Stochastic Models in Reliability Theory: Proceedings of a Symposium Held in Nagoya, Japan, April 23–24, 1984

222 Pages·1984·4.599 MB·English

by  Masami Miyakawa (auth.), Prof. Shunji Osaki, Prof. Yukio Hatoyama (eds.)

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Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
235
Stochastic Models
in Reliability Theory
Proceedings of a Symposium
Held in Nagoya, Japan, April 23-24,1984
Edited by S. Osaki and Y. Hatoyama
Springer-Verlag
Berlin Heidelberg New York Tokyo 1984
Editorial Board
H. Albach M. Beckmann (Managing Editor) P. Dhrymes
G. Fandel J. Green W. Hildenbrand W. Krelle(Managing Editor) H.P. KOnzi
G.L Nemhauser K. Ritter R. Sato U. Schittko P. Schonfeld R. Selten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W. Krelle
Institut fOr Gesellschafts-und Wirtschaftswissenschaften
der Universitat Bonn
Adenauerallee 24-42, 0-5300 Bonn, FRG
Editors
Prof. Shunji Osaki
Department of Industrial and Systems Engineering
Faculty of Engineering, Hiroshima University
Higashi-Hiroshima 724, Japan
Prof. Yu kio Hatoyama
Faculty of Business Administration, Senshu University
Kawasaki 214, Japan
ISBN-13: 978-3-540-13888-4 e-ISBN-13: 978-3-642-45587-2
001: 10.1007/978-3-642-45587-2
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© by Springer-Verlag Berlin Heidelberg 1984
PREFACE
In Japan there are many research workers who are especially inter
ested in stochastic models in reliability theory. On the above back
ground, it was a nice occasion for the Editors to organize the Reliabil
ity Symposium with emphasis on "Stochastic Models in Reliability Theory."
The Reliability Symposium was held in Nagoya, Japan, April 23-24, 1984.
In the Symposium there were 14 contributions of the recent research works
on stochastic models in reliability theory in Japan. We briefly sketch
all the contributions by the following classifications just for conveni
ence.
The first three papers discuss coherent structure theory: Miyakawa
considers stochastic coherent systems where the system's state can not
be completely determined by the states of its components. He shows that
such models will be used in identifying the failure mechanism based on
failure pattern data. Ohi and Nishida propose multistate systems in
which all the state spaces are not necessarily the same, but somewhat
strong coherency is assumed. Nakashima and Yamato discuss multivalued
output systems and su~gest the type of redundancy effective for improv
ing- fail-safe characteristics of such systems.
The next two papers are concerned with maintenance and replacement
problems: Nakagawa summarizes seven replacement models with discrete
variables. He shows that each optimal number is given by a unique solu
tion to equation. Kaio and Osaki consider optimum inspection policies
for a system whose failure can be detected effectively. They discuss
three inspection models and give their optimum inspection policies.
The following two papers discuss reliability and availability mod
eling: Sugasawa and Murata investigate a two-unit standby redundant
repairable system and propose two preventive maintenance policies for
such a system. Kodama considers a system of consisting of two subsys
tems connected in series with a single repair facility. He derives ex
plicit expressions for the steady state availability and the mean time
to the first system failure.
Osaki and Nakamura discuss fault-tolerant computing systems: They
propose a multi-processor system with a buffer. Calculating several
measures numerically, they show that some measures give an optimum
storage capacity of the buffer.
The following two papers are concerned with software reliability
modeling: Yamada and Osaki discuss nonhomogeneous error detection rate
models for software reliability growth by introducing two types of er
rors. They show that the model parameters can be estimated by a method
IV
of maximum likelihood. They also discuss the optimum release policies
for such models. Ohba proposes inflection S-shaped software reliability
growth modeilis and shows the applicability of the observed data.
The final four papers discuss Markovian deterioration and replace
ment modeling: Ohashi deals with a coherent system of n components un
der jump deterioration. He considers a continuous-time replacement
model for components in the coherent system with minimal repair, and in
vestigates the structural properties of the optimal replacement policy.
Kawai discusses an optimal ins~ection and replacement policy for con
tinuous-time Markovian deterioration system. He shows a control limit
rule for replacement. Ohnishi, Mine and Kawai investigate an optimal
inspection and replacement policy for a discrete-time Markovian deterio
ration system under imcomplete state information. They show that there
exists an optimal inspection and replacement policy in the o.lass of
monotonic four region-policies. Hatoyama, Fukuoka and Suzuki discuss a
maintenance problem of a road-type construction. They show that the problem
is formulatedusing Markovian deterioration models and an optimal policy
is a simple strucured control limit policy.
"We wish to express our gratitude to Dr. W.A. Mliller, Economics
Editor, Lecture Notes in Economics and Mathematical Systems, for his
kind advice, and to the referees who review all the contributions. We
finally acknowledge the kind assistance of Professor T. Nakagawa, Meijo
University, Nagoya, Japan, and Professor N. Kaio, Hiroshima Shudo Uni
versity, Hiroshima, Japan, for the preparation of the symposium and the
proceedings.
Higashi-Hiroshima, Japan Shunji Osaki
Kawasaki, Japan Yukio Hatoyama
The Editors
August 1984
LIST OF CONTRIBUTORS
Hiroshi Fukuoka, Railway Technical Research Institute, Japanese National
Railways, Tokyo 185, Japan
Yukio Hatoyama, Faculty of Business Administration, Senshu University,
Kawasaki 214, Japan
Naoto Kaio, Department of Management Science, Hiroshima Shudo University,
Hiroshima 731-31, Japan
Hajime Kawai, School of Economics, University of Osaka Prefecture, Sakai
591, Japan
Masanori Kodama, Department of Economic Engineering, Kyushu University,
Fukuoka 812, Japan
Hisashi Mine, Department of Applied Mathematics and Physics, Kyoto Uni
versity, Kyoto 606, Japan
Masami Miyakawa, Department of Industrial Engineering and Management,
Tokyo Institute of Technology, Tokyo 152, Japan
Koichi Murata, College of Industrial Technology, Nihon University, Chiba
275, Japan
Toshio Nakagawa, Department of Mathematics, Meijo University, Nagoya
468, Japan
Masahiro Nakamura, Department of Industrial and Systems Engineering,
Hiroshima University, Higashi-Hiroshima 724, Japan
Kyoichi Nakashima, Department of Electronics, Himeji Institute of Tech
nology, Himeji 671-22, Japan
Toshio Nishida, Department of Applied Physics, Osaka University, Osaka
565, Japan
Mamoru Ohashi, Anan Technical College, Anan 774, Japan
Mitsuru Ohba, Science Institute, IBM Japan, Ltd., Tokyo 102, Japan
Fumio Ohi, Department of Applied Physics, Osaka University, Osaka 565,
Japan
Masamitsu Ohnishi, Department of Applied Mathematics and Physics, Kyoto
University, kyoto 606, Japan
Shunji Osaki, Department of Industrial and Systems Engineering, Hiroshima
University, Higashi-Hiroshima 724, Japan
Yoshio Sugasawa, College of Industrial Technology, Nihon University,
Chiba 275, Japan
Kazuyuki Suzuki, Department of Mathematical Science, Tokai University,
Hiratsuka 259-12, Japan
Shigeru Yamada, Graduate School of Systems Science, Okayama University
of Science, Okayama 700, Japan
Kazuharu Yamato, Department of Electronics, Himeji Institute of Tech
nology, Himeji 671-22, Japan
TABLE OF CONTENTS
ON STOCHASTIC COHERENT SYSTEMS
Masami Miyakawa 1
MULTI STATE SYSTEMS IN RELIABILITY THEORY
Fumio Ohi and Toshio Nishida 12
IMPROVING FAIL-SAFE CHARACTERISTICS OF MULTIVALUED-OUTPUT SYSTEMS
THROUGH THE USE OF REDUNDANCY
Kyoichi Nakashima and Kazuharu Yamato 23
DISCRETE REPLACEMENT MODELS
Toshio Nakagawa
ANALYTICAL CONSIDERATIONS ON INSPECTION POLICIES
Naoto Kaio and Shunji Osaki 53
RELIABILITY AND PREVENTIVE MAINTENANCE OF A TWO-UNIT STANDBY REDUNDANT
SYSTEM WITH DIFFERENT FAILURE TIME DISTRIBUTIONS
Yoshio Sugasawa and Koichi Murata 72
RELIABILITY AND MAINTAINABILITY OF A MULTI COMPONENT SERIES-PARALLEL
SYSTEM WITH SIMULTANEOUS FAILURE UNDER PREEMPTIVE REPEAT REPAIR
DISCIPLINE
Masanori Kodama 85
PERFORMANCE/RELIABILITY MODELING FOR MULTI-PROCESSOR SYSTEMS WITH
COMPUTATIONAL DEMANDS
Shunji Osaki and Masahiro Nakamura 105
NONHOMOGENEOUS ERROR DETECTION RATE MODELS FOR SOFTWARE RELIABILITY
GROWTH
Shigeru Yamada and Shunji Osaki 120
INFLECTION S-SHAPED SOFTWARE RELIABILITY GROWTH MODEL
Mitsuru Ohba 144
OPTIMAL REPLACEMENT POLICY FOR COMPONENTS IN A COHERENT SYSTEM UNDER
JUMP DETERIORATION
Mamoru Ohashi 163
AN OPTIMAL INSPECTION AND REPLACEMENT POLICY OF A MARKOVIAN
DETERIORATION SYSTEM
Hajime Kawai 177
AN OPTIMAL INSPECTION AND REPLACEMENT POLICY UNDER INCOMPLETE STATE
INFORMATION: AVERAGE COST CRITERION
Masamitsu Ohnishi, Hisashi Mine and Hajime Kawai 187
APPLICATION OF MARKOVIAN DECISION THEORY TO THE PROBLEM OF HIGHWAY
MAINTENANCE
Yukio Hatoyama, Hiroshi Fukuoka and Kazuyuki Suzuki 198
ON STOCHASTIC COHERENT SYSTEMS
Masami Miyakawa
Department or Industrial Engineering and Management
Tokyo Institute of Technology
Oh-okayama, Meguro-ku, Tokyo, Japan
ABSTRACT
As an idea of non-coherent systems, such systems where the system's
state can not be completely determined by the states of its components
are considered. This structual model will be used in identifying the
failure mechanism based on failure pattern data. The'properties of such
systems are also investigated.
1. INTRODUCTION
For a quantitative consideration of system's.reliability and meas
ures needed for reliability improvements, identification of the system's
failure mechanism is essential. In general, a system failure consists
of many failure modes, and the failure pattern ( i.e. the history of
failure components until the system's failure ) corresponding to each
failure mode should be identified at the design and testing stages.
However, sometimes during the use or operation of the system, failure
modes that were not anticipated or expected can emerge, or a failure
mode due to an unanticipated failure pattern results. In a situation
like this, a re-examination of the system's failure mechanism is neces
sary, and the failure pattern data observed at that time will provide
us with the useful information.
We can consider "coherent structure" [2J,[3J to be a rational de
terministic failure mechanism model. In this model, the cut set as a
result of failure pattern can be made to correspond to a particular
failure mode on the one hand, and on the other hand, assuming coherent
structure for a failure mode, the relationship between the mode and a
plural number of failure pattern could also be expressed. A special case
of the series system model is equivalent to the competing risks model
that is popularly used in the medical statistical fields [5J. Also, for
the stochastic failure mechanism model, many deterioration models with a
2
number of discrete deteriorating states for the system or unit have
been considered, where inter-state transition probabilities were given.
Many research have been carried out relating to the inspection and re
placement rule employed in this type of Markov deterioration model,
where the transition probabilities are taken to be constants [8J,[10J.
Furthermore, in the analysis of life data, the mixture model where sys
tems having different failure modes are being mixed has been used as a
failure mechanism model in contrast to the competing risks model.
In this paper, for the purpose of an uniform representation of
these linking failure mechanisms, the concept of "stochastic coherent
structure" is introduced, and some definitions and the basic properties
of the structure are shown. In addition, against this structure set
ting, a methodology to identify failure mechanisms from failure pattern
data will also be discussed.
2. DEFINITIONS AND PROPERTIES OF STOCHASTIC COHERENT SYSTEMS
2.1 BACKBROUND
The special characteristics of coherent system model are, firstly,
there exists a deterministic functional relationship between the system
and components states. This is also true in the case of multi-state
coherent systems [9J. Second one is monotonicity, i.e. there exist no
components which exert a negative effect. Thirdly, all the components
are relevant, that is to say, there are no components which are com
pletely unrelated with the system state.
In recent years, although research on "non-coherent" systems have
been carried out [4J,[7J, their attention were mainly focused on contra
dicting the second characteristic mentioned above. That is to say, it
represents the case where the combination of failure states of a plural
number of components exerts a positive effect on the system state. Cer
tainly, according to the way how the component state is defined, the ex
istence of a case of that kind seems possible. However, when the coher
ent system model is applied to a realistic system, it is thought that
the biggest obstacle that stands in the way would be rather the first
characteristic mentioned above, i.e. the deterministic functional rela
tionship. If the physical dynamics of component states and the system
state can be continuously grasped, and the environmental conditions sur
rounding the system can be completely understood, a physically determin
istic functional relationship should exist between the two states.
3
However, in reality, these states can be grasped as discrete finite or
countable states and discrimination of the states remains ambiguous.
Also, a complete understanding of the environmental conditions is almost
impossible. Due to this reason, a situation may arise where reproduci
bility does not exist for the relationship between the components state
and the system state. Accordingly, in this study, the structural model
where the system state is stochastically related to the components state
will be considered for the system's failure mechanism taken as a non
coherent structure.
2.2 DEFINITIONS
Consider a system made up of m components. To indicate the state
of ith component, a binary indicator variable xi is defined,
1 if component i is functioning,
o if component i is failed,
for i = l,---,m. Similarly, the binary variable ~ is defined to indi
cate the state of the system as follows,
t if the system is functioning,
~
= 1
o if the system is failed.
For ~ = (xl,---,xm) which indicate the state of the sets of com
ponents, we define CO(~) = { i I xi=O } and Cl(~) = { i I xi=l }.
Consider the stochastic behavior of the states of components and
the system. Let { Xi(t), t~O } be a stochastic process which takes val
ues in { 0, I} and the path is right-continuous and nonincreasing.
That is to say, {X.(t), t>O } represents the behavior of the ith com-
l -
ponent. Similarly, let { ~(t), t~O } be a stochastic process which
takes values in { 0, 1 } and the path is also right-continuous and non
increasing. {~(t), t>O } represents the behavior of the system. Fur
thermore, denote the multivariate stochastic process which represents
the behavior of m components by { !(t)=(Xl(t),---,Xm(t», t~O}. In
this paper, it is assumed that {~(t), t~O } depends on { !(t), t~O }
in the following way. Firstly, the decrement of { ~(t), t>O } can occur
only at the jump points of {!(t), t~O}. Secondly, the probability
that the decrement of { ~(t), t~O } occurs at one of the jump points of
{ !(t), t~O } depends only on the states ~ of !(t) at the time pOint,
and is independent of the history of { !(t), t~O } and its elasped time.
That is to say, if we denote a jump point of { !(t), t>O } by t*, this