsystem mean lifetimes in the states are: M(1) =__0.08 years, M ( 2 ) ~ 0.12 years, M(3) ~ 1.25 years. If a critical state is r = 2, then from (3.18) the subsystem risk function is given by

r(t) ~ 1 - exp[-0.4168?]. Hence, from (3.19), the moment when the subsystem risk exceeds the permitted level 8 = 0.05 is r = r - l ( 6 ) = [-(1/0.4168)1og(1 - 6)] u2 ~ 0.35 years. Subsystem $5 is composed of one belt conveyor, which consists of a ribbon belt, a drum driving the belt, a reversible drum, 162 channelled rollers and 53 rollers supporting the belt. It means that subsystem $5 consists of one conveyor composed of n - 218 components. In the conveyor there is one belt with reliability functions R(t)(t, 1) = exp[-0.012t2], R(t)(t,2) = exp[-0.022t2], R(l)(t,3) = exp[-0.049t 2] for t > 0,

Port and Shipyard Transportation Systems

188

two drums driving the belt with reliability functions R(2)(t, 1) = exp[-0.0019t2], R(2)(t,2) = exp[-0.0024t2], R(2)(t,3) = exp[-0.0029t 2] for t > 0, 162 channelled rollers with reliability functions R(3)(t,1) = exp[-0.0074t2], R(3)(t,2) = exp[-0.012t2], R(3)(t,3) = exp[-0.021t 2] for t > 0 and 53 supporting rollers with reliability functions R(*)(t,1) = exp[-O.002t2], R(4)(t,2) = exp[-0.0025t2], R~

= exp[-0.003t 2]

for t >_0. Thus, according to Definition 3.17, it is a non-homogeneous multi-state series system with parameters n = 218, a = 4 , ql = 1/218, q2 = 2/218, q3 = 162/218, q4 = 53/218. and according to (3.33)--(3.34) its exact multi-state reliability function is given by R ' 2 18 (t,.) = [1,exp[-1 . 3206t2],exp[-2.1033t2],exp[-3.6158t2]] for t > 0.

Next, applying Corollary 6.1, according to (6.3), we have a(1) = min{2, 2, 2, 2} = 2 , fl(1) =

1

218

.0.012 +

2

218

.0.0019 +

162 218

.0.0074 +

53 218

90.002 = 0.006057798,

a(2) = min{2, 2, 2, 2} = 2 , ,3(2) =

1

218

.0.022+

2

90.0024+

218

162 218

9149 +

53 218

.0.0025 =0.009648165,

a(3) = rain{2, 2, 2, 2} = 2 ,

,8(3) =

1

218

90.049 +

2

218

90.0029 +

162 218

90.021 +

53

90.003 = 0.016586238,

218

and according to (6.2) a,(1) = (0.006057798.218) -1/2 = 0.870190543, b,(1) = O,

Chapter 6

189

a~(2) = (0.009648165.218) -1/2 = 0.689524008, b~(2) = 0, an(3) = (0.016586238. 218) -1/2 = 0.525893504, bn(3) = 0, and we conclude that the subsystem limit reliability function is given by ~ ' 2 (t,.) = [1,exp[-t2],exp[-t2],exp[-t2]] for t > O. Therefore, from (3.49), the approximate reliability function (the formula is exact in this case) takes the form m

R'218 (t,.) ~ "~'2 ((t- bn(u))/an(u)) = [1,exp[-1.3206t2],exp[-2.1033t2],exp[-3.6158t2]] for t > O. The expected values of the subsystem lifetimes in the state subsets and their standard deviations, according to (3.13)-(3.15), are: M(1) ~ 0.77 years, M(2) ~ 0.61 years, M(2) ~ 0.47 years. o(1) ~ 0.40 years, 0(2) --- 0.32 years, o(3) _=0.24 years. Hence, from (3.17), the subsystem mean lifetimes in the states are: m

~

m

M(1) --- 0.16 years, M(2) ~ 0.14 years, M(3) ~ 0.47 years. If a critical state is r = 2, then from (3.18) the subsystem risk function is given by r(t) _~ 1 - exp[-2.1033?]. Hence, from (3.19), the moment when the subsystem risk exceeds permitted level 8 = 0 . 0 5 is r = r-l(b") = [-(1/2.1033)1og(1

-

Oe)]1/2 ~ 0.16 years.

Subsystem $6 is a set of belt conveyors placed on the moving tracks and having the possibility of moving in relation to each other. It is composed of a rotary mechanism, two stable belt conveyors and one moving belt conveyer. One of the stable conveyors consists of one ribbon belt, a drum driving the belt, a reversible drum, 34channelled rollers and 10 rollers supporting the belt. The second stable conveyor consists of one ribbon belt, a dram driving the belt, a reversible drum, 15charmelled rollers and five rollers supporting the belt. The moving conveyor consists of one ribbon belt, a drum driving the belt, a reversible drum, 15channelled rollers and five rollers supporting the belt. Thus subsystem $6 is a non-homogeneous multi-state series system composed of n

Port and Shipyard Transportation Systems

190

= 93 components of four types. In the subsystem there are three belts with reliability functions R(l)(t, 1) = exp[-0.012t2], R(1)(t,2) = exp[-0.022t2], R(l)(t,3) = exp[-0.049t a] for t > 0, six drums driving the belt with reliability functions R(2)(t, 1) = exp[-0.0019t2], R(2)(t,2) = exp[-0.0024t2], R(2)(t,3) = exp[-O.0029t 2] for t > 0, 64 channelled rollers with reliability functions R(3)(t, 1) = exp[-0.0046t2], R(3)(t,2) = exp[-O. 0075 t2], R(3)(t,3) = exp[-0.012t 2] for t > 0 and 20 supporting rollers with reliability functions R(4)(t, 1) = exp[--0.0012t2], R(4)(t,2) = exp[-O.0018t2], R(4)(t,3) = exp[-0.0024t 2] for t > 0. Thus, according to Definition 3.17, it is a non-homogeneous multi-state series system with parameters n =93, a = 4 , q l = 3/93, q2 = 6/93, q3

=

64/93 q4

=

20/93.

and according to (3.33)--(3.34) its exact multi-state reliability function is given by R'93 (t,') "- [1,exp[--0.3658tZ],exp[-0.5964tZ],exp[-0.9804t2]] for t >__0.

Next, applying Corollary 6.1, according to (6.3), we have a(1) = min{2, 2, 2, 2} =2, 6 64 20 /7(1) = . 3 .0.012+ . . 0.0019+ . . .0.0046+ .0.0012 =0.003933333, 93 93 93 93 a(2) = min{2, 2, 2, 2} = 2, 6 64 20 / 7 ( 2 ) - 3 0.022+ 0.0024+ -0.0075+ .0.0018 93 93 93 93

0.006412903

a(3) = min{2, 2, 2, 2} - 2 , 6 64 20 ,0(3) = . 3 .0.049+ . . . 0.0029+ .0.012+~.0.0024 93 93 93 93

= 0.010541935,

Chapter 6

191

and according to (6.2) a,(1) = (0.003933333.93) -~/2 = 1.6534000893, b~(1) = 0, a,(2) = (0.006412903.93) -1/2 = 1.294884971, b,(2) = 0, a,(3) = (0.010541935.93) -]/2 = 1.009946454, b,(3) = 0, and we conclude that the subsystem limit reliability function is given by -~'2 (t,-) = [1,exp[-t2],exp[-fl],exp[-t2]] for t >_0. Therefore, from (3.49), the approximate reliability function (the formula is exact in this case) takes the form m

R t

93 ( t , . )

~

~

t

2

((t-bn(u))/a,,(u))

= [1,exp[-0.3658t2],exp[-0.5964t2],exp[-0.9804t2]] for t > 0. The expected values of the subsystem lifetimes in the state subsets and their standard deviations, according to (3.13)-(3.15), are: M(1) ~ 1.47 years, M(2) ~ 1.15 years, M(3) ~ 0.90 years. o(1) ~ 0.77 years, o(2) ~ 0.60 years, 0(3) ~_0.47 years. Hence, from (3.17), the subsystem mean lifetimes in the states are: m

m

M (1) ~ 0.32 years, M (2) ~_0.25 years, M (3) =_0.90 years. If a critical state is r - 2, then from (3.18) the subsystem risk function is given by r(t) -_-_1 - exp[-0.5964t2]. Hence, from (3.19), the moment when the subsystem risk exceeds the permitted level 6 = 0.05 is r = r - l ( ~ = [-(1/0.5964)1og(1

-

Or 1/2 ~ 0.29 years.

Since, according to Definition 3.17, all subsystems create a series reliability structure, then from (3.33)-(3.34) the reliability function of the whole transportation system is given by

Port and Shipyard Transportation Systems

192

R' (t,.) _= [ 1, R' (t,1), R' (t,2), R' (t,3) ], where R' (t,1) = 3exp[-4.0026t 2] - 3exp[-4.3769fl] + exp[-4.7512fl], R" (t,2) = 3exp[-6.0294P]- 3exp[-6.4462fl] + exp[-6.8630fl], R' (t,3) = 3exp[-9.8884fl] - 3exp[-10.38727] + exp[-10.8860?]. The expected values of the subsystem lifetimes in the state subsets and their standard deviations, according to (3.13)-(3.15), are: M(1) --- 0.45 years, M(2) -= 0.37 years, M(3) ~ 0.29 years, o(1) _--_0.27 years, o(2) -= 0.20 years, o(3) _=_0.15 years. Hence, from (3.17), the subsystem mean lifetimes in the states are: u

m

w

M (1) ~ 0.08 years, M (2) ~ 0.08 years, M (3) --- 0.29 years. If a critical state is r = 2, then from (3.18) the subsystem risk function is given by

r(t) _=_1 - R'(t,2). Hence, by (3.19), the moment when the subsystem risk exceeds the permitted level 8 = 0.05 is r = ~ l ( ~ =_0.10 years. The system exact and approximate reliability functions are identical. Their behaviour and the system risk function are presented in Table 6.3 and Figure 6.6.

Chapter 6

193

Table 6.3. The values of the multi-state reliability function and the risk function of the port bulk transportation system

t

'

R' (t,1)

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 i 1.20 1.30 1.40 1.50

9

.

.

.

.

.

.

.

R' (t,2)

1.00000 0.96437 0.86490 0.72138 0.55949 0.40342 0.27031 0.16820 0.09712 0.05198 0.02575 0.01180 0.00499 0.00195 0.00070 .

.

.

.

.

.

.

.

i

.

0.00023

.

.

.

1.00000 0.94542 0.79891 0.60339 0.40727 0.24558 0.13223 0.06351 0.02719 0.01036 0.00351 0.00105 0.00028 0.00007 0.00001 0.00000 .

.

.

.

.

.

.

r(t)

R' (t,3) 1.00000 0.91038 0.68683 0.42949 0.22251 0.09546 0.03389 0.00994 0.00241 0.00048 0.00008 0.00001 0.00000 ~ 0.00000 0.00000 0.00000

9

.

.

.

.

.

1norm 0.00000

i ,

0.05458 0.20109 0.39661 0.59273 0.75442 0.86777 0.93649 0.97281 0.98964 0.99649 0.99895 0.99972 0.99993 0.99999 1.00000

,

I

~ R'(t,.) u =0

1.00 I~ 0.80 0"601

~

0"4017'

'

u=1

0"20J/u,,,u

O.Ou ~ 000.

, 0.50

, 1.00

, 1.50

r

J

Fig. 6.6. Graphs of the multi-state reliability function and the risk function of the port bulk cargo transportation system

194

Port and Shipyard Transportation Systems

6.5. Reliability of a shipyard rope transportation system Ship-rope elevators are used to dock and undock ships coming to shipyards for repairs. The elevator utilised in the Naval Shipyard in Gdynia, with the scheme presented in Figure 6.7, is composed of a steel platform-carriage placed in its syncline (hutch). The platform is moved vertically with 10 rope-hoisting winches fed by separate electric motors, each rope having a maximum load of 300 tonnes. During ship docking the platform, with the ship settled in special supporting carriages on the platform, is raised to the wharf level (upper position). During undocking, the operation is reversed. While the ship is moving into or out of the syncline and while stopped in the upper position the platform is held on hooks and the loads in the ropes are relieved. Since the platformcarriage and electric motors are highly reliable in comparison to the ropes, which work in extremely aggressive conditions (salt water, wind, pollution and so on), in our further analysis we will discuss the reliability of the rope system only. The system under consideration is in order if all its ropes do not fail. Thus we may assume that it is a series system composed of 10 components. Each of the ropes is composed of 22 strands: 10 outer and 12 inner. The outer strands of ropes are composed of 26 steel wires. Their inner strands are composed of 19 steel wires and they form the rope's steel core, which is covered by a plastic layer. The cross-section of the rope is shown in Figure 6.8.

Fig. 6.7. The scheme of the ship-rope transportation system Thus, considering the strands as basic components of the system, according to Definitions 3.14-3.16 we conclude that the rope elevator is a parallel-series system

Chapter 6

195

composed of k,, = 10 series-linked subsystems (ropes) with l,, = 22 parallel-linked components (strands).

Fig. 6.8. The cross-section of the rope

According to safety standards and to the requirements for approximately comparable reliability (durability of outer and inner strands, after considering technical norms ([22], [ 102]) and expert opinion ([79]), the following reliability states of all strands have been distinguished: state 3 - a strand is new, without any defects, state 2 - the number of broken wires in the strand is greater than 0% and less than 25% of all its wires, or corrosion of wires is greater than 0% and less than 25%, state 1 - the number of broken wires in the strand is greater than or equal to 25% and less than 50% of all its wires, or corrosion of wires is greater than or equal to 25% and less than 50%, state 0 - otherwise (a strand is failed). Considering these component reliability states, according to Definitions 3.14-3.16, we conclude that the rope system is a homogeneous regular four-state parallel-series system. Thus, from (3.30)-(3.31), its reliability function is defined by Rlo,z 2 (t,.) = [ 1, Rio,2 2 (t,1), Rio,2 2 (t,2), Rlo,22 (t,3) ], where R~0,22 (t, u) = [1 -

[F(t,u) ]22 ]10, t e (--oo,oo), u = 1,2,3.

(6.7)

According to experts' opinions ([79]) the mean numbers of ship docking and undocking are equal to 80 per year. It means that the elevator is active 160 times per year. Moreover, the rope elevator system reliability depends strongly on the tonnage of the docking ships and therefore the following states of the elevator system loading have been distinguished: state 4 - loading from 2250 to 1750 tonnes, state 3 - loading from 1750 to 1250 tonnes,

196

Port and Shipyard Transportation Systems

state 2 state 1 state 0 The loading

loading from 1250 to 750 tonnes, loading from 750 to 250 tonnes, without loading. states i, the frequencies n i of loading states per year, the duration times t i

of the loading states and the total duration times Ti of the loading states per year of the rope elevator system are given in Table 6.4.

Table 6.4. The rope elevator loading states characteristics

, , ,

i

ni

ti(h )

Ti(h )

4 3 2 1 0 Total

20 80 40 20 160

8 6 3 2 -

160 480 120 40 7960 8760 ,,,

,,

,

Using these data and the formula

re

Pi = ~ , i = 0,1,2,3,4, 8760 it is possible to evaluate the probabilities of the elevator loading states. They are as follows Po = 0.9087, Pl = 0.0046, P2

=

0.0137, P3 = 0.0548, P4

=

0.0182.

(6.8)

Considering these particular elevator loading states we obtain the following formula for its reliability function R 10,22

(t,.) = [1 , R 10,22 (t,1) , R' 10,22 (t,2) , R' 10,22 (t,3) ],

(6.9)

where e--, 10,22

~(o) (t, u) + Pl ~(l) ~ (2) (t, u) (t, u) = Po "10,22 10,22 (t, u) + P2 ~'L10,22 + P3 ~(3) 10,22 (t, u) + P4 ~(4) ""10,22 (t, u) ' t ~ (-oo,oo), u = 1,2,3,

(6.10)

and according to (6.7)

-~(i) (t,u) = [1 - [ F (i) (t,u) 10,22

]22 ]10

i =

0,1,2,3,4,

are its multi-state reliability functions in particular loading states, while

(6.11)

Chapter 6 R (i) (t,u) = 1 -

F (i)

197

(t,U), t ~ (--oo,oo), u = 1,2,3, i = 0,1,2,3,4,

are component (strand) multi-state reliability functions in particular elevator loading states. According to rope reliability data given in their technical certificates ([22]) and experts' opinions ([79]) based on the nature of strand failures it has been assumed that the strands have multi-state Weibull reliability functions

R r (t,u) = 1 for t < O, R (i) (t,u) = exp[

- ~i

(u) tai(u) ] for t >_ O,

u = 1,2,3, i = O, 1,2,3,4, with the following parameters

ai(u ), fli(u), i = 0,1,2,3,4, in the particular elevator loading states: a o (u) = 3, flo (u) = O.O03u, a I (u) = 3, fll (u) = O.O06u, Ct2 (u) = 3, f12 (U) = O.O08u , Ct 3 (u) = 2,

f13 (U) = 0.090u,

ct 4 (u) = 1.5, f14 (u) = 0.250u, u = 1,2,3. Hence, after considering (6.10), (6.11) and (6.8), the exact elevator reliability function is given by the formula (6.9) with

R' 10,22 (t, u) = 0.908711-[1 - exp[-0.003ut 3 ]]22 ]10 + 0.004611 -[1 - exp[-O.OO6ut 3 ]]22 ]10 + 0.013711 - [1 - exp[-0.008ut 3 ]]21 ]lo + 0.054811 -[1 - exp[-0.09ut 2 ]]22 ]10 + 0 . 0 1 8 2 [ 1 - [ 1 - e x p [ - O . 2 5 u t 3/2 ]]22 ]10, U "- 1,2,3.

(6.12)

Port and Shipyard Transportation Systems

198

Since the number of parallel subsystems in the system is k n = 10 and the number of components in each subsystem is In = 22, then considering that 1n = 22 >> logk n = loglO --- 2.3 it seems reasonable to apply in finding the limit reliability function of the rope elevator system either Corollary 6.2 or Corollary 5.11. We will use both of them separately; first at different loading states for estimating the system reliability functions given by (6.11), and approximating the reliability function of the system by combining the results according to (6.12). Next, we will compare and comment on the achieved results.

Application of Corollary 6.2 According to (6.6), assuming normalising constants

b(i)(u)= [ 1 log I. ]l/a,(.) fli(U)

logkn

a (i) (u) = (b (i) (u)) l-ai(u)/(ai(u)i~i(u) logkn),u = 1,2,3, i = 0,1,2,3,4,

(6.13)

we conclude that ~ ( i ) (t,') = [1, ~ ( i ) (t,1), ~(i) (t,2), ~3 (i) (t,3) ], t ~ (--00,00), where

.~(3i) (t,u) = exp[-exp[t]] for t ~ (-oo,oo), u = 1.2,3, i = 0,1,2,3,4, is the multi-state reliability function of the rope elevator system in the ith loading state. According to (6.13) and (3.49) for particular elevator loading states we have: loading state 0 b~~ (I) = 9.0950, a~~ (1) = 0.5834, b (~ (2) = 7.2187, a~~ (2)=0.4630, b~~ (3) = 6.3061, a~~ (3) = 0.4045,

~(o) 10,22(t,1) =exp[-exp[1.7141t-15.5909]] ~-(o) 10,22(t,2) = exp[-exp[2.1598t-15.5909]]

Chapter

•

0,22

(

o

6

199

)

(t,3) -_- exp[-exp[2.4722t-15.5909]]

loading state 1 bn0) (1) = 7.2187, an0) (1) = 0.4630, bnO) (2) = 5.7295, a(nl) (2) = 0.3675, bO) (3) = 5.0052, n

a n(1) (3)

= 0.3210,

~ 010,22 ) (t,1) ~exp[-exp[2.1598t-15.5909]] R10,22--(1) _ (t,2)

)

_=exp[- exp[2.721 l t - 15.5909]],

~ 'lO,zz 0 ) (t,3) -___exp[-exp[3.1153t-15.5909]], loading state 2 b,(2) (1) = 6.5587, a~ 2) (1) = 0.4207, b(2) n (2) = 5.2056, a(nz) (2) = 0.3339, bn(2) (3) =4.5475, a(n2) (3) = 0.2917, 'I(2) 0,22 (t,1) ~. exp[-exp[2.3771t-15.5909]] 'I(2) 0,22 (t,2) --- e x p [ - e x p [ 2 . 9 9 5 0 t - 15.5909]]

~(2) lO,z2 (t,3) --- exp[- exp[3.4282t - 15.5909]], loading state 3 bn(3) (1) = 5.0078, a(n3) (1) = 0.4818,

bn ( 3 ) (3.5410, 2)=

a n(3)(2)=0.3407,

b~a) (3) = 2.8912, a(n3) (3) =0.2782,

)

200

Port and Shipyard Transportation Systems ~(3) 10,22 (t,1)

--- exp[-exp[2.0756t- 10.3939]]

~(3) 10,22 (t,2) =exp[-exp[2.9351t-10.3939]] ~(3) 10,22 (t,3) -_-exp[-exp[3.5945t-10.3939]] loading state 4 b (4) (1)= 4.3357, a~4) (1)= 0.5562, b~4) (2) = 2.7313, a~4) (2) = 0.3504, bn~4)(3) = 2.0844, a n(4)(3)=0.2674, R1(4) 0,22 (t,1) - exp[-exp[1.7979t- 7.7954]] ~ (10,22 4 ) (t,2)

--- exp[-exp[2.8539t-7.7954]]

~(4)_ 10,22 (t,3)

--- exp[- exp[3.7397t- 7.7954]].

Combining the achieved results of reliability approximation, according to (6.9) and (6.10), we get the approximate reliability function of the rope elevator system given by R lO,22 (t,.) = [1 R' 10,22 (t,1) ~R' 10,22 (t,21 ~R' 10,22 (t,3) 1, t ~ (--~,oo),

where R 10,22 (t,1)

~ 0.9087 exp[-exp[1.7141t-15.5909]] + 0.0046 exp[- exp[2.1598t - 15.5909]] +0.0137 exp[- exp[2.377 It - 15.5909]] + 0.0548 exp[- exp[2.0756t - 10.3939]] + 0.0182 exp[- exp[1.7979t - 7.7954]],

(6.14)

Chapter 6

201

m

R' 10,22 (t,2) ~ 0.9087 exp[-exp[2.1598t-15.5909]] + 0.0046 exp[- exp[2.721 It - 15.5909]] + 0.0137 exp[- exp[2.9950t - 15.5909]] + 0.0548 exp[- exp[2.935 It - 10.3939]] +0.0182 exp[- exp[2.8539t- 7.7954]], m

R' 10,22 (t,3) --. 0.9087 exp[-exp[2.4722t - 15.5909]] +0.0046 exp[- exp[3.1153t-15.5909]] +0.0137 exp[- exp[3.4282t - 15.5909]] + 0.0548 exp[- exp[3.5945t- 10.3939]] + 0.0182 exp[- exp[3.7397t - 7.7954]]. The approximate expected values of the system lifetimes T(u) in the state subsets, according to (3.13), are ([73]): M(1) ~ 8.40 years, M(2) v_6.70 years, M(3) ~ 6.30 years. Hence, according to (3.17), the mean lifetimes of the system in the states are: m

m

M(1) ~ 1.70 years, M(2) ~ 0.40 years, M(3) ~ 6.30 years. If a critical state is r = 2, then a system risk, by (3.18), is given by r(t) = 1 - R lo,22 (t,2) ~ 1-0.9087 exp[-exp[2.1598t-15.5909]] --'1

-0.0046 exp[- exp[2.721 I t - 15.5909]] -0.0137 exp[- exp[2.9950t -15.5909]] - 0.0548 exp[- exp[2.935 It - 10.3939]] -0.0182 exp[-exp[2.8539t- 7.7954]]. The time when the risk exceeds a permitted level 8 = 0.05, according to (3.19), is

(6.15)

Port and Shipyard Transportation Systems

202

r = r-l(o") = 3.50 years.

Application of Corollary 5.11 According to (5.75), letting

1

b (i) (u) = [ fli (U)

log

ln

]z/ai(.)

a(n0 (u) = (b(n0 (u)) 1-a'(u) /(ai(u)~i(u)),

U = 1,2,3, i = 0,1,2,3,4,

(6.16)

we conclude that --(i) . . . . . . ~1o (t,.) = [1, .q/l~ ) (t,1), .$/1~) (t,2), ~ 1 ~ ) (t,3) 1, t ~ (-o%oo/, where

9l--l(iO) (t,u) = [1 - exp[-exp[-t]]* for t ~ (-oo,oo), u = 1,2,3, i = 0,1,2,3,4, is the multi-state limit reliability function of the rope elevator system in the ith loading state. Hence, considering (6.16) and (3.4), for the particular states we have: loading state 0 bn(~ (1) = 10.1002, an(~ (1) = 1.0892, b (~ (2) = 8.0165, an(~ (2) =0.8645, b~~ ( 3 ) = 7.0031, a (~ (3) =0.7552, ~(o)_ 10,22 (t,1) =_[1 - exp[-exp[-0.9181t + 9.2731]]] 1~

• 10,22 (t,2)(

= [ 1 - e x po[ - e x p [ - 1 ) ' 1568t +9.2731]]] 1~'

~(o) 10,22 (t,3) =[1-exp[-exp[-1.3242t loading state 1 b~ 1) (11 = 8.01650, a , O, u = 1,2 .... ,z, and d(t,u)

= ~qi i-1

Xi(u) .... ;~(u)

for t > O, u = 1,2,...,z,

is its limit reliability function.

Proposition 7.2 If c o m p o n e n t s o f the n o n - h o m o g e n e o u s multi-state parallel s y s t e m h a v e e x p o n e n t i a l reliability functions

R (i) (t,.) = [ R (i) (t,1),..., R (i) (t, z) ], t ~ (--oo,oo), where

RO)(t,u) = 1 for t < O, R(~

= exp[-A~(u)t] for t > O, i = 1,2, .... a, u = 1,2 .... ,z,

1 1 a , ( u ) - ,~()-'u" bn(u)= ,~()-'u"l o g n , u = 1,2, .... z,

where 2 ( u ) = m i n { A i ( u ) } , u = 1,2,...,z, l O, i = 1,2,...,a, u - 1,2,...,z,

215

216

Large Multi-State Exponential Systems

is its limit reliability function.

Proposition 7.5 If components of the non-homogeneous regular multi-state parallel-series system have exponential reliability function

R (i'j) (t,.) = [ R (i'j) (t,1) ..... R (i'y) (t, z) ], t ~ (--oo,oo), where

R(tJg(t,u) = 1 for t < O, R(iJ)(t,u) - exp[-2o(u)t ] for t ___O, i = 1,2,...,a,j = 1,2,...,e~, u = 1,2,...,z, and

Case l. k n = n,

I n --->l, l ~ (O, ao),

1 a,(u) = A,(u)nl/ln , bn(u) = O, u = 1,2,...,z, where

Ai(u) = f l A p~ (u), A(u)= max{Ai(u)} , u = 1,2,...,z, l>s,

an(U) =

2(u) log n

c>O,

s>O,

,b,(u) = 2 ~ u ) l o g ( i / g n ) , U = l , 2 ,

.... z,

where 2i(u) = max {2o.(u)}, 2 ( u ) = max{2i(u)}, u = 1,2,...,z, loo) I(n+l)m logn+l (n - m + 1 m

= constant (m / n -~ 1 as n - o oo)

A(u) = n$(u), u = 1,2,..., z

Chapter 7 Results

223

Case 1. m = constant (m / n --~ 0 as n ~ ~ ) system reliability function R (m) (t,.) ~ ~ o ) ( ( t - bn(u))/an(u),')

m-1 exp[-iA(1)t + iB(1)]

=[1, 1- E

....

i=o

exp[exp[-A(1)t + B(1)]] ,...,

i!

l - Ym-1 , exp[-iA(z)t + iB(z)] exp[exp[-A(1)t + B(1)]] ]

i=o i! system mean lifetimes in state subsets M(u) = (a geometric integration), u = 1,2,...,z standard deviations of system lifetimes in state subsets or(u) = (a geometric integration), u = 1,2,...,z system mean lifetimes in particular states M (u) = M(u) - M(u + 1), u = 1,2,...,z - 1, M (z) = M(z) system risk function m-1 exp[-iA(r)t + iB(r)] exp[exp[-A(r)t + B(r)]] r(t) = ~, i=0 i! exceeding moment of admissible system risk level d Z'= r - l ( t )

Results

Case2. m / n --> lt , ( O < lt < l as n-->oo) system reliability function

e(n m) (t,.) ~ 91(6u) ( ( t - b,(u))/a,(u),.) 1 2~

=[1,1

1

1 ~ m

~

A(1)t-B(1) x2 -oo~ e 2 d x , . . . , m

~

x2

A(z)t-B(z) -~ol e 2 d x ]

system mean lifetimes in state subsets M ( u ) = B(u) / A(u), u = 1,2,...,z, system mean lifetimes in particular states M (u) = M(u) - M(u + 1), u = 1,2,...,z - 1, M (z) = M(z) system risk function 1 A(r)t-n(v) _x~ r=_ J e 2 dx ,r z Tr exceeding moment of admissible system risk level 6

r(t)-

--00

224 Results

Large Multi-State Exponential Systems

Case 3. n - m = ~ = constant (m / n --, 1 as n --> oo) system reliability function R (n~) (t,.) _~ ~ 9 O) ( ( t - bn(u))/a.(u),.) =

[1, E [A(1)]i ti ~ exp[-A(1)t], i=O i[ [A(2)] i t i Z ~ exp[-A(2)t], .... /=o i!

[A(z)] i t i ~, ~ e x p [ - A ( z ) t ] for t > 0 i=o i! system mean lifetimes in state subsets M(u) = (a geometric integration), u = 1,2,...,z standard deviations of system lifetimes in state subsets o'(u) = (a geometric integration), u = 1,2,...,z system mean lifetimes in particular states g ( u ) = M(u) - g ( u + 1), u = 1,2,...,z- 1, g ( z ) = M(z) system risk function [A(r)] i t i r(t) = 1 - E exp[-A(r)t] i=o i[ exceeding moment of admissible system risk level 6

r= r-~(t)

.

.

.

.

.

.

.

.

.

Chapter 7

Table 7.4. Algorithm of reliability evaluation of a series-parallel system

A series-parallel system

System type Data

number of series subsystems linked in parallel

k. number of components in series subsystems

I. number of types of series subsystems a fractions of series subsystems of particular types ql,...,qa

numbers of types of components in series subsystems el,...,ea

fractions of components of particular types in series subsystems Pll,..., Plq Pal,..., Paea

number of component and system states g component transition rates between state subsets '~11 (U),..., ~le I (U),

~al (U),..., ,~,aea (U), U = 1,2,..., Z

i

Calculations

system critical state r system risk admissible level 6 C a s e 1. system shape: k, --->oo, 1, > 0 2i(u ) =

j=l

poA.ij(u), i = l,2,...,a ,

2(u) = min{2i(u)}, u = 1,2,...,z l s,

c>0, s>0 ei

Ai(u) = E~,~ # (u), i = 1,2,...,a, A(u) = max{Ai(u)} j-I

d(t,u) =

l 0, s e (--0%00) A(u) = (n l/t,, _ 1)A(u)ln ' B(u)

Calculations

= (n l/l

-

1)l n log(1 - n -1/l" )

Case 4. system shape: k~ --~ oo, I n - c log n >> s, c > O, s > 0 A i ( U ) -- m a x {A# ( u ) } , l s, c > 0, s > 0

R knln (t,.) ~ 91 3((t-bn(u))/an(U),.)

= [ 1,exp[-exp[A (1)t + B(1)]],..., exp[-exp[A(z)t + B(z)]]] mean system sojourn times in state subsets M(u) = [-0.5772 - B(u)]/A(u), u = 1,2,...~ standard deviations of system sojourn times in state subsets

or(u) = x / ( A ( u ) , f 6 ) , u = 1,2,...,z mean system sojourn times in particular states M ( u ) = M ( u ) - M ( u + 1), u = 1,2 .... , z - 1, M ( z ) = M ( z ) system risk function r(t) = 1 -exp[-exp[A(r)t + B(r)] exceeding moment of admissible system risk level 6 r _=_[log(- log(1 - 6)) - B(r)] / A(r) ....

Results

Case 5. system shape: k, ~ k, I, --~ oo system reliability function R knln (t,.) = 91'1o ((t-bn(u))/an(u),')

= [1,

I-I[1- exp[- exp[-A(1)t + B(1)]]] q'* (i:Xi (1)=;t(1))

[ I [ 1 - e x p [ - e x p [ - A ( z ) t + B ( z ) ] ] ] qik ] (i:~,.(z)=,~(z )) system mean sojourn times in state subsets o0

M(u) = I[

1111- exp[-exp[-A(u)t + B(u)]]] qi* ]dt, u = 1,2,...,z

0 (i:2i (1)-2(1))

standard deviations of system sojourn times in state subsets

cr(u) = ~ N ( u ) - M 2 (u) oO

N(u) = 2 I t [

I-I[1- exp[-exp[A(u)t + B ( u ) ] ] qik ]dt , u = 1,2, .... z

0 (i:2i(u)=2(u))

mean system sojourn times in particular states M ( u ) = M ( u ) - M ( u + 1), u = 1,2,...,z- 1, M ( z ) = M ( z ) system risk function

r(t) = 1 -

I-I[1-exp[-exp[-A(r)t + B(r)]]] q'* (i:A~.(r)=2(r))

exceeding moment of admissible system risk level 8

f'(o3

Large Multi-State Exponential Systems

232

7.3. Algorithms application to reliability evaluation of exponential systems The examples of the system reliability evaluation presented here are an illustration of the usage of the algorithms, addressed directly to reliability practitioners.

Example 7.1. (a piping system) Let us consider a piping system composed of n = 80 four-state pipe segments of four types such that 20 pipe segments have exponential reliability functions with transition rates between state subsets /~l(u) = 4 u-4, u = 1,2,3, 20 pipe segments have exponential reliability functions with transition rates between state subsets ,~2(u) = (4.5) u-4, u = 1,2,3, 10 pipe segments have exponential reliability functions with transition rates between state subsets 23(u) = (6.5) u-4, u = 1,2,3, and 30 pipe segments have exponential reliability functions with transition rates between state subsets &4(u) = 8u-4, u = 1,2,3. According to Definition 3.17, the considered piping is a non-homogeneous multi-state series system with parameters n = 80, a = 4, ql = 2/8, q2 = 2/8, q3 = 1/8, q4 = 3/8. Its reliability evaluation may be performed using the algorithm presented in Table 7.1. Sequential steps of the procedure are given in Table 7.6.

Table 7.6. Reliability evaluation of the piping system

System type Data

A non-homogeneous series system number of system components n = 80 ,,,

Chapter 7

233

number of component types a=4 fractions of different component types ql = 2/8, q2 = 2/8, q3 = 1/8, q4 = 3/8 number of component and system states z=3 component transition rates between state subsets 21 (1) = 4 -3 , 22 (1) = (4.5) -3 , 2 3 (1) = (6.5) -3 , )].4 (1) = 8 -3 '~,1 (2) = 4-2, '~'2 (2) = (4.5) -2 , )]'3 (2) = (6.5) -2 , ~4 (2) = 8 -2 ,~1 (3) = 4 -1 , ~2 (3) = (4.5) -1 , )]'3 (3) = (6.5) -1 , )~4 (3) = 8 -1

system reliability critical state r=2 admissible level of system risk 6 =0.05 Calculations

A(1) = 80[(2 / 8)4 -3 + (2 / 8)(4.5) -3 + (1 / 8)(6.5) -3 + (3 / 8)8 -3 ] =0.627 A(2) = 80[(2 / 8)4 -2 + (2 / 8)(4.5) -2 + (1 / 8)(6.5) -2 + (3 / 8)8 -2 ] = 2.943 A(3) = 80[(2 / 8)4 -1 + (2 / 8)(4.5) -1 + (1 / 8)(6.5) -1 + (3 / 8)8 -1 ] = 14.733

Results

system approximate reliability function R t

8o (t,.)

=

.9i" 2 ( ( t - bn(u))/an(u),')

= [1,exp[-O.627t],exp[-2.943t],exp[-14.733t]]

for t > 0

system mean lifetimes in state subsets M(1) = 1/0.627 --- 1.59, M(2) = 1/2.943 _=_0.34, M(3) = 1/14.733 _-- 0.07 standard deviations o f system lifetimes in state subsets o(1) --__1.59, 0(2) --- 0.34, o(3) --__0.07 system mean lifetimes in particular states M ( 1 ) ) = 1.59-0.34--__ 1.25, M ( 2 ) =- 0.34 - 0.07 _--0.27,

M(3) --0.07 system risk function r(t) = 1 - exp[-A(2)t] = 1 - exp[-2.943t] exceeding moment of admissible system risk level fi r = - ( l o g ( l - 0.05)) / 2.94 _=_0.017 The behaviour o f the multi-state reliability function and risk function of the piping system is illustrated in Table 7.7 and Figure 7.1.

Large Multi-State Exponential Systems

234

Table 7.7. The values of the multi-state reliability function and the risk function of the piping system -q?'2 ( t _- ~b n (1) ,1) ] -~"2(~t - b n (2) an(l) ]i an(2) ' 0.0' 1.00000 1.00000 ' 0 ' 2 ....... 0.88214 ' 0.5551'0 . . . . i 0.4 ' 0.77818 ' 0.30814 0.17105 0.6 0.68647 0160556 . . . . . . . . . . . 0.09495 0.8 0105271 1.0 0153419 0.02926 . . . . j 1.2 0,47123 0.01624 1.4 0'41570 0100902 '1.6 .... 0,36670 ! ' 1 8. . . . 0.32349 i 0100500 9 i 0100278 ' 2.0 ' 0.28'536 0.00154 ' 2.21' 0.25173 ' 0.00086 2.4 0.22206 o.ooo48 ......2.6" ' 0.1~)589 ........ 0.17280 ' 0.00026 l 2 .

ffl'2 ( t - b n (3) an~ ,3)

t

92

9

.

. . . . . .

,

/,

.

.

.

)

' i "

1,6oooo

=

. . . . . . . .

=

|

|

|

.8-

1.0

,

|

,

~

.

.

.

01oooo0 0.44456 "0'69149 0.82864 '0.90482' " 0'947i3 0.97064 !0'98369' ' 0.~J9094 ' 0.9949'7 0.99721 '0.99845' 0'99914 0.99952 i 0.99974 i

0.0:5252 0.00276 0.00014 0.00001 ~00000' 0.00000 0.00000 0.00000

.

'

i . i ' " '

9

o.oooo0 o.ooooo 0.00000 0.00000 0.oo0oo

i

|

t

|

0.ooooo

.

r(t)

,

,

12 =

|

0

0.8

0.6

U=I 0.4

0.2

U=2 O0 '0.0

0.2

0.4

0.6

0.8

1.O

1.2

1.4

1.6

' 1.8

t

Fig. 7.1. Graphs of the multi-state reliability function and the risk function of the piping system

Chapter 7

235

Example 7.2. (a model parallel system) Let us consider a h o m o g e n e o u s six-state parallel system such that n = 30, 2(u) = 10 u - 6 h -l, u

-

-

1,2,3,4,5.

We will p e r f o r m its reliability evaluation based on Table 7.2. The procedure is given in Table 7.8.

Table 7.8. Reliability evaluation of the homogeneous parallel system A homogeneous parallel system

System type Data

n u m b e r o f system components n=30 n u m b e r o f different component types a=l fractions o f different component types ql=l n u m b e r o f component and system states z=5 component transition rates between state subsets Al (1) = 10 -5 2 a (2) = 10 -4 '~,1(3) = 10 -3 '~1 (4) = 10 -2 A1(5) = 10 -l system reliability critical state r=2 admissible critical level of system risk 8=0.05

Calculations

2(1) = 10 -5

A(2) = 10 -4

A(3) = 10 -3

2(4) = 10 -2

A(5) =1o -~

d(u) =

1, u -

1,2,3,4,5

A(1)= 10 -5 ,A(2) = 10 -4 , A(3) = 10 -3 , A(4) = 10 -2 , A(5) = 10 -1

B(u) =

log 80 ~ 3.4012, u = 1,2,...,z

.

.

.

.

Large Multi-State Exponential Systems

236 Results

system reliability function R30 (t,-) ~ .~i'3 ((t - bn(u))/an(U),') = [1,1 -exp[-exp[-0.00001t + 3.4012]], 1 - exp[-exp[-0.0001t + 3.4012]], 1 - exp[-exp[-0.001t + 3.4012]], 1 - exp[-exp[-0.01t + 3.4012]], 1 - exp[-exp[-0. It + 3.4012]]] for t ~ (-oo, oo) system mean lifetimes in state subsets M(1) --- 397840, M(2) ~ 39784, M(3) ~ 3978.4, M(4) ~ 397.84, M(5) ~ 39.784, standard deviations of system lifetimes in state subsets o(1) =_ 128190, o(2)= 12819, o(3)~ 1281.9, o(4)= 128.19, o(5) = 12.819, system mean lifetimes in particular states M(1) ___-358056, M(2) ~ 35806, M(3) ~ 3581, M(4) ~ 358, .....

M(5) --__40 system risk function r(t) =_exp[-exp[--O.0001t + 3.4012]], exceeding moment of admissible system risk level 6 r = [3.4012 - log(- log 0.05)] / 0.0001 ~ 23040 Example 7.3. (a piping system) Let us consider a piping system composed of k, = 3 pipelines linked in parallel, each of them composed of In = 100 six-state pipe segments of two types linked in series. Two of the pipelines consist of 40 pipe segments that have exponential reliability functions with transition rates between state subsets

;ql(U) = (2.5) "-6 , u = 1,2,3,4,5, and 60 pipe segments that have exponential reliability functions with transition rates between state subsets 212(u) = 2 u-6, u = 1,2,3,4,5, The third pipeline consists of 50 pipe segments that have exponential reliability functions with transition rates between state subsets 221(u) = (1.9) u-6, u = 1,2,3,4,5, and 50 pipe segments that have exponential reliability functions with transition rates between state subsets ~22(u) = (2.1)u - 6 u = 1,2,3,4,5 ,

9

Chapter 7

237

Thus the considered piping system is a non-homogeneous regular six-state seriesparallel system with parameters kn = k = 3, In = 100, a = 2, ql = 2/3, q2

=

1/3.

q~ = 2,pll = 0.4,p12 =0.6, e2 = 2, P21 = 0.5, P22 = 0.5. We will perform its reliability evaluation according to the procedure given in Table 7.4 (Case 2). This procedure is presented in Table 7.9.

Table 7.9. Reliability evaluation of the piping system System type Data

A non-homogeneous regular series-parallel system system shape k n ~ k, I. ~ oo

number of series subsystems linked in parallel k.--3 number of components in series subsystems 1. = 100 number of subsystem types a=2 fractions of different subsystem types ql = 2/3, q2 = 1/3 numbers of component types in series subsystems e l = 2 , e2 = 2 fractions of different component types in series subsystems Pll = 0.4, Pl2 = 0.6 P21 = 0.5, P22 = 0.5 number of component and system states z=5 component transition rates between state subsets 211 (u) = (2"5) u-6 ' ~12 (U) ---- 2 u-6 Azl(U ) = (1.9) u-6, 212 (u)= (2,1) u-6, u = 1,2,3,4,5 system reliability critical state r=2 admissible level of system risk 6=0.05

Large Multi-State Exponential Systems

238

Calculations

...... c a s e 2 , At(U) 22(u) A1(u) A2(u)

Results

= = = =

kn ~k

l i n ...~oo

0.4"(2.5) u-6 + 0.6"2 u-6 0.5"(1.9) u-6 + 0.5"(2.1) u-6 40"(2.5) u-6 + 60"2 u-6 50"(1.9)~-6 + 50"(2.1) ~-6, u = 1,2,3,4,5

Case2. k,, ~ k , l n ~ oo system reliability function R'3,1o o (t,.) = fli" 9 ( ( t - bn(u))/an(U),') = [ 1,1 - [ 1 1 - [1 1 - [1 1 - [1 1 - [1 -

exp[-2.285t]]2[ 1 - exp[-3.244t]], exp[-4.774t]]2[ 1 - exp[-6.408t]], exp[-10.060t]]2[1 - exp[-19.096t]], exp[-21.400t]]2[ 1 - exp[-25.188t]], exp[--46.000t]]2[1 - exp[-50.125t]]]

for t>_ 0 system m e a n lifetimes in state subsets oo

M(u) =

f[1 - [1 - exp[-,4t

(u)t]] 2 [1 - exp[-A2

(u)]]]dt

0

= 3~[2At(u)] + 1/A2(u) + 1/[2Al(u) + A2(u)] - 2/[Al(u) + A2(u)], i.e. M(1) = 0.74, M(2) = 0.35, M(3) = 0.16, M(4) = 0.05, M(5) = 0.04 standard deviations of system lifetimes in state subsets o(u) = ~ ] N ( u ) - M 2 (u) (*) oO

N(u) = 2 ~ t [ 1 - [ 1 - e x p [ - A 1(u)t]]2 [1- e x p [ - A 2 (u)t]]]dt, 0

i.e. o(1)=0.43, 0(2)=0.20, o(3)=0.10, 0(4)=0.05, 0(5)=0.02, system m e a n lifetimes in particular states M(1) = 0.39, M ( 2 ) = 0.19, M ( 3 ) = 0.11, M ( 4 ) = 0.01, M ( 5 ) = 0.04 system risk function r(t) = [1 - exp[-4.774t]] 2[1 - exp[-6.408t]] exceeding m o m e n t of admissible system risk level r = r-t(6) = 0.09 years

(,) tr = ~ 5 / [ 4 A 1]2 ] + 1/[A 2 ]2 + 1/[2A1 + A2 ] 2 + 4/[(A 1 + A 2)(2A 1 + A 2)] - 8/[A 1 + A 2 ]2 The behaviour of the system multi-state reliability function and risk function is illustrated in Table 7.10 and Figure 7.2.

239

Chapter 7

Table 7.10. The values of the piping system multi-state reliability function and its risk function

"~'9 ( ( t - b n t 0.00 0.05

|

.

u=l 1.0000 0.9983

m

.

m

.

m

0.10 0.20

i

,

,

m

l

u=2 1.0000 0.9876

m

,

0.9884 0.9358

m

0.25

u =4

u=3 1.0000 0.9039

m

.

0.9318 0.7267 9

0.8948

i

_

. . . . . .

.

|

'0.30'0.8468'0.5053'0.0983 ' 0 . 3 5 ' 0.7943 ........().4108 "-0.40" 0.7391 " 0.3303 ' 0.45 0.6832 0.2634 0.50 0.6279 0.2088 , 0.55 , 0.5741 'j 0.1649 , 0.60 0.5228 0.1298 0.65 0.4743 0.1020 0.70 0.4289 0.0800 .

m

m

I~

m

|

m

m

m

.

,

.........

.

.

0.0594 0.0359

m

,

l

0.0217 0.0131 0.0079 0.0048 0.0029 0.0017

.

9

.

.

.

.

i

0.0265 0.0002

.

1

0.1623

.

.

0.2565 |

i

0.6123

.

.

9

0.0338

m

.

1.0000

|

0.6909 0.2842

.

l

, .

m

.

r(t) 0.0000 0.0124

i

1.0000

0.6572 0.2660

:

u =5 J

m

m

-

,,

(u)) / an (u),u)

0.0113 0.0038 0.0013 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000

_

..

.

9

, .

.

9

.

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

|

|

0.0682 0.2733

__

1

m

i

i

0.3877 0.4947 0.5892 0.6697 0.7366 0.7912 0.8351 0.8702 0.8980 0.9200

m_

_

m

....

|

|

|

|

,,,

|

.

_

m

|

|

|

|

|

.

-~'9 I'-b"':"> ~, an(u) ,U)

u=0

1.0

0.8

0.6

u=l

0.4

u=2 0.2

u=3 4 0.0 ~-. 0.0

.

. . 0.1 !

. 0.2 u'

i

0.3

|

'

0.4

,

0.5

0.6

,

0.7

......, 0.8

,

0.9

r

t

Fig. 7.2. The graphs of the multi-state reliability function of the piping system and its risk function

Large Multi-State Exponential Systems

240

Example 7.4. (a bus transportation system) A bus transportation company has k, = 100 transportation lines. In each of the lines passengers have at their disposal 1, = 3 buses on which they may travel. Buses have sixstate reliability functions, i.e. z = 5. Forty of the lines in the considered system have two buses that have exponential reliability functions with transition rates between state subsets ~ll(U)

=

(2.5)(2u-12)/3, u - 1,2,3,4,5,

and one bus that has exponential reliability functions with transition rates between state subsets ,~,12(U) ._

2(2u- 12)/3

1,2,3,4,5

U =

The remaining 60 lines have two buses that have exponential reliability functions with transition rates between state subsets u = 1,2,3,4,5,

/~21(U ) = ( 1 . 9 ) ( 2 u - 1 2 ) / 3 ,

and one bus that has exponential reliability fimctions with transition rates between state subsets )],22(U) -"

(2.1)(2u -

12)/3, u :--

1,2,3,4,5.

The bus transportation system is able to perform its transportation tasks if at least one bus on each of the lines is not failed. Thus, according to Definition 3.21, it is a multistate non-homogeneous regular parallel-series system with parameters k, = n = 100, In = 1 = 3, a = 2, ql = 0.4, q2 = 0.6, el = 2, Pll = 2/3, PI2 = 1/3,

e2

2, P2!

=

2/3, P22 = 1/3.

=

Its reliability evaluation is performed in Table 7.11, according to the algorithm given in Table 7.5 (Case 1).

Table 7.11. Reliability evaluation of the bus transportation system System type

A non-homogeneous regular parallel-series system

Data

.

system shape: number of parallel subsystems linked in series kn =100 number of components in parallel subsystems /,=3 number of types of parallel subsystems a=2 .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Chapter 7

241

fractions of parallel subsystems of particular types ql = 0.4, q2 = 0.6 numbers of component types in parallel subsystems e l = 2, e2 = 2 fractions of components of particular types in parallel subsystems Pll = 2/3, Pt2 = 1/3 P2t = 2/3, P22 = 1/3 number of component and system states z=5 component transitions rates between state subsets 211(U) = (2.5) (2u- 12)/3,/~I2(U) = 2 (2u -12)/3 '~21(U) = (1.9) (2u- 12)/3, ~22(U) __ (2.1)(2u-12)/3, u - 1,2,3,4,5 a system critical state r=2 system risk admissible level 6 = 0.05 Calculations

Case 1. system shape: kn ---) oo, In -") 1 21(u) = (2.5) (2u- 12)2/9"2(2u- 12)1/9 22(u) = (1.9) (2u- 1 2 ) 2 / 9 . ( 2 . 1 ) ( 2 u - 12)1/9 A(u) = 40(2.5) (2u- 12)2/3.2(2u-12)/3+ 60(1.9)(2u- 12)2/3.(2 , 1) (2u- 12)/3

Results

Case 1. system shape: k, ---) oo, l, ---) 1 system reliability function R 100,3 (t,.) ~ ~?'2 ((t-bn(u))/an(U),.) -- [ 1 ,exp[--0.079t 3],exp[-0.318t3],exp[-1.300t3], exp[-5.408t3],exp[-22.974t3]] for t ___0 mean system sojourn times in state subsets M(u) = F(4 / 3) [A (u)] -u3 , u = 1,2,...,5, i.e.

M(1) = 2.08, M(2) = 1.31, M(3) = 0.82, M(4) = 0.51, M(5) = 0.31 standard deviations of system sojourn times in state subsets o(u) =IF(5 / 3) - F 2 (4 / 3)]l/2[A(u)] -1/3 , u =1,2,...,5, i.e. o(1) = 0.57, o(2) = 0.23, 0(3) = 0.09, o(4) = 0.03, 0(5) = 0.01, mean system sojourn times in particular states M(1) = 0.77, M(2) = 0.49, M (3) = 0.31, M (4) = 0.20, M(5) = 0.31 system risk function r(t) = 1 - exp[-0.318t 3] exceeding moment of admissible system risk level 5 r = r-~(b") = [(,log(1 - 6))/0.318] v3= 0.5.4 years The behaviour of the approximate multi-state reliability function components and the risk function of the bus transportation system is presented in Table 7.12 and Figure 7.3.

Large Multi-State Exponential Systems

242

Table 7.12. Values of the multi-state reliability function and the risk function of the bus transportation system

~'2 ((t- b.(u))/a.(u),.) m

|

t 0.0 0.4 0.6 0.8 1.0 1.2 il.4 ~1.6 1.8 2.0 ,

i

,

'2.2 |

2.4

'2.6

u=l 1.0000 0.9950 0.9811 0.9604 0.9240 0.8723 0.8051 0.7236 0.6308 0.5315 0.4312 0.3355 0.2494

'

u=2 1.0000

iioooo

0.9799 0.9336 0.8497 0.7276 0.5772 0.4179 0.2718 0.1565 0.0786 0.0338 0.0123 0.0037

0.9222 0.7609 0.5233 0.2822 0.1124 0.0311 0.0056 0.0006 0.0000 0.0000 010000 0.0000

u=3

u=4 1.0000 0.7092 0.3136 0.0640 0.0047

u=5 1.0000 0.8317 0.2289 6.0069

o.ooo~

0.0000 0.0000 0.0000 Oib000 0.0000 0.0000 0.0000 0.0000

o.oooo

0.0000

o.oooo 0.0000

o.oooo 0.0000 0.0000 0.0000

....

r(t) ' o.ooo0

|

!

0,0191 0.0629 0.1428 0.2599 0.4056 0.5622 0.7086 "0.8272 0.9100 0.9594 0.9844 0.9950'

|

i

ff?'2 ((t- b.(u))/a.(u), u) 1.0

u =0

0.8

0.6

O4

Oo. Fig. 7.3. Graphs of the multi-state reliability function and the risk function of the bus transportation system

CHAPTER 8

RELATED AND OPEN PROBLEMS Domains of attraction for limit reliability functions of two-state systems are introduced. They are understood as the conditions, that the reliability functions of the particular components of the system have to satisfy in order that the system limit reliability function is one of the limit reliability functions from the previously fixed class for this system. Exemplary theorems concerned with domains of attraction for limit reliability functions of homogeneous series systems are presented and the application of one of them is illustrated. ,4 practically important problem of accuracy of the asymptotic approach to large systems reliability evaluation concerned with the speed of convergence of system reliability sequence is discussed. This problem is illustrated by analysing the speed of convergence of the homogeneous series-parallel system reliability sequences to its limit reliability function. Series-"m out of n " systems and "m out of n "-series systems are defined and exemplary theorems on their limit reliability functions are presented and applied to the reliability evaluation of an illumination system and a rope elevator. Hierarchical series-parallel and parallel-series systems of any order are defined, their reliability functions are determined and limit theorems on their reliability functions are applied to reliability evaluation of exemplary hierarchical systems of order two. Applications of the asymptotic approach in large series systems reliability improvement are also presented. The chapter is completed by showing the possibility of applying the asymptotic approach to the reliability analysis of large systems placed in their operation processes. In this scope, the asymptotic approach to reliability evaluation for a large port grain transportation system related to its operation process is performed.

8.1. Domains of attraction for system limit reliability functions The problem of domains of attraction for the limit reliability functions of two-state systems considered in this book is solved completely in [23], [71]-[72] and [80]-[81 ]. We will illustrate this problem partly for two-state series homogeneous systems only.

244

Related and Open Problems

From Theorem 4.1 given in Chapter 4 it follows that the class of limit reliability functiom for a homogeneous series system is composed of three functions, ~t',(t), i = 1,2,3, defined by (4.3)-(4.5). Now we will determine domains of attraction D~,

for these fixed functions, i.e. we will determine the conditions which the reliability functions R(t) of the particular components of the homogeneous series system have to satisfy in order that the system limit reliability function is one of the reliability functions 9i',(t), i = 1,2,3. P r o p o s i t i o n 8.1 If R(t) is a reliability function of the homogeneous seres system components, then

R(t)~ D ~ if and only if lim 1- R ( r ) = t a f o r t > O . r-,-oo 1 - R ( r t ) P r o p o s i t i o n 8.2 If R(t) is a reliability function of the homogeneous seres system components, then

R(t)~ D~2 if and only if (i) S y ~ (-o% oo) R(y) = 1 and R(y + 6) < 1 for e > 0, (ii) lim 1- R ( r t + y ) = t a f o r t > O. r-~O+ 1 - R ( r + y )

P r o p o s i t i o n 8.3 If R(t) is a reliability function of the homogeneous seres system components, then

R(t)~ D~3 if and only if lira n[1 - R ( a n t + b n )] = e t for t e (-oo, oo) tl--~aO

with

Chapter 8

245

b n = inf{t 9R(t +0) < 1 - 1 < R(t-O)}, n

a n = i n f { t ' R ( t ( 1 + O ) + b n ) < 1- e

< R(t(1-O)+b.)}. n

Example 8.1 If components of the homogeneous series system have reliability functions

I

ll,

R(t) =

-t,

t 1,

then R(t) e D~2 .

Motivation: Since (i) R(0) = 1 and R(6) < 1 for each 6 > 0,

(ii) lira 1 - R(rt + y) = lim 1 - (1- rt) =t for t > 0 , r-~o+ 1 - R ( r + y) r-~o+ 1 - (1 - r) then by Proposition 8.2, R(t) ~ D~2, where

-{1, t O, a > O,/3 > O, and

k n ~ k, l n = n, m = constant (m / n ~ 0 as n ~ oo),

bn glogn'

bn = [ log ,n,],,• fl '

then m

[.0/3(~ (t)] k = [ 1 - e x p [ - e x p [ - t ] ]

m-I

"

~] exp[-tt]]k, i=o i!

t ~ (-oo, oo),

is its limit reliability function. Motivation: W e will use L e m m a 4.9 with unmodified condition (4.30) in the form

V(t) = lim n[R(ant + bn) ] for t e C z . n .-9, oo

Since for sufficiently large n, we have 1

ant+bn=

lo~ n [- "fl ] a [ l +

t alogn

]>0 for t~(-~o, oo),

(8.5)

Chapter 8

261

then

R(a.t + b. ) = exp[-fl(a.t + b n )a ] = exp[- log n[1 +

t ]a ] cr log n

= exp[- log n - t - o(1)] for t ~ (-oo, oo). Hence

V(t) = lim n[R(ant + bn) ] n-.~oo

= lim n exp[- log n - t - o(1)] n--~oo

= lim exp[-t - o(1)] = exp[-t] for t ~ (-o%oo). n--~oo

Thus, from Lemma 4.9, [~--3(0)(t)] k given by (8.5) is the system limit reliability function. U

Example 8.4 Let us consider the ship-rope elevator used to dock and undock ships coming in to shipyards for repairs. The elevator is composed of a steel platform-carriage placed in its syncline (hutch). The platform is moved vertically with 10 rope hoisting winches fed by separate electric motors. During ship docking the platform, with the ship settled in special supporting carriages on the platform, is raised to the wharf level (upper position). During undocking, the operation is reversed. While the ship is moving into or out of the syncline and while stopped in the upper position the platform is held on hooks and the loads in the ropes are relieved. In our further analysis we will discuss the reliability of the rope system only. The system under consideration is in order if all its ropes do not fail. Thus we may assume that it is a series system composed of 10 components (ropes). Each of the ropes is composed of 22 strands. Thus, considering the strands as basic components of the system and assuming that each of the ropes is not failed if at least m = 5 of its strands are not failed, according to Definitions 8.4-8.5, we conclude that the rope elevator is the two-state homogeneous and regular ,,5 out of 22"series system. It is composed of kn = 10 series-linked "5 out of 22" subsystems (ropes) with In = 22 components (strands). Assuming additionally that strands have Weibull reliability functions with parameters a = 2, fl = 0.05, i.e.

R(t) = exp[-0.05t / ] for t >__0, from (8.4), we conclude that the elevator reliability function is given by

Related and Open Problems

262 lO,22(t) = [1 - 2

)exp[-i0.05t ][1 - exp[-0.05t 21122-i ]1o for t >_0.

i=0

Next, applying Proposition 8.6 with 7.8626 = 1.2718, b n = [ log22, 89~ 7.8626, 2 log 22 0.05 I and (1.1) we get the following approximate formula for the elevator reliability function [.W--3(~

(5) (t)= 10,22

l~

4 exp[-0.7863it +6.1821i]]10 = [1 - exp[- exp[-0.7863t + 6.1821]] Z

i=o

i!

for t ~ (-oo, oo),

8.5. Reliability of large hierarchical systems Prior to def'ming the hierarchical systems of any order we once again consider a seriesparallel system presented in Figure 8.9. This system here is called a series-parallel system of order 1.

.....

I .... I

Ii

EII~

]....

Ell

El2

E21

E22

E2l2

Eknl

Ek, 2

kntknl

Fig. 8.9. The scheme of a series-parallel system of order I

It is made up of components

Chapter 8

263

Ell A, i I = 1,2 ..... kn, Jl = 1,2,...,1i I , with the lifetimes respectively

Til•, il = 1,2,'",kn, Jl = 1,2,"',li l 9 Its lifetime is given by

T= max { min {Ti,A } }.

(8.6)

1_ 0.

Next applying Proposition 8.7 with normalising constants

an~

1 1 1 1 =11.1, bn = ~.-~ (-~ + ~) log 200 = 235.5, 0.01.9

we conclude that the system limit reliability function is given by 9?3(0 = 1 - exp[-exp[-t]] for t ~ (-oo, oo), and from (1.1), the following approximate formula is valid

Chapter 8

269

R 2,200,3(t) ~ ~3 (0.09t - 21.2) = 1 - exp[- exp[-0.09t + 21.2]] for t ~ (-oo, oo).

The accuracy of this approximation is illustrated in Table 8.4 and Figure 8.11.

Table 8.4. Values of exact and approximate reliability functions of a hierarchical regular series-parallel homogeneous system of order 2 t

.~i'3(0.09t - 21.2)

R 2,200,3 (t)

LI = R 2,200,3( t ) - gi'3 (0.09t - 21.2)

i

200 210 ....220 230 240 250 260 270 280 290 300 310 .....320 330 340 .

,

,,

.

0.999996 0.997'281 0.934547 0.705338 0.414313 0.205450 0.092891 0.040069 0.016873 0.007014 0.002895 0.001189 0.000487 0.000199 0'000081 .

i

l

....

.

' .

i.oooo06 ' 0.999953 61982668 0.807704 .... 0.488455 i 0.238551 ' 0.104885 . . . . . . 61044050 " 01018149 ' 0.007419 0.003023 ..... 0.001230 0.000500 ' 0.000203 0.000083 1

. . . . .

-o.ooooo;i -0.002672 -0.048121 -01102366 -0.074142 -0.033101 -0.011994 -0.003981 -0.001276 -0.000405 --0.000128 -0.000041 -0.000013 -0.000116 -0.000002

R 2,2,,,,,3 ( t ) . . . . . . . . . . . . .

..............................................................................................................................................................

exact reliability function approxlmate form

200

250

300

t ,._

350

Fig. 8.11. Graphs of exact and approximate reliability functions of a hierarchical regular series-parallel homogeneous system of order 2

Related and Open Problems

270 Definition 8.10

A two-state system is called a parallel-series system of order r if its lifetime T is given by T = min { max {

mi

,{

ma~.

.[

"

max

1 0 is its limit reliability function. Motivation: Since for sufficiently large n, we have 1 a n t + b n = 2,

1

t

1 < 0 for t < O

~+...+--

In

Ir

kn

and 1 ant + bn = -A

t 1 1 --+...+w In Ir

>- 0 f o r t >-O ,

k.

then F ( a n t + b n ) = O for t < O

and t

F(ant + b,,)= 1- cxp[-

~ ] for t >_0,

~+...+~

In

kn

Ir

(8.10)

Chapter 8

273

and further

V(t)-

= lira ktnn'-t+'''+lFg (ant+bn) = 0 for t < 0 n--~r

and

V'(t) = lira k/r-' +...+1FIT, (ant + bn ) n --> oo

t

= lim k~7'-'+'''+1 [1- exp[n.-),oo

1

r

1 ]]t.

~+...+~

t, kn

t~

t t~'

l r-I

1

)] = = lim k,~ +...+l [ ~r - I + o ( l r~- I n--~oo "'nkln +...+1 kn n +...+1

t lr

for t > O.

m

Hence from Theorem 8.2, gi' 2 (t) given by (8.10) is the system limit reliability function. D

Example 8.6 We consider a hierarchical regular parallel-series homogeneous system of order r = 2 such that k n = 200, I n = 3, whose components have identical exponential reliability functions with the failure rate 2 = 0.01. Its exact reliability function, according to Corollary 8.4, is given by R2,200, 3 (t) = 1 for t < 0

and R2,200,3 (t) =[1 -[1 - [ 1-[1-exp[-0.01t]]3] 2~176 ]3]200 for t > 0. Next applying Proposition 8.8 with normalising constants

1 1 a n = ~" = 9.4912, b n = 0, 0.01 2001/3+1/9 we conclude that ~t--2(t) = 1 for t < 0 and 9i'-2(t) = exp[-t 9 ] for t > 0

274

Related and Open Problems

is the system limit reliability function, and from (1.1), the following approximate formula is valid R2,2oo,3 (t) -=- -~-'2(0.1054t) = exp[-(0.1054t) 9 ] for t > 0. The accuracy of this approximation is illustrated in Table 8.5 and Figure 8.12. Table 8.5. Values of exact and approximate reliability functions of a hierarchical regular parallel-series homogeneous system of order 2 t

R-"2,2oo,3(t)

.

.

.

.

0 2

.

.

4

.

6 8 10 12 14 16 .

.

.

.

.

.

.

.

.

.

.

.

.

A =R2,2oo,3(t)-,9/2(0.1054t)

.

1.000000 0.999999 ..... 0.999656 0.988445 0.876948 0.451036 0.040806 0.000070 0.000000 .

.

.

.9i~2(0.1054t)

.

.

.

.

.

.

.

.

1.000000 0.999999 0.999579 0.983952 0.806167 0.200822 0.000253 0.000000 0.000000 .

.

.

.

.

.

.

.

.

.

.

.

.

0.000000 0.000000 0.000077 0.004493 0.070781 0.250214 0.040553 0.000070 0.000000

.

.

.

.

exact reliability function 9. approximate form t ,.~

O

.

0

5

10

.

.

.

.

.

.

.

.

.

.

15

.

.

.

.

.

.

.

.

.

20

Fig. 8.12. Graphs of exact and approximate reliability functions of a hierarchical regular parallel-series homogeneous system of order 2

275

Chapter 8

8.6. A s y m p t o t i c a p p r o a c h to systems reliability improvement We first consider the homogeneous series system illustrated in Figure 8.13.

E21

Ell

Fig.

8.13.

The

scheme

of a series

system

It is composed of n components Ell, i = 1,2,..., n, having lifetimes T,a, i = 1,2,...,n, and exponential reliability functions R(t) = 1 for t < 0, R(t) = exp[-At] for t > 0, A > 0.

Its lifetime and its reliability function respectively are given by

T = rain{T,1}, l 0. Case 4. A mixed reservation of the system components 1

H

'

,

,,

,

- ' -

:

Fig. 8.16. The scheme of a series system with components having mixed reservation

Chapter 8

277

2

T=min{min{Y'.T/i

l 0 and

TCS)=E[T]=

3 . 22n

Case 6. If

1 an = -'~n , bn = O,

then .~6)(t) = 1 for t < 0 and .~6)(t) = [1 + t]exp[-t]

for t > O,

is the limit reliability function of the homogeneous exponential series system with cold reservation, i.e.

R On6)(t) = .0i'c6) (Ant) = [1 + Ant]exp[-Ant] for t _>0 and

T(6)= E[T]

=

2

,In

Motivation: We will prove the proposition parts concerned with the considered system limit reliability functions only. The approximate formulae for the system reliability functions follow directly from (1.1) and the expressions for the system lifetimes are easy to obtain by direct integration of the system approximate reliability functions.

Chapter 8

281

Case 1. Since t

t

a n t + b n = , ~ p n < O for t < 0 and ant+b n = A p n > 0

for t > 0 ,

then

F(ant + b n ) = 0 for t < 0 and F(ant + b n ) = 1 - exp[ - t ] for t > 0. n Hence V (t) = lim nF(ant + b n ) = 0 for t < 0 and V(t) = lim nF(ant + bn) = lira nil - e x p [ - t ] ] n ---~

n--)oo

= lim n[ t-- + n~oo n

o(1)] n

/~

= t for t > 0,

which from Lemma 4.1 completes the proof in this case. Case 2. Since

t ant +bn = A ~

t

< 0 for t < O and a n t + b n = ~ - - ~ > O A4n

for t>O,

then

F(ant+bn)=O

for t < 0

andF(ant+bn)=l-exp[

Hence V(t) =

lim n[F(ant + bn)] 2 = 0 for t < 0 n--~

and

V(t) = n-,~limn[F(ant + b n )]2 = n-,,olimn[1 - exp[- ~ n ]]2

- t--L] ~ n for t>_O.

Related and Open Problems

282 t 2

= lim n [ - - + o(1)] = t 2 for t _>O, n--~oo

/,/

r/

which from Lemma 4.18 completes the proof in this case. Case 3. Since

R(t) = 1 for t < 0 and R(t) = [1 + At]exp[-At] for t >_0 and

ant+b n

, ~ , ~ < 0 for t0 for t>O,

then

F(ant+b,)=O

f o r t O.

Hence B

g (t) = lim nF(ant + b n ) = 0 for t < 0 1'/-9,oo

and

V(t) = lim nF(ant +bn) = limn[1-[1 + n--~oo

= limn[2-t 2 - 1 . n--~oo

]exp[-

]]

n---)oo

rl

2

2-t2 + o ( 1 ) ] = t 2 for t > 0 , n

n

which from Lemma 4.1 completes the proof in this case. Case 4. We will motivate this case using the expressions for the system reliability sequence. Since R (4) (t) = 1 for t < 0 and R (4) (t) = [1 + ,~t] m r n n

- e x p [ - A t ] ] n-m for t > 0

and

ant + bn = -2t I 2"n2-" m _O, then

Chapter 8 lim

R (4) tl

(ant +

283

b. ) = 1 for t < 0

tl---}oO

and lim R (n4)(ant + bn ) tl .--} oO

= lim[1 + no,o

-m

-t] m exp[-n

= lim exp[m log[1 + .-+oo

2n'm

+ (n - m) log[1 + [1 - exp[-

= lim exp[m ~ 2 n 2 m n~oo

+(n-m)[1-exp[-~]

t

I

2n

2,_m t] [ 2 -

t]- n

n-m

exp[-

I

.... t]] n-m

2n-m

.....t

2 ..........t]]] 2n - m

~ mt

2n - m

2" 2 + 0(1) - n 2n - m t

2. . ,,t]] . . . (n. . m) . .l ~' 2 ' 2 2n - m -~[1 exp[ . . . . .2. n. .-.m. .t ] ] + o(1)]

= 2limn -e x~p [ om ~oi n - m t - - - - m - - - t 2 -n.12~i 2 2n-m n-m

+(n-m) =

2 re.t- n - m t2 - n - m t2 + o(1)] :~n - m 2n - m 2n - m

lim exp[-t 2 + 0(1)] = exp[-t 2 ] for t >_.O, n.-~oo

which completes the proof in this case. Case 5. Since t ant+b n=---O,

then R(ant+bn)=l

fort_O. n

Related and Open Problems

284 Hence

V(t) = l i m [ R ( a . t + b.)] n = 1 for t < 0

and V(t) = lim[R(ant + b n)]n = l i m [ e x p [ _ t ] ] . n--~oo

n---~oo

= exp[-t] for t > O,

n

which from Lemma 4.13 completes the proof in this case. Case 6. We will motivate this case using the expressions for the system reliability sequence. Since R ~6)(t) = 1 for t < 0 and R (6) (t) = [1 + nat] exp[-n2t] for t > 0. and

ant + bn = - -t' - < 0 An

for t O An

for t>O,

then

limR(n6)(ant+bn)=l

for t 0, is the limit reliability function of the system, i.e.

288

Related and Open Problems

R(n5) (t) = .gi~s)(0.397t) = 1 - [ 1 - exp[-0.397t]] 2 for t > 0 and 3 _=_3.78 h. T (5) = E[T] = 2.0.0049.81 Comparing the system reliability functions for considered cases of improvement, from Corollary 8.5, results in the following values of the factor p : p = p(t) = 0.0049t for i = 2,

p = 0.0340t for i = 4, p = p(t) = 1 - log[2- exp[-O.397t]] for i = 5,

while comparison of the system lifetimes results respectively in: p=0.1254

for i = 2 ,

p=0.1043

for i =4,

p = 0.6667 for i = 5. Methods of system reliability improvement presented here supply practitioners with simple mathematical tools, which can be used in everyday practice ([83]). The methods may be useful not only in the operation processes of real technical objects but also in designing new operation processes and especially in optimising these processes. Only the case of series systems made up of components having exponential reliability functions with a single reservations of their components and subsystems is considered. It seems to be possible to extend these results to systems that have more complicated reliability structures, and made up of components with different from the exponential reliability functions.

8.7. Reliability of large systems in their operation processes This section proposes an approach to the solution of the practically very important problem of linking systems' reliability and their operation processes. To connect the interactions between the systems' operation processes and their reliability structures that are changing in time a semi-markov model ([37], [92]) of the system operation processes is applied. This approach gives a tool that is practically important and not

Chapter 8

289

difficult for everyday use for evaluaffng reliability of systems with changing reliability structures during their operation processes. Application of the proposed methods is illustrated here in the reliability evaluation of the port grain transportation system. We assume that the system during its operation process is performing a repertory of tasks. Namely, the system at each moment t, t ~ < 0, 0 >, where 0 is its operation time, is performing at most w tasks. We denote the process of change of the system task repertory by Z(t) = [ Z l ( t ) , Z 2 ( t ) , . . . , Zw(t)],

where = fl, if the system is executing the jth task at the moment t Zj (t)

[ 0 , / f the system is not executing the jth task at the moment t, j = 1,2.... , w.

Thus Z(t) is the process with continuous time t, t ~< 0, 0 >, and discrete states from the set of states {0,1}w. We number the states of the process Z(t) assuming that it has v different states from the set Z={z

I ~Z 2 ~...~Zv }

and they are of the form z k = [z~, z2k ,...Zkw], k = 1,2,..., v,

where Zj

k

6

{0,1}, j

=

1,2,... ,W.

If the process of change of the system task repertory Z(t) is semi-markov ([37], [92]) with its conditional sojourn time 0 kt at the state z k when its next state is z t, k, 1 = 1,2,..., v, k ~ 1, then it may be described by: - the vector of probabilities of the initial states

[pk (o)]~•

=

[p~ (o), p 2 (o), ..., p ~ (o)],

9where pk (0)= P(Z(O)= z ~ ) for k = 1,2,...,v,

- the matrix of probabilities of its transitions between states

290

Related and Open Problems

Ipll pl2 ... pW [pkl ]vxv

p21 p22.."'.P2V. =

L'p ~l p V2

p VV

where p kk = 0 for k = 1,2,..., v, - the matrix of conditional distribution functions of the sojourn times 0 kt I H I I (t) H 12(t)... H iv (t) ]

!t,

[H kl (t)]vx v =

H

vl

(t) H

v2

I

(t)... H vv (t)J

where H kt (t) = P(O kt < t) for k, 1 = 1,2,..., v, k r 1,

and H~(t)=0

for k =l,2,...,v.

Then, the sojourn time t9 kt mean values are given by co

E[okl] = ~tdHkt(t), k,l=l,2,...,v, k r o

(8.11)

The unconditional distribution functions of the sojourn times O k of the process Z(t) at the states z k, k = 1,2,..., v, are given by v p ktHk t (t), k = 1,2,..., v. H k (t) = ]~ l=l

The mean values E[ 0 k ] of the unconditional sojourn times O k are given by

(8.12)

291

Chapter 8 E[O k] = ~ p kt E[0 kt ], k = 1,2,..., v,

(8.13)

l=1 where E[ 0 kt ] are defined by (8. I 1).

Limit values of the transient probabilities at the states p k (t) = P(Z(t) = z k ) are given by zc ~ E[O k ]

pk = l i m p k ( t )

=

, k=l,2,...,v,

(8.14)

1=1 where the probabilities x k of the vector [z k ]lxv satisfy the system of equations

[:k ] = [~.k ][p

f

kl ]

l~__l~t =1.

In the case when the sojourn times 0 kt , k, 1 = 1,2,...,v,

k ~: 1, have exponential

distributions with the transition rates between the states y kt, i.e. if H k t ( t ) = P ( O kt < t ) = 1 - e x p [ - y k t t ] ,

t>O,

(8.15)

for k, l = 1,2,..., v, k ~ 1, then their mean values are determined by

~-,

k , l = 1,2,... v, k ~ l,

E[Okt ] = 1

(8.16)

and the probabilities of transitions between the states are given by kl

pkl =

Y , k,l=l,2,...,v, y'y~Y

k r l.

(8.17)

j~k

The unconditional distribution functions of the process Z ( t ) sojourn times 0 k at the states z k , k = 1,2,...,v, according to (8.12) and (8.15) are given by

v

H k (t) ffi 1 - ~ p l=l

k/

exp[-y k/t], t > 0, k = 1,2,..., v,

(8.18)

292

Related and Open Problems

and their mean values, from (8.13) and (8.16), are M k =E[0k] =~pkt

1 , k=l,2,...,v.

l=l

(8.19)

-'~

Limit values of the transient probabilities p k (t) at the states z k , according to (8.14), are given by n: k . M k pk = lim Pk (t) t-,,|

=

, k = 1,2,...,v,

(8.20)

~[z' .M t ] 1=1

where the probabilities ~rk of the vector [rr k ]lxv satisfy the system of equations

I[,rk ] v

=

[,rk ][pkl ]

1

with [p kt ] and M k given by (8.17) and (8.19) respectively. As an example we will analyse here the reliability of the port grain transportation system in its operation process. This system, described in Chapter 6, is composed of four non-homogeneous series-parallel subsystems. Therefore we will need the following modifications of Proposition 7.4 (Case 2) reduced to two-state systems.

Proposition 8.9 If components of the non-homogeneous regular two-state series-parallel system have exponential reliability functions R(iJ)(t) = 1 for t < 0, R~

= exp[-A0.t] for t > 0, i = 1,2 .... ,a,j = 1,2,...,e;,

and kn---> k, l,, --->oo, an -

1 , bn = 0, ,,2,1n

where ei

2~ = E PUAU, 2 = min{2 i}, j-I

l t), t ~< O, oo),

where T is the unconditional lifetime of the system, is given by R(t) = ~pk(t)R(k)(t),

t~ O, i = 1,2,...,a,j = 1,2,...,e,

(8.24)

according to (2.14) and (2.15), we get a

k

R kn Ck),ln (t) = 1 - l-I[1-exp[-lnPo.Ao.t]]q'kn i=l

~

t --> 0.

(8.25)

In this case the mean value and the variance of the regular non-homogeneous seriesparallel system lifetime are v

k

k

m = E P E[T ],

(8.26)

k=l

and

o.2 = ~ p k 0.2 [T k ],

(8.27)

k=l

where for k = 1,2,..., v, E[T k]

-~

~ R (k) (t)dt, t>O, k n ~ ?l

(8.28)

0

and oo

0 .2 [T k ] = 2~ t RCk) kn,ln (t)dt - [E[T k ]]2 , t ~ O, 0

(8.29)

Chapter 8

295

and pk are given either by (8.14) or by (8.20). The grain elevator described in Chapter 6 is composed of the following transportation subsystems: S l -horizontal conveyors of the first type, S 2 - vertical bucket elevators, S 3 - horizontal conveyors of the second type, $4 - worm conveyors. To illustrate the problem we simplify our considerations by assuming that all elevator components have exponential distributions of their lifetimes. Subsystem S l is composed of two identical belt conveyors. In each conveyor there is one belt with a reliability function

R(~'~)(t) = exp[-0.0125t]

for t > 0,

two drums with reliability functions R~

1) = exp[-0.0015t] for t >_0,

117 channelled rollers with reliability functions RO'3)(t, 1) = exp[-0.005t] for t > 0, and nine supporting rollers with reliability functions R(~'4)(t,1) = exp[-0.004t] for t > 0. Subsystem $2 is composed of three identical bucket elevators. In each elevator there is one belt with a reliability function

R(l'~)(t) = exp[-0.025t]

for t > 0,

two drums with reliability functions R(l'2)(t) = exp[-0.0015t] for t > 0, 740 buckets with reliability functions R(t'3)(t) = exp[-0.03t] for t > 0. Subsystem S 3 is composed of two identical belt conveyors. In each conveyor there is one belt with a reliability function R(~'~)(t) = exp[-0.0125t] for t > 0,

Related and Open Problems

296

two drums with reliability functions R(l'2)(t) = exp[-0.0015t],

for t _>0,

117 channelled rollers with reliability functions R(l'3)(t) = exp[-0.005t] for t > 0, and 19 supporting rollers with reliability functions R(l'4)(t) = exp[-0.004t] for t > 0. Subsystem S 4 is composed of three chain conveyors, each composed of a wheel

driving the belt, a reversible wheel and 160, 160 and 240 links respectively. The subsystem consists of three conveyors. Two of them have 162 components and the remaining one has 242 components. Thus it is a non-homogeneous non-regular multistate series-parallel system. In order to make it a regular system we conventionally complete two first conveyors that have 162 components with 80 components that do not fail. After this supplement the subsystem consists of kn = 3 conveyors, each composed of In = 242 components. In two of them there are two driving wheels with reliability functions R(l'l)(t) = exp[-0.005t]

for t > 0,

160 links with reliability functions R(l'2)(t) = exp[---0.012t] for t > 0, and 80 components with "reliability functions" R~

= exp[-At ] for t > 0, where 2 = 0.

The third conveyor is composed of two driving wheels with reliability functions R(2'l)(t) = exp[-0.022t] for t __.0, and 240 links with reliability functions R(2'2)(t) = exp[-0.034t] for t _>0. Taking into account the operation process of the considered transportation system we distinguish the following as its three tasks: task 1 - the system operation with the largest efficiency when all components of the subsystems S 1, S 2, S 3 and S 4 are used,

Chapter 8

297

task 2 - the system operation with less efficiency system when the first conveyor of subsystem Si, the first and second elevators of subsystem $2, the first conveyor of subsystem S 3 and the first and second conveyors of subsystem $4 are used, task 3 - the system operation with least efficiency when only the first conveyor of subsystem S l, the first elevator of subsystem S 2, the first conveyor of subsystem S 3 and the first conveyor of subsystem S 4 are used. Since the system tasks are disjoint then its operation states belong to the set Z = { z I ,z 2 ,z 3 },

where z' =[1,0,0], z 2 =[0,1,0], z 3 =[0,0,1]. Moreover, we arbitrarily assume the following matrix of the conditional distribution functions of the system sojourn times 0 kt , k, l = 1,2,3, 0

[H kl (t)] = 1 -

e-40t

- e -lOt

1 - e -St

1 - e -lOt

0

1 - e -50t

1 - e -20t

.

0

Hence, from (8.17), the probabilities of transitions between the states are given by 1 0

~

[p,t] =

2 ~

3

3

405 9

9 1

2

"-

~

3

0

3

and further, according to (8.18), the unconditional distribution functions of the process Z ( t ) sojourn times O k in the states z k , k = 1,2,3, are given by

H 1(t) = 1 - 31 exp[-5t] - -~ 2 exp[-10t] H 2 (t) = 1 - 4 e x p [ - 4 0 t ] - ~5 exp[-5Ot] H 3 (t) = 1 - -~ 1 e x p [ - 1 0 t ] - -~ 2 exp[-2Otl

298

Related and Open Problems

and their mean values, from (8.19), are

M

11

21 6 3 10 45 51 1 + ____ 9 40 9 50 45 1 1 21 3 310 3 2 0 45

E'8~ ] 3 5 41 2 = E[82] = _ _ _

M1

M3

[

[

]

E'O3j

Since from the system of equations

o_1 2 3

3

40_5

[7/.1, ~2, ~3 ] ._ [y/.l ~7/.2, ~3

9

9

! .2 0 3

3

7/.1 + ~ 2 +7/.3 =1 we get ~rI

17 61

z

21 61

x3

23 61'

then the limit values of the transient probabilities pk (t) at the operational states z k , according to (8.20), are given by pl

34 =-~'

2 7 3 =~.23 P =6-4' P 64

(8.30)

The structure of the port grain transportation system at operation state 1 is given in Figure 8.19.

Chapter 8

STORAGE ,

299

MAINDISTRIBUTIONSTATION 9thfloor BALANCE 6:h~oo:

//i | |

[ FLAPS I [ 4th floor //I , ',t, 4, § IS2l I II , H~21

H,,

//

I=.oorlll }I~H 2 1 ~ --Ii6211 i ~-'][-'t[ It 1 H2 ] "-'4 ~.[16211 I 'll'J [i, H=I q=?ll

....

1[ : ~ : ]1

~ i39[ [

=11129

S' J129 I

BASE~NTI

I ~ILWAVT~CK I

Fig. 8.19. The scheme of the grain transportation system structure at operation state I At system operational state 1, subsystem S~ becomes a non-homogeneous regular series-parallel system with parameters k, = k = 2, In = 129, a = 1, ql = 1, el = 4, Pit = 1/129, Pt2 = 2/129, PI3 = 117/129, PI4 = 9/129, 211 = 0.0125, 212 = 0.0015, 2t3 = 0.005, 214 = 0.004. Since 4

/21 = ~ Plj/]aj j=l

1

= i29

2 0.0125 +

129

117

0.0015

+ 129 0.005 +

2 = min{O.O049} = 0.0049, then applying Proposition 8.9 with normalising constants

1 an

--

0.0049.129

we conclude that

~_ ....,.......,,,.__

0.6365' b,, = 0,

9 129

0.004 = 0.0049,

Related and Open Problems

300

.~'C91)(t) = 1 - [1 - exp[-t]] 2 for t > 0 is the subsystem S 1 limit reliability function and from (1.1), we get R (i) 2,129 (t) = ~ I ) (0.6365t) = 1 - [ 1 - exp[-0.6365t]] 2 for t >__0. At system operational state 1, subsystem

S 2 becomes a n o n - h o m o g e n e o u s regular

series-parallel system with parameters k, = k = 3 , 1, = 7 4 3 , a = 1, ql = 1, el = 3 , Pll =

211 =

1/743,

PI2 =

2/743,

PI3 =

740/743,

0.025,/~t2 = 0.0015, ~13 = 0.03.

Since 3

1

j=l

743

/l I = ~., PljA~j =

0.025 +

2

740 0.0015 + .. 0.03 = 0.0299, 743 743

= min{0.0299} = 0.0299,

then applying Proposition 8.9 with normalising constants 1 an

=

0.0299. 743

""

1 2 2 . 2 2 8 ' b, = 0,

we conclude that .q/(91)(t) = 1 - [1 - exp[-t]] 3 for t > 0 is the subsystem S 2 limit reliability function and from (1.1), we get R 0) 3,743 (t) = 9i'(91)(22.228t) = 1 - [1 - exp[-22.228t]] 3 for t > 0. At system operational state 1, subsystem S 3 is a n o n - h o m o g e n e o u s regular seriesparallel system with parameters

kn = k = 2, l, = 139, a = 1, ql = 1, el = 4, Pll =1/139, Pl2 ----2/139, PI3 = 117/139, Pt4 = 19/139,

Chapter 8 '~ll

"-

0.0125,

'~'12 "-

301

0.0015, 213 = 0.005, 214 = 0.004.

Since

4 ,~ = ~, p U 2 U = j=~

1

2 117 19 0.0125 + ......... 0.0015 + 0.005 + 139 0.004 = 0.00487, 139 139 139

2 = min{0.00487} = 0.00487,

then applying Proposition 8.9 with normalising constants

an =

1 0.00487.139

1 -~ " - ' - - ' - , 0.6765

bn = O,

we conclude that gi'~l) (t) = 1 - [1 - exp[-t]] 2 for t > 0 is the s u b s y s t e m S 3 limit reliability function and from (1.1), we get

R (1).,, 2,13, (t) -- gi'~1) (0.6765t) = 1 - [ 1 - exp[-O.6765t]] 2 for t >_ O. At s y s t e m operational state 1, s u b s y s t e m S 4 b e c o m e s a n o n - h o m o g e n e o u s regular series-parallel s y s t e m with parameters k, = k = 3, In = 242, a = 2, ql = 2/3, q2 = 1/3, el = 3, p l l = 2/242, pl2 211 m 0 . 0 0 5 , e2 =

2,

P21 =

2,21 = 0.022,

=

160/242, PI3

'~12 ---" 0.012,

213 = O,

2/242,

240/242,

P22 =

=

80/242,

'~'22 ----"0.034.

Since 3 2q = ~ p l j 2 U j=~

2 160 80 = ,,,~, 0.005 + 0.012 + - - - - 0 z4z 242 242

= 0.007975,

2 2 240 22 = Y', p2j/~2j = ........ 0.022 + 0.034 = 0.0339, j=l 242 242

Related and Open Problems

302

A = rnin{O.O07975, 0.0339} = 0.007975,

then applying Proposition 8.9 with normalising constants

1 an =

0.007975. 242

1 .-='93 b,, = O, 1

we conclude that .qi'(90 (t) = 1 - [1 - exp[-t]] 2 for t > 0 is the subsystem $4 limit reliability function and from (1.1), we get

R(1)3,242(t) ~

'Jr 9-(1) (1.93t) = 1 - [ 1 - exp[-1.93t]] 2 for t >_ 0.

Since the considered subsystems create a series structure in a reliability sense, then the reliability function of the whole transportation system, for t > O, is given by

(I) (t) ~ R (I) 2,129 (t) R (I) 3,743(t) R (I)39 2,1 (t) R (I) 3,242 (t) = 24exp[-25.471t] - 24exp[-47.699t] - 12 exp[-26.1075t] - 12 exp[-27.401t] + 12 exp[-48.3355t] + 12 exp[-49.629] -12exp[-26.1475t]

+ 12exp[-48.3755t] + 8exp[-69.927t]

+ 6 exp[-28.0375t] - 6 exp[-50.2655t] - 6 exp[-50.3055t] + 6exp[-26.784t] + 6exp[-28.0775t] - 6exp[-49.012t] - 4exp[-70.6035t] - 4exp[-70.5635t] - 4exp[-71.857t] + 3exp[-50.942t]-

3exp[-28.714t] + 2exp[-72.4935t]

+ 2exp[-71.24t] + 2 e x p [ - 7 2 . 5 3 3 5 t ] - exp[-73.17t]

(8.31)

and according to (8.28) and (8.29) the system lifetime mean value and its standard deviation are E[T (0 ] ~ 0.0807, cr(T (1)) ___-0.057.

(8.32)

Chapter 8

303

The structure o f the port grain transportation system at operation state 2 is given in Figure 8.20.

MAIN DISTRIBUTIONSTATION 9th. .floor . . . + ... § r . . . . ] [BALANCE 6th floor !

sthnoor !

I 2 ~

I

4th floor

- r162

2'floor[

1 H' 2 l_. 1H

...........

2 I

d'162- ] s 7, q

.

I BAsEMEN i

I RAILWAYTRACK

Fig. 8.20. The scheme of the grain transportation system structure at operation state 2 At system operational state 2, subsystem S 1 becomes a non-homogeneous regular series-parallel system with parameters k, = k = 1, 1, = 129, a = 1, qt = 1, et = 4, Pll = 1/129, PI2 21t = 0.0125,

= 2/129,

'~12 =

PI3

0.0015,

=

117/129, PI4

'~13 "-" 0 . 0 0 5 ,

= 9/129,

214 = 0.004.

Since

4

1

2

117

9

129

129

2~ = ~ Plj2~j = i 2 9 0.0125 + "="lz~0.0015 + ...... 0.005 + ..... 0.004 = 0.0049, j=~

2 = min{0.0049} = 0.0049,

then applying Proposition 8.9 with normalising constants

0.0049.129

1 0 . 6 3 6 5 ' b, = O,

Related and Open Problems

304 we conclude that

9i'~2) (t) = 1 - [1 - exp[-t]] = e x p [ - t ] for t > 0 is the subsystem S~ limit reliability function and from (1.1), we get R (2) _ (t) 1,129

= .0i'{92)(0.6365t) --- exp[--0.6365t] for t > 0.

At system operational state 2, subsystem

S 2 becomes a n o n - h o m o g e n e o u s regular

series-parallel system with parameters k, = k = 2, In = 743, a = 1, ql = 1, el = 3, Pll

=

1/743,

P 1 2 "-

2/743,

PI3 -

740/743,

2tl = 0.025, A,t2 = 0.0015, 2t3 = 0.03. Since 3 1 2 740 21 = ~" PtJAtJ = 743 0.025 + 743 0.0015 + - - - - 0 . 0 3 = 0.0299, j---1 743 = min{O.0299} = 0.0299,

then applying Proposition 8.9 with normalising constants

1 an --

0.0299. 743

---

1 22.228

, bn = 0,

we conclude that .~i'(92)(t) = 1 - [1 - exp[-t]] 2 for t >_ 0 is the subsystem S 2 limit reliability function and from (1.1), we get R (2) (t) = 9/(92) (22.228t) = 1 - [1 - exp[-22.228t]] 2 for t > 0. 2,743 At system operational state 2, subsystem S 3 is a n o n - h o m o g e n e o u s regular seriesparallel system with parameters k, = k = 1, 1, = 139, a = 1, qt = 1, el = 4,

Chapter 8 Pll

305

=1/139, P I 2 = 2/139, P I 3 = 1 1 7 / 1 3 9 , P I 4 = 19/139,

/~ll =

0.0125, Al2 = 0.0015, 213 = 0.005, '~14 = 0.004.

Since 4 1 2 19 0.004 = 0.00487, 21 = 5-'.ply2~j = 0.0125 + 0.0015 + 117 0.005 + j=l 139 139 139 139

2, = min{0.00487} = 0.00487,

then applying Proposition 8.9 with normalising constants

an =

1

0.00487.139

=

1

0.6765

, b, = 0,

we conclude that ~ ( 29) (t) -" 1 - [ 1 - e x p [ - t ] ] = e x p [ - t ]

for t > 0

is the subsystem S 3 limit reliability function and from (1.1), we get R (2) _ 1,139 ( t )

= ,~(2)

(0.6765t) = exp[-0.6765t] for t > 0.

At system operational state 2, subsystem S 4 becomes a non-homogeneous regular series-parallel system with parameters kn = k = 2, In = 242, a = 1, ql = 1, el = 3, Pll

=

2/242, PI2

=

')]'11 "- 0 . 0 0 5 , ~12 ~--"0.012,

160/242, Pl3

213 =

=

80/242,

0.

Since 3

/1,1 = ~ P l j ~ l j = j=~

2

160 80 0.005 + ....... 0.012 + 0 = 0.007975, 242 242 242

2 = min{O.O07975} = 0.007975, then applying Proposition 8.9 with normalising constants

306

Related and Open Problems

an =

1 0.007975. 242

2

1 1.93

, bn = 0,

we conclude that .qi'~2) (t) = 1 - [1 - exp[-t]] 2 for t >_0 is the subsystem $4 limit reliability function and from (1.1), we get R ~2) 2,242 (t) - .q/'~92)(1.93t) = 1 - [1 - exp[-1.93t]] 2 for t > 0. Since the considered subsystems create a series structure in a reliability sense, then the reliability function of the whole transportation system is given by

-~(2)(t)=R(2) (2) (2) ( t ) R ( 2 ) (t) 1,129 (t) R 2,743 (t) R 1,139 2,242 = 4 exp[-25.471t] - 2 exp[-27.401t] - 2 exp[-47.699t] + exp[-49.629t] for t > O,

(8.33)

and according to (8.28) and (8.29) the system lifetime mean value and its standard deviation are E [ T (2) ] = 0.0623, o'(T (2)) = 0.0466.

(8.34)

The structure of the port grain transportation system at operation state 3 is given in Figure 8.21. At system operational state 3, subsystem S~ becomes a non-homogeneous regular series-parallel system with parameters k, = k = 1, In = 129, a = 1, qt = 1, el = 4, Pll = 1/129, Pl2 = 2/129, P13 = 117/129,p14 = 9/129, ;hi = 0.0125, 2t2 = 0.0015, 2ta = 0.005, 214 = 0.004. Since 4 1 2 117 9 2~ = ~, p~j2~j = 0.0125 + . . . . 0.0015 + 0.005 + 0.004 = 0.0049, j=l 129 129 129 129 2 = min{O.O049} = 0.0049,

Chapter 8

.....STORA'GE .... 8th floor .....

307

MAIN DISTRIBUTION STATION 9th floor !~ ...... ~ ~ +

I BALANCE

I

, i

I

[ FLAPS I 4th floor . . . . iiir r ~

I ]

IBASENTI

i

"

RAILWAY TRACK- [

Fig. 8.21. The scheme of the grain transportation system structure at operation state 3 then applying Proposition 8.9 with normalising constants

an

--

1

1

0.0049" 129

0.6365'

bn = 0,

we conclude that ~i'(93) (t) = 1 - [1 - exp[-t]] = e x p [ - t ] for t _ 0 is the subsystem S t limit reliability function and from (1.1), we get

Re3)^ l,t2u (t)

= .qi'c93)(0.6365t) = exp[-0.6365t] for t _> 0.

At system operational state 3, subsystem S 2 becomes a non-homogeneous regular series-parallel system with parameters kn = k = 1, In -- 743, a = 1, ql = 1, el = 3, Pll = 1/743, PI2 = 2/743, PI3 = 740/743, 2tl = 0.025, '~'t2 = 0.0015,

'~13 --'-

0.03.

308

Related and Open Problems

Since 3 1 2 740 ~1 = j=l ~ Plj/~1J = ....... 743 0.025 + 743 0.0015 + 743 0.03 = 0.0299,

2. = rnin{0.0299} = 0.0299,

then applying Proposition 8.9 with normalising constants

an =

1

1

0.0299" 743

~ "-----,

22.228

bn = 0,

we conclude that gi'g3) (t) = 1 - [1 - exp[-t]] = e x p [ - t ] for t >_.0 is the subsystem S 2 limit reliability function and from (1.1), we get R (3) 1,743 (t) = .~(93) (22.2280 = exp[-22.228t] for t > 0. At system operational state 3, s u b s y s t e m S 3 is a non-homogeneous regular seriesparallel system with parameters kn = k = 1, In = 139, a = 1, ql = 1, el = 4,

pit =1/139, p12 = 2/139, p13 = 117/139, p14 = 19/139, 211 = 0.0125, 212 = 0.0015, 213 = 0.005, 214 = 0.004. Since 4

~1 = ~ P l j 2 ~ j j=l

1 2 117 19 = .~;. 0.0125 + - - - 0 . 0 0 1 5 + 0.005 + 0.004 = 0.00487, lJ~ 139 139 139

= min{0.00487} = 0.00487,

then applying Proposition 8.9 with normalising constants

an =

1 0.00487" 139

we conclude that

"~

1 0.6765

, bn = 0,

Chapter 8

309

9?~3) (t) = 1 - [1 - exp[-t]] = e x p [ - t ] for t > 0 is the subsystem S 3 limit reliability function and from (1.1), we get R (3) 1,139 (t) = 91(93)(0.6765t) = exp[-0.6765t] for t > 0. At system operational state 3, subsystem S 4 becomes a non-homogeneous regular series-parallel system with parameters kn = k = 1, 1, - 242, a = 1, ql = 1, el = 3, Pll = 2/242, PI2 = 160/242, PI3 = 80/242, /]'11 = 0 . 0 0 5 , /~'12 =

0.012, A13=

0.

Since 3 2 160 80 21 = ~ PljAlj = 242 0.005 + .... 0.012 + 0 = 0.007975, j=l 242 242

= rnin{0.007975} = 0.007975,

then applying Proposition 8.9 with normalising constants 1 an =

0.007975. 242

1 Z

1.93' bn = 0,

we conclude that 9i't93)(t) = 1 - [1 - exp[-t]] = e x p [ - t ] for t > 0 is the subsystem S 4 limit reliability function and from (1.1), we get R O) 1,242 (t) - .9i'~93)(1.93t) = exp[-1.93t] for t > 0.

Since the considered subsystems create a series structure in a reliability sense, then the reliability function of the whole transportation system is given by ~'(3)(t)

= R (3) (t) R 0

and according to (8.28) and (8.29) the system lifetime mean value and its standard deviation are E[T (3) ] = 0.0393, cr(T (3)) = 0.0393.

(8.36)

Finally, considering (8.22) and (8.30), the system unconditional reliability is given by R(t)

34 ~0) 7 23 (t) + - - ~(2)(t) + - - - R(3)(t), 64 64 64

~ ~

where ~0) (t), ~(2)(t) and ~(3)(t) respectively are given by (8.31), (8.33) and (8.35). Hence, applying (8.26)-(8.27) and (8.32), (8.34) and (8.36), we get the mean value and the standard deviation of the system unconditional lifetime respectively given by 34 7 23 m _=.~ . 0.0807 + ~ . 0.0623 + - - . 0.0393 ---0.0638 years, 64 64 64 or---I~4 (0"057)2~+-~ (0"0466)2 +23(0"0393)64

_=0.0502 years.

The reliability data concerned with the operation process and component reliability functions of the port grain transportation system are not precise. They come from experts and are concerned with the mean lifetimes of the system components and with the conditional sojourn times of the system in the operation states under the arbitrary assumption that their distributions are exponential. By further development of the proposed method it seems to be possible to obtain more general results useful in the complex technical systems and their operation processes reliability and availability evaluation, improvement and optimisation.

SUMMARY

In this book the asymptotic approach to the reliability evaluation of homogeneous and non-homogeneous series and parallel systems, homogeneous "m out of n" systems and homogeneous and non-homogeneous regular series-parallel and parallel-series systems has been completely analysed. For these systems, in the case where their components are two-state as well in the case where they are multi-state, the classes of limit reliability functions have been fixed. Moreover, the auxiliary theorems useful for finding limit reliability functions of real technical systems composed of components having any reliability functions have been formulated and motivated. The series-parallel and parallel-series systems have been considered in the case where their reliability structures are regular. However, this fact does not restrict the completeness of the performed analysis, since by conventional joining of a suitable number of failed components in parallel subsystems of the non-regular parallel-series systems we get the regular nonhomogeneous parallel-series systems considered in the book. Similarly, conventional joining of a suitable number of components which do not fail, in series sub-systems of the non-regular series-parallel systems, leads us to the regular non-homogeneous seriesparallel systems considered in the book. Thus the problem has been analysed exhaustively. In addition to the general solutions, a practically important case when the components of the considered systems have exponential reliability functions has been considered separately. In this case the class of limit reliability functions for the considered multistate systems has been fixed and other practically useful theorems have been proposed. The results obtained in this case may play the role of an easy-to-use guide necessary in quick reliability evaluations of real large technical systems, as well as during their operations and when they are designed. There are proposed algorithms presented in the form of tables giving simple sequential steps in systems reliability evaluation. To make these algorithms an easy and useful tool for reliability practitioners their usage is illustrated by various examples of their application to the evaluation of the real system reliability characteristics. Thinking about the practitioners using computers, on the basis of the algorithms the computer program has been elaborated and the results of its practical use have been illustrated in the book. Theoretical results have been illustrated by many practical examples of their application in reliability evaluation of an extensive range of large systems. The evaluations of the real technical systems presented here have been obtained on the basis of non-precise component reliability data and therefore first of all they should be treated as an illustration of a wide possibility of applications of the proposed asymptotic approach in system reliability analysis. Reliability data come from experts and from the literature

312

Summary

and are concerned with component's mean lifetimes and their expected reliability function types. These evaluations, despite not being precise may be a very useful, simple and quick tool in approximate reliability evaluation, especially during the design of large systems, and when planning and improving their safety and effectiveness operation processes. Optimisation of the reliability structures of large systems with respect to their safety and costs is complicated and often not possible to perform by practitioners because of the mathematical complexity of the exact methods. The proposed method offers enough simplified formulae to allow significant simplifying of reliability optimising calculations. This is testified by the recent partial and preliminary results presented in the last chapter. Especially important are results concerned with exponential systems. Their joining with non-precise data coming from experts concerned with the component mean lifetimes and expected reliability function only allow the constructors to evaluate and to optimise the system structures and their operation processes during designing and before including them in everyday practice ([44]). The results presented in the book have become the basis of investigations on domains of attraction of system limit reliability functions and initiated the problem of the speed at which system reliability function sequences reach their limit reliability functions. The problem of the domains of attraction for fixed limit reliability functions, presented partly in the book, has been completely solved for two-state systems in [81] and generalised to multi-state systems in [71]. The solutions deliver the necessary and sufficient conditions that the reliability functions of the system particular components have to satisfy in order that the system limit reliability function is one of the reliability functions from the fixed class of possible limit reliability functions for this system. This way, on the basis of data about the types of system component reliability functions and the system shape it is possible to expect which of the reliability functions is its limit reliability function. Another significant problem in applying the proposed method to the reliability evaluation of real technical system is its accuracy. This problem has been completely solved for the considered two-state systems in [71 ]. Practical examples presented in the book testify that the mistakes in the approximation of the exact system reliability functions by their limiting forms are not significant in practice, often for not very large systems. Moreover, in the asymptotic approach to system reliability evaluation it is possible to get lower and upper bounds of the exact system reliability characteristics, which is illustrated in the investigation of the ship-rope elevator. The complete solution of the proposed method, i.e. the evaluation of the speed of convergence of the system exact reliability sequences to their limiting forms presented partly in the book for twostate systems, is easy to transfer to multi-state systems. The way of proceeding in this transfer is commented on in the book and solved in [71]. The asymptotic method accuracy evaluations are complicated, so it is probably not possible to use them in everyday practice by reliability practitioners. However, it seems to be practically possible to make the accuracy evaluation supported by computer calculations. Additionally, the main results of the book have initiated and become the basis for further investigations on limit reliability functions. Especially the investigations on limit reliability functions of practically important large series-"m out of n" and "m out of n"series systems and hierarchical systems have been recently significantly developed ([95], [251).

Summary

313

Another problem concerned with the methods of the improving of large systems reliability, which are briefly presented in the last chapter, has been recently completely solved for series systems in [83]. The results presented in the book suggest that it seems reasonable to continue the investigations focusing on: -finding the classes of limit reliability functions for series-"m out of n" and "m out of n"-series systems, series-parallel and parallel-series hierarchical systems and systems with cold reserve, methods of improving reliability for large two-state and multi-state systems, of reliability optimisation for large two-state and multi-state systems related to costs and safety of the system operation processes, availability and maintenance of large systems, -elaboration of universal practical tools in the form of computer program package addressed to large systems operators, allowing them to evaluate and optimise automatically large systems reliability, availability and safety, -elaboration of the offer of training courses in the scope of safety, reliability and availability of large system operation processes based on the computer program package. -

- m e t h o d s

-

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BIBLIOGRAPHY

[1] Abouammoh A., Al-Kadi M.: Component relevancy in multi-state reliability models. IEEE Transactions on Reliability 40, 1991, 370-375. [2] Amari S. V., Misra R. B.: Comment on: Dynamic reliability analysis of coherent multi-state systems. IEEE Transactions on Reliability 46, 1997, 460-461. [3] Anderson C. W.: Extreme value theory for a class of discrete distributions with applications to some stochastic processes. J. Appl. Prob. 7, 1970, 99-113. [4] Aven T.: Reliability evaluation of multi-state systems with multi-state components. IEEE Transactions on Reliability 34, 1985, 473--479. [5] Aven T, Jensen U.: Stochastic Models in Reliability. Springcr-Verlag, New York, 1999. [6] Aven T.: On performance measures for multi-state monotone systems. Reliability Engineering and System Safety 41, 1993, 259-266. [7] Barlow R. E., Proschan F.: Statistical Theory of Reliability and Life Testing. Probability Models. Holt Rinehart and Winston, Inc., New York, 1975. [8] Barlow R. E., Wu A. S.: Coherent systems with multi-state components. Mathematics of Operations Research 4, 1978, 275-281. [9] Bamdorff-Nielsen O.: On the limit behaviour of extreme order statistics. Ann. Math. Statist. 34, 1963, 992-1002. [10] Berman S. M.: Limiting distribution of the maximum term in sequences of dependent random variables. Ann. Math. Statist. 33, 1962, 894-908. [ 11] Berman S. M.: Limit theorems for the maximum term in stationary sequences. Ann. Math. Statist. 35, 1964, 502-516. [12] Block H. W., Savitis T. H.: A decomposition for multi-state monotone systems. J. Applied Probability 19, 1982, 391-402. [13] Bobrowski D.: Mathematical Models in Reliability Theory (in Polish). WNT, Warsaw, 1985. [14] Boedigheimer R., Kapur K.: Customer-driven reliability models for multi-state coherent systems. IEEE Transactions on Reliability 43, 1994, 45-50. [15] Bausch A.: Calculation of critical importance for multi-state components. IEEE Transactions on Reliability 36, 1987, 247-249. [ 16] Brunelle R. D., Kapur K. C.: Review and classification of reliability measures for multi-state and continuum models. IEEE Transactions 31, 1999, 1117-1180. [17] Butler D.: A complete importance ranking for components of binary coherent systems with extension to multi-state systems. Naval Research Logistics 26, !979, 556-578.

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ACKNOWLEDGEMENTS The author would like to thank all his friends and colleagues involved in the research activity of ESREL, KONBiN and MMR conferences and the Winter School of Reliability for their inspiration to write this book. Especially, the author would like to thank Prof. Jerzy Ja~wifiski, the outstanding Polish leader in the reliability field, for his amicable assistance and support in preparing the book.

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INDEX

Ageing see degrading components Agreements 1, 3-4 Algorithms exponential reliability evaluation 232-42 homogeneous parallel system evaluation 235-6 "m out o f n " systems reliability evaluation 222-4 multi-state exponential system evaluation 219-42 non-homogeneous series system evaluation 232-3 parallel systems reliability evaluation 221 parallel-series systems reliability evaluation 227-31 piping systems example 232-3 series systems reliability evaluation 220 series-parallel systems reliability evaluation 225-6 usage evaluation 232-42 Asymptotic approach see also exponential systems basic notions 1-7 homogeneous two-state systems 278-88 multi-state systems 87, 153 speed of convergence 245-51 system reliability improvement 275-88 two-state series systems 245-51,278-88 Auxiliary results 153-6 Auxiliary theorems exponential systems 211-19 multi-state systems 87-152, 211-19 two-state systems 39-86 Baltic Grain Terminal 156-68 Basic notions 1-7 multi-state systems 23-37 two-state systems 9-22 Belt conveyors 181, 189-90, 295-6 Bucket elevators 295 Bulk cargo transportation systems 178-93 scheme 179 Buses cases 241 service example 285-8 transportation systems 96-9, 240-2 Cable example 49-52 see also ropes

Cargo transportation systems 178-93 reliability/risk function values 193 Cases see also examples algorithms 223-4, 225-6 bus transportation systems 241 exponential systems 213-19, 223-4, 225-6, 227-31 homogeneous series systems 275-4, 286-8 multi-state "m o u t o f n " systems 213-14, 223-4 multi-state parallel-series systems 216-19, 227-31 multi-state series-parallel systems 127-8, 137-8 algorithms 225-6 exponential 214-16, 225-6 piping systems 237-8 two-state series systems 278-80 two-state series-parallel systems 61-2 Chain conveyors 163, 296 Changing reliability structures 289 Cold system reservation 276-7 Components, reservations 276-7 Conditional reliability functions 293 Conveyors 157 see also elevators belt type 181, 189-90, 295-6 bulk 178-9 chain type 163, 296 Cross-sections, ropes 195 Degenerate reliability functions 2 Degrading components 23-37 Distribution, electrical energy example 148-9 Domains of attraction 243-5 Durability, steel ropes 113-17, 122-5 Electrical energy distribution example 148-9 Elevators examples 261-2 loadings 196, 198-200, 202-4 operating process reliability 295 port grain systems 156-7, 159 ship-rope type 194-210 Energetic cable example 49-52, 108-10 Energy distribution system example 148-9

324

Index

Erlang's distribution 94 Examples see also cases bus service companies 285-8 bus transportation systems 96-9, 240-2 distribution systems 65-7, 148-9 electrical energy distribution systems 148-9 elevators 261-2 energetic cables 49-52, 108-10 energy distribution systems 148-9 gas distribution system 65-7 gas piping system 44-7 lighting systems 58-60, 256-7 multi-state series systems 90-9 parallel-series systems 273-4 piping systems 90-3, 103-5, 131-4, 141-4 exponential 232-42 rope durability 113-17 ship-rope elevators 261-2 sports hall illumination 256-7 steel rope durability 122-5 telecommunication networks 93-6 three-stratum rope durability 113-17 transportation systems 90-3, 96-9, 103-5, 240-2 Exponential systems see also multi-state exponential systems algorithms 232-42 auxiliary theorems 211-19 cases 213-19, 223-4, 225-6, 227-31 distribution 94 homogeneous 213-14, 235-6 reliability evaluation 232-42 reliability functions 57, 121,288 Gas

distribution system 65-7 piping system example 44-7 Gdynia (port) 153, 156-68, 285-8 Grain transportation systems 156-68, 289-310 multi-state series-parallel 158, 160, 161-2, 164 operation state 1 298-302 operation state 2 303-6 operation state 3 306-9 reliability/risk function values 167-8 series-parallel 158, 160, 161-2, 164 Graphs cable reliability functions 110 cargo transportation reliability/risk functions 193 energetic cable reliability/risk functions 110 grain transportation reliability/risk functions 168 hierarchical systems 269, 274 homogeneous series-parallel systems 249-51 multi-state reliability functions 92, 105, 110, 117, 134 non-homogeneous Weibull systems 105

oil transportation reliability/risk functions 177 parallel-series hierarchical systems 274 piping systems examples 92, 134 reliability/risk functions 234, 239 rope durability example 117 series-parallel hierarchical systems 269 ship-rope transportation reliability/risk functions 207-10 sports hall reliability functions 257 Hierarchical systems 262-74 parallel-series 270-4 series-parallel 262-9 Homogeneous... exponential systems 213-14, 235-6 "m out ofn" systems 12-13, 30, 54-8, 117-25, 213-14 model systems 235-6 multi-state systems exponential 235-6 " m o u t o f n " 30, 117-25,213-14 parallel 28-9, 106-10 parallel-series 32, 144-9, 154-5 series 28, 88-96 series-parallel 31,125-34 parallel model systems 235-6 series systems 275-88 cases 275-7, 278-80, 281-4, 286-8 reliability improvement 275-88 series-parallel systems 249-51 two-state systems asymptotic approach 278-88 hierarchical 272 "m out ofn" 12-13, 54-8 parallel 11, 47-9 parallel-series 16, 77-82, 270-3 series 10, 39-40, 243-5,278-88 series-parallel 14, 62-7, 266-9 Hot system reservation 276-7 Illumination, sports halls 256-7 Lighting system example 58-60, 256-7 Limit reliability functions auxiliary theorems 39-86, 87-152 gas distribution system 65-7 gas piping example 46-7 homogeneous two-state systems "m out ofn" systems 54-8 series-parallel systems 65-7 multi-state 36-7 non-homogeneous two-state parallel systems 54 Linking systems reliability 288-310 Loadings, ship-rope elevators 196, 198-200, 202-4

Index

"m out ofn" systems 252-7 homogeneous 30, 54-8 multi-state systems 29-30, 34, 117-25 non-homogeneous 18-19, 34 two-state 11-13, 18-19, 54-8 "m out ofn"-series systems 257-62 definitions 257-9 schemes 258, 259 ship-rope elevators 261-2 M-80-200-10 type rope 113-17 Mean values rope lifetime 116 system component lifetimes 46 Mixed system reservations 276-7 Model systems multi-state series-parallel 129-31 telecommunication networks 93-6 two-state parallel-series 80-2 water supply 74 Multi-state exponential systems 211-42 algorithms 219-42 auxiliary theorems 211-19 homogeneous 213-14, 235-6 non-homogeneous 237-8 parallel model systems 235-6 piping systems example 232-42 series-parallel 237-8 Multi-state "m out ofn" systems 117-25 exponential 213-14, 222-4 homogeneous 30, 117-25, 213-14 non-homogeneous 34 reliability evaluation algorithms 222-4 steel rope example 122-5 Multi-state parallel systems 106-17 exponential 212-13 homogeneous 28-9, 106-10 non-homogeneous 33-4, 110-17, 212-13 reliability evaluation algorithms 221 rope durability example 113-17 Weibull reliability functions 107, 112 Multi-state parallel-series systems 144-52 bus transportation 240-1 electrical energy distribution example 148-9 exponential 216-19 homogeneous 32, 144-9 non-homogeneous 35--6, 149-52 bus transportation 240-1 exponential 216-19 regular 32-3 reliability evaluation algorithms 227-31 Multi-state series systems 88-105 bus transportation system 96-9 cargo transportation 182, 184, 186, 188, 190 examples 90-9

325

homogeneous 28, 88-96 non-homogeneous 33, 99-105 cargo transportation 182, 184, 186, 188, 190 exponential 211-12 piping example algorithms 232-3 piping transportation systems 90-3, 103-5 reliability evaluation algorithms 220 shipyard transportation 154 telecommunication network model systems 93-6 Weibull reliability functions 101 Multi-state series-parallel systems 125-44 algorithms 225-6 cargo transportation 180 cases 127-8, 137-8 grain transportation 158, 160, 161-2, 164 homogeneous 31,125-34 model systems 129-31 non-homogeneous 135-44 cargo transportation 180 exponential 214-16 oil transportation 173 oil transportation 173 piping system examples 141-4 regular 31-2 reliability evaluation algorithms 225-6 Weibull reliability functions 128, 129, 130 Multi-state systems 87-152 asymptotic approach 87, 153 auxiliary results 153-6 basic notions 23-37 exponential 211-42 graphs 92, 105, 110, 117, 125, 134 "m out ofn" systems 117-25 parallel 28-9, 33-4, 106-17 parallel-series 32-3, 35-6, 144-52 port systems 153-210 reliability functions 24-5 graphs 92, 105, 110, 117, 125, 134 risk 27 series 27-8, 33, 88-105 series-parallel 31-2, 125-44 shipyards 153-210 transfer from two-state systems 248 transportation 153-210 Naval Shipyard, Gdynia 153 Non-degenerate limit reliability functions 56-7 Non-homogeneous "m out ofn" systems 18-19, 34 multi-state exponential systems 237-8 multi-state "m out ofn" systems 34 multi-state parallel systems 33-4, 110-17 exponential 212-13 multi-state parallel-series systems 35-6, 149-52

326 Non-homogeneous (Continued) bus transportation 240-1 exponential 216-19 multi-state series systems 33, 99-105, 154 cargo transportation 182, 184, 186, 188, 190 exponential 211-12 piping example algorithms 232-3 multi-state series-parallel systems 135-44 cargo transportation 180 exponential 214-16 grain transportation systems 158, 160, 161-2, 164 oil transportation 173 non-regular conveyors 296-310 regular two-state systems series-parallel 292-6 series-parallel systems 35,237-8 two-state parallel systems 52-4 two-state parallel-series systems 20-1, 82-6 two-state series systems 17-18, 40-4 two-state series-parallel systems 66-76, 292-6 Weibull parallel systems 117 Non-regular non-homogeneous conveyors 296-310 Oil transportation systems pipelines 171 ports 168-78 scheme 169 Operating processes 288-310 operation state 1 298-301 operation state 2 303-6 operation state 3 306-9 Order 1, series-parallel systems 262-3 Order 2 parallel-series systems 273-4 series-parallel systems 263-4 Order r parallel-series systems 270-4 series-parallel systems 265-7 Parallel reliability functions 50-1 Parallel systems 10-11, 33-4 see also multi-state parallel systems; two-state parallel systems Parallel-series systems hierarchical 270-4 homogeneous 270, 271-3 multi-state systems 32-3 order r 270-4 scheme 15 two-state systems 15-17, 20-1 Piping systems 90-3, 103-5 cases 237-8 examples 131-4, 141-4 multi-state exponential 232-42 oil 171

Index

reliability function values 133-4 reliability/risk function graphs 92, 105 reliability/risk functions values 234, 239 Ports bulk cargo transportation systems 178-93 Gdynia 153-210 grain transportation systems 156-68, 289-310 multi-state systems 153-210 oil transportation systems 168-78 reliability/risk functions 176-7 transportation systems 153-210, 289-310 Probability vectors 25-6, 30-1 Rayleigh reliability function 246-7 Regular systems exponential 216-19 multi-state parallel-series 32-3, 216-19 multi-state series-parallel 31-2, 125-44 non-homogeneous exponential 216-19 multi-state parallel-series 216-19 series-parallel 14-15, 19-20 two-state series-parallel 292-6 series-parallel 14-15, 19-20, 31-2 two-state parallel-series 16-17, 77-86 two-state series-parallel 14-15,19-20, 60-76, 292-6 Reliability functions 1-22 degenerate 2 energetic cable example 50-1 exact/approximate energetic cable example 50-1 grain transportation systems 167-8 hierarchical systems 269, 274 lighting system example 59-60 model two-state parallel-series systems 81-2 parallel-series systems 274 pipeline systems 133-4 two-state parallel-series systems 81-2 two-state series-parallel systems 66-7 water supply system 76 exponential 57, 121 gas piping example 46-7 grain transportation systems 167-8 homogeneous series-parallel systems 249-51 lighting system example 59-60 model two-state parallel-series system 81-2 multi-state systems 24-5 oil transportation values 176-7 pipeline systems 92, 133-4 speed of convergence 249-51 two-state systems 9-22 value differences example 46-7 water supply system 76 Reliability improvement 275-88 Reliability structure linking 288-310

Index

Risk functions energetic cable values 109-10 multi-state systems 27 oil transportation values 176-7 piping systems 92, 105, 133-4 rope durability example 117 steel ropes durability values 124-5 transportation system graphs 92 Ropes see also Ship-rope transportation systems cross-sections 195 durability example 113-17 durability graphs 117 shipyard transportation systems 194-210 Weibull reliability functions 114-15 Semi-markov models 288-310 Series systems component reservations 276 definition 9-10 reservations 276-7 Series-"m out ofn" systems 252-7 definitions 252-4 example 256-7 scheme 252 Series-parallel systems hierarchical 262-9 homogeneous examples 268-9 order 1 262-3 order 2 263-4 order r 265-7 scheme 264 two-state 13-14 Ship-rope transportation systems 194-210 elevators 194-210, 261-2 graphs 207-10 loadings 196, 198-200, 202-4 "m out ofn"-series systems 261-2 reliability function values 206-10 results comparisons 206--10 risk function graphs 209-10 scheme 194 Shipyard transportation systems 153-210 multi-state series systems 154 rope type 194-210 Weibull reliability functions 154-5 Sojourn times 94, 98, 132, 290-2 Speed of convergence asymptotic approach 245-51 reliability functions 249-51 system reliability function sequences 245-51 Sports hall illumination 256-7 Standard deviation 25, 26-7 Steel ropes 113-17, 122-5 System components

327

lifetimes 46 mean values 46 reservations 276-7 System limit reliability functions basic notions 1-7 multi-state 36-7 System operating processes 288-310 System reliability function sequences 245-51 System reliability improvement 275-88 Telecommunication network model systems 93-6 Three-stratum rope durability example 113-17 Trade Norm 114 Transportation systems bulk cargoes 178-93 buses 96-9, 240-2 Gdynia 153-210 grain 156-68, 289-310 multi-state 153-210 oil 168-78 piping 90-3, 103-5 ports bulk cargoes 178-93 Gdynia 153-210 grain 156-68, 289-310 oil 168-78 ship-rope elevators 194-210 ship-rope elevators 194-210 shipyards 153-210 Two-state "m out ofn" systems 11-13, 18-19, 54-60 homogeneous 12-13, 54-8 Two-state parallel systems 10-11, 47-54 homogeneous 11, 47-9 non-homogeneous 52-4 reliability evaluation 47-54 Two-state parallel-series systems 15-17, 77-86 hierarchical 272 homogeneous 16, 77-82, 270-3 model system 80-2 non-homogeneous 20-1, 82-6 order r 270-1 regular 16-17 Two-state series systems 39-47 homogeneous 10, 39-40, 243-5 non-homogeneous 17-18, 40-4 reliability evaluation 40-4 speed of convergence 245-51 Two-state series-"m out ofn" systems 252-7 Two-state series-parallel systems 13-14, 60-76 cases 61-2 homogeneous 14, 62-7, 246-51,266-9 non-homogeneous 66-76, 292-6 order r 265-7 regular 14-15 Weibull reliability functions 64

328

Index

Two-state systems see also individual systems

definitions 9-22 domains of attraction 243-5 reliability evaluation 39-86 reliability functions 9-22 transfer to multi-state systems 248 Type M-80-200-10 rope 113-17 Values bus transportation reliability/risk functions 242 cargo transportation reliability/risk functions 193 energetic cable reliability/risk functions 51, 109-10 grain transportation reliability/risk functions 167-8 oil transportation reliability/risk functions 176-7 parallel-series hierarchical systems 274 pipeline system reliability/risk functions 133-4, 234

rope multi-state reliability/risk functions 124-5 ship-rope transportation reliability/risk functions 206-10 Water supply systems 73-6 Weibull reliability functions energetic cables 108 gas distribution system 65-6 model series-parallel systems 129 multi-state parallel systems 107, 112, 117 multi-state series systems 101 multi-state series-parallel systems 128, 129, 130 pipeline system example 131 rope durability 114-15, 117 shipyard transportation systems 154-5, 197 two-state parallel systems 48-9 two-state parallel-series systems 79 two-state series systems 42 two-state series-parallel systems 64, 71