Stochastic Models in Reliability and Maintenance

by Osaki, Shunji

• ISBN13: 9783540431336
• ISBN10: 3540431330
• Format: Hardcover
• Copyright: 2002-06-01
• Publisher: Springer Verlag

Summary

This book contains 12 contributions on stochastic models in reliability and maintenance. Written by the leading researchers on each topic, each contribution surveys the current status on stochastic models emphasizing mathematical formulation and optimization applications. Each contribution is self-contained and has a thorough bibliography. The topics include renewal processes, semi-Markov processes, Markovian deterioration models, maintenance and replacement models, software reliability models and Monte-Carlo simulation. This book provides researchers, reliability engineers and graduate students with the current status of the field and future developments of the subject.

Table of Contents

Renewal Processes and Their Computational Aspects	1	(30)
T. Dohi
N. Kaio
S. Osaki
Introduction	2	(1)
Basic Renewal Theory	3	(4)
Continuous renewal theory	3	(3)
Discrete renewal theory	6	(1)
Some Useful Properties of the Renewal Function	7	(2)
Specific examples	7	(1)
Asymptotic properties	8	(1)
Analytical Approximation Methods	9	(5)
Phase renewal processes	9	(1)
Gamma approximations	10	(3)
Methods based on equilibrium distribution	13	(1)
Bounds	14	(2)
Numerical Methods	16	(7)
Laplace inversion technique	17	(1)
Cubic spline algorithm	18	(1)
Discritization algorithm	19	(1)
Approximation by rational functions	20	(3)
Concluding Remarks	23	(8)
Stochastic Orders in Reliability Theory	31	(34)
M. Ohnishi
Introduction	31	(1)
Definitions and Basic Properties	32	(10)
Stochastic orders generated from univariate functions	33	(4)
Conditional stochastic orders	37	(4)
Bivariate characterization of stochastic orders	41	(1)
Applications in Reliability Theory	42	(21)
Notions of aging	42	(5)
Useful stochastic inequalities in reliability theory	47	(1)
Stochastic comparisons of system reliabilities	48	(3)
Redundancy improvement	51	(2)
Stochastic comparisons of maintenance policies	53	(1)
Replacements upon failures	53	(1)
Age replacement	54	(1)
Block replacement	55	(1)
Minimal repair	56	(1)
Minimal repair with block replacement	56	(1)
Stochastic comparison of different maintenance policies	57	(6)
TP2 Functions	63	(2)
Classical Maintenance Models	65	(24)
N. Kaio
T. Dohi
S. Osaki
Introduction	65	(2)
Block Replacement	67	(4)
Age Replacement	71	(5)
Order Replacement	76	(3)
Inspection Strategies	79	(3)
Conclusions	82	(7)
A Review of Delay Time Analysis for Modelling Plant Maintenance	89	(36)
A. Christer
Introduction	89	(4)
Maintenance Practice	93	(1)
The Delay Time Concept	94	(2)
Basic Delay Time Maintenance Model: Complex Plant	96	(1)
Basic Maintenance Model: Component Tracking	97	(1)
Relaxation of Assumptions	98	(1)
Non-perfect Inspection	98	(1)
Non-steady-state Condition	99	(1)
Non-homogeneous Defect Arrival Rate λ	100	(2)
Condition-dependent Cost and Downtime for Repair	102	(1)
Case Experience Using Subjective Data: Case Experience	103	(3)
Revision of Subjectively Estimated Delay Time Distribution	106	(1)
Correction for Sampling Bias	106	(1)
Subjective Estimation of the Delay Time Distribution Directly	107	(1)
Objective Estimation of Delay Time Parameters	107	(3)
Case Experience Using Objective Data: HPP of Defect Arrival	110	(5)
Discussion of Further Developments in Delay Time Modelling	115	(1)
Conclusions	116	(9)
Imperfect Preventive Maintenance Models	125	(20)
T. Nakagawa
Introduction	125	(1)
Sequential Imperfect Preventive Maintenance	126	(5)
Introduction	126	(1)
Model A - age	127	(1)
Model B - failure rate	128	(1)
Numerical examples	129	(2)
Shock Model with Imperfect Preventive Maintenance	131	(8)
Introduction	131	(1)
Model and expected cost	132	(3)
Optimal policies	135	(4)
Conclusions	139	(6)
Generalized Renewal Processes and General Repair Models	145	(20)
M. Kijima
Background and Motivation	145	(4)
Generalized Renewal Processes	149	(4)
g-Renewal Processes in Discrete Time	153	(2)
Monotonicity and Asymptotic Properties of the g-Renewal Density	155	(1)
On the g-Renewal Function	156	(3)
A General Repair Model	159	(6)
Two-Unit Redundant Models	165	(28)
T. Nakagawa
Introduction	165	(2)
Two-Unit Standby System	167	(6)
Model and assumptions	167	(2)
First-passage time distributions	169	(1)
Expected numbers of visits to state	170	(1)
Transition probabilities	171	(2)
Preventive Maintenance of Two-Unit Systems	173	(6)
Model and analysis	173	(2)
Optimum preventive maintenance policies	175	(3)
Replacement of a two-unit parallel system	178	(1)
Other Two-Unit Systems	179	(14)
Two-unit parallel system	179	(2)
Two-unit priority standby system	181	(1)
Two-unit standby system with imperfect switchover	182	(2)
Other models	184	(9)
Optimal Maintenance Problems for Markovian Deteriorating Systems	193	(26)
H. Kawai
J. Koyanagi
M. Ohnishi
A Basic Optimal Replacement Problem for a Discrete Time Markovian Deteriorating System	193	(2)
Some conditions on transition probabilities and cost structure	194	(1)
Formulation by Markovian decision process (MDP)	194	(1)
Optimality of control limit rule	195	(1)
An Optimal Inspection and Replacement Problem	195	(4)
Transition probability	196	(1)
Formulation by semi-Markov decision process (SMDP)	196	(1)
Structure of optimal inspection and replacement policy	197	(2)
An Optimal Inspection and Replacement Policy with Incomplete Information	199	(8)
Some notations and conditions	200	(1)
Formulation by partially observable Markov decision process (POMDP)	200	(2)
Some properties of TP2 order	202	(2)
Some properties of optimal function	204	(2)
Structure of optimal inspection and replacement policy	206	(1)
A Continuous Time Markovian Deteriorating System	207	(4)
A continuous time Markovian deteriorating system	207	(1)
Transition probability	208	(1)
Formulation by semi-Markov decision process	209	(1)
Structure of optimal policy	209	(2)
An Optimal Maintenance Problem for a Queueing System	211	(8)
Model description	211	(3)
Formulation by semi-Markov decision process	214	(1)
Properties of value function	214	(1)
Structure of optimal policy	215	(4)
Transient Analysis of Semi-Markov Reliability Models - A Tutorial Review with Emphasis on Discrete-Parameter Approaches	219	(34)
A. Csenki
Introduction	219	(1)
Modelling Framework	220	(2)
Dependability Measures	222	(4)
Methods of Analysis	226	(8)
Continuous-parameter models	226	(7)
Discrete-parameter models	233	(1)
Equations for the Dependability Measures	234	(5)
Numerical Solution Techniques	239	(3)
Solving the integral equations	239	(1)
Discrete-parameter approximations	240	(2)
Recent Developments, Conclusions and Further Work	242	(11)
Software Reliability Models	253	(28)
S. Yamada
Introduction	253	(1)
Definitions and Software Reliability Model	254	(2)
Software Reliability Growth Modeling	256	(4)
Imperfect Debugging Modeling	260	(5)
Imperfect debugging model with perfect correction rate	262	(1)
Imperfect debugging model for introduced faults	263	(2)
Software Availability Modeling	265	(3)
Model description	265	(1)
Software availability measures	266	(2)
Application of Software Reliability Assessment	268	(13)
Optimal software release problem	269	(1)
Maintenance cost model	269	(2)
Maintenance cost model with reliability requirement	271	(1)
Statistical software testing-progress control	272	(2)
Optimal testing-effort allocation problem	274	(7)
Reliability Models in Data Communication Systems	281	(26)
K. Yasui
T. Nakagawa
H. Sandoh
Introduction	282	(1)
SW ARQ Model with Intermittent Faults	283	(5)
Intermittent faults	283	(2)
ARQ policy	285	(2)
Optimal retransmission number	287	(1)
Numerical examples and remarks	288	(1)
SR ARQ Model with Retransmission Number	288	(6)
Model and analysis	289	(2)
Optimal policy	291	(2)
Numerical examples and remarks	293	(1)
Hybrid ARQ Models with Response Time	294	(13)
Type-I hybrid ARQ	295	(1)
Type-II hybrid ARQ	296	(3)
Comparison of type-I and type-II hybrid ARQs	299	(1)
Numerical examples and remarks	300	(7)
Quick Monte Carlo Methods in Stochastic Systems and Reliability	307	(28)
C. Papadopoulos
N. Limnios
Introduction	307	(1)
The Problem with Direct Simulation	308	(1)
Importance Sampling	309	(1)
The Optimal Change of Measure	310	(4)
Remarks	310	(1)
Preliminary definitions	311	(1)
The recursive approach	312	(2)
Exact calculation of γ(x)	314	(1)
Cases of Application of the Recursive Approach	314	(2)
System Model	316	(2)
Regenerative Simulation	318	(1)
Failure Biasing Methods	319	(4)
Simple failure biasing (SFB)	319	(1)
Balanced failure biasing (BFB)	320	(1)
Bias2 failure biasing	321	(1)
Failure distance biasing (FDB)	322	(1)
Balanced 1 failure biasing (B1FB)	322	(1)
Balanced 2 failure biasing (B2FB)	323	(1)
Bounded relative error and failure biasing	323	(1)
Unreliability Estimation	323	(3)
One-component system	323	(1)
General case	324	(2)
Example	326	(1)
Analytical-Statistical Methods	326	(3)
Concluding Remarks	329	(6)
Index	335