Full Description
Our daily lives can be maintained by the high-technology systems. Computer systems are typical examples of such systems. We can enjoy our modern lives by using many computer systems. Much more importantly, we have to maintain such systems without failure, but cannot predict when such systems will fail and how to fix such systems without delay. A stochastic process is a set of outcomes of a random experiment indexed by time, and is one of the key tools needed to analyze the future behavior quantitatively. Reliability and maintainability technologies are of great interest and importance to the maintenance of such systems. Many mathematical models have been and will be proposed to describe reliability and maintainability systems by using the stochastic processes. The theme of this book is "Stochastic Models in Reliability and Main tainability. " This book consists of 12 chapters on the theme above from the different viewpoints of stochastic modeling. Chapter 1 is devoted to "Renewal Processes," under which classical renewal theory is surveyed and computa tional methods are described. Chapter 2 discusses "Stochastic Orders," and in it some definitions and concepts on stochastic orders are described and ag ing properties can be characterized by stochastic orders. Chapter 3 is devoted to "Classical Maintenance Models," under which the so-called age, block and other replacement models are surveyed. Chapter 4 discusses "Modeling Plant Maintenance," describing how maintenance practice can be carried out for plant maintenance.
Contents
1. Renewal Processes and Their Computational Aspects.- 1.1 Introduction.- 1.2 Basic Renewal Theory.- 1.2.1 Continuous renewal theory.- 1.2.2 Discrete renewal theory.- 1.3 Some Useful Properties of the Renewal Function.- 1.3.1 Specific examples.- 1.3.2 Asymptotic properties.- 1.4 Analytical Approximation Methods.- 1.4.1 Phase renewal processes.- 1.4.2 Gamma approximations.- 1.4.3 Methods based on equilibrium distribution.- 1.5 Bounds.- 1.6 Numerical Methods.- 1.6.1 Laplace inversion technique.- 1.6.2 Cubic spline algorithm.- 1.6.3 Discritization algorithm.- 1.6.4 Approximation by rational functions.- 1.7 Concluding Remarks.- 2. Stochastic Orders in Reliability Theory.- 2.1 Introduction.- 2.2 Definitions and Basic Properties.- 2.2.1 Stochastic orders generated from univariate functions.- 2.2.2 Conditional stochastic orders.- 2.2.3 Bivariate characterization of stochastic orders.- 2.3 Applications in Reliability Theory.- 2.3.1 Notions of aging.- 2.3.2 Useful stochastic inequalities in reliability theory.- 2.3.3 Stochastic comparisons of system reliabilities.- 2.3.4 Redundancy improvement.- 2.3.5 Stochastic comparisons of maintenance policies.- 2.3.5.1 Replacements upon failures.- 2.3.5.2 Age replacement.- 2.3.5.3 Block replacement.- 2.3.5.4 Minimal repair.- 2.3.5.5 Minimal repair with block replacement.- 2.3.5.6 Stochastic comparison of different maintenance policies.- 2.A TP2 Functions.- 3. Classical Maintenance Models.- 3.1 Introduction.- 3.2 Block Replacement.- 3.3 Age Replacement.- 3.4 Order Replacement.- 3.5 Inspection Strategies.- 3.6 Conclusions.- 4. A Review of Delay Time Analysis for Modelling Plant Maintenance.- 4.1 Introduction.- 4.2 Maintenance Practice.- 4.3 The Delay Time Concept.- 4.4 Basic Delay Time Maintenance Model: Complex Plant.- 4.5 Basic Maintenance Model: Component Tracking.- 4.6 Relaxation of Assumptions.- 4.7 Non-perfect Inspection.- 4.8 Non-steady-state Condition.- 4.9 Non-homogeneous Defect Arrival Rate ?.- 4.10 Condition-dependent Cost and Downtime for Repair.- 4.11 Case Experience Using Subjective Data: Case Experience.- 4.12 Revision of Subjectively Estimated Delay Time Distribution.- 4.13 Correction for Sampling Bias.- 4.14 Subjective Estimation of the Delay Time Distribution Directly.- 4.15 Objective Estimation of Delay Time Parameters.- 4.16 Case Experience Using Objective Data: HPP of Defect Arrival.- 4.17 Discussion of Further Developments in Delay Time Modelling.- 4.18 Conclusions.- 5. Imperfect Preventive Maintenance Models.- 5.1 Introduction.- 5.2 Sequential Imperfect Preventive Maintenance.- 5.2.1 Introduction.- 5.2.2 Model A — age.- 5.2.3 Model B — failure rate.- 5.2.4 Numerical examples.- 5.3 Shock Model with Imperfect Preventive Maintenance.- 5.3.1 Introduction.- 5.3.2 Model and expected cost.- 5.3.3 Optimal policies.- 5.4 Conclusions.- 6. Generalized Renewal Processes and General Repair Models.- 6.1 Background and Motivation.- 6.2 Generalized Renewal Processes.- 6.3 g-Renewal Processes in Discrete Time.- 6.4 Monotonicity and Asymptotic Properties of the g-Renewal Density.- 6.5 On the g-Renewal Function.- 6.6 A General Repair Model.- 7. Two-Unit Redundant Models.- 7.1 Introduction.- 7.2 Two-Unit Standby System.- 7.2.1 Model and assumptions.- 7.2.2 First-passage time distributions.- 7.2.3 Expected numbers of visits to state.- 7.2.4 Transition probabilities.- 7.3 Preventive Maintenance of Two-Unit Systems.- 7.3.1 Model and analysis.- 7.3.2 Optimum preventive maintenance policies.- 7.3.3 Replacement of a two-unit parallel system.- 7.4 Other Two-Unit Systems.- 7.4.1 Two-unit parallel system.- 7.4.2 Two-unit priority standby system.- 7.4.3 Two-unit standby system with imperfect switchover.- 7.4.4 Other models.- 8. Optimal Maintenance Problems for Markovian Deteriorating Systems.- 8.1 A Basic Optimal Replacement Problem for a Discrete Time Markovian Deteriorating System.- 8.1.1 Some conditions on transition probabilities and cost structure.- 8.1.2 Formulation by Markovian decision process (MDP).- 8.1.3 Optimality of control limit rule.- 8.2 An Optimal Inspection and Replacement Problem.- 8.2.1 Transition probability.- 8.2.2 Formulation by semi-Markov decision process (SMDP).- 8.2.3 Structure of optimal inspection and replacement policy.- 8.3 An Optimal Inspection and Replacement Policy with Incomplete Information.- 8.3.1 Some notations and conditions.- 8.3.2 Formulation by partially observable Markov decision process (POMDP).- 8.3.3 Some properties of TP2 order.- 8.3.4 Some properties of optimal function.- 8.3.5 Structure of optimal inspection and replacement policy.- 8.4 A Continuous Time Markovian Deteriorating System.- 8.4.1 A continuous time Markovian deteriorating system.- 8.4.2 Transition probability.- 8.4.3 Formulation by semi-Markov decision process.- 8.4.4 Structure of optimal policy.- 8.5 An Optimal Maintenance Problem for a Queueing System.- 8.5.1 Model description.- 8.5.2 Formulation by semi-Markov decision process.- 8.5.3 Properties of value function.- 8.5.4 Structure of optimal policy.- 9. Transient Analysis of Semi-Markov Reliability Models — A Tutorial Review with Emphasis on Discrete-Parameter Approaches.- 9.1 Introduction.- 9.2 Modelling Framework.- 9.3 Dependability Measures.- 9.4 Methods of Analysis.- 9.4.1 Continuous-parameter models.- 9.4.2 Discrete-parameter models.- 9.5 Equations for the Dependability Measures.- 9.6 Numerical Solution Techniques.- 9.6.1 Solving the integral equations.- 9.6.2 Discrete-parameter approximations.- 9.7 Recent Developments, Conclusions and Further Work.- 10. Software Reliability Models.- 10.1 Introduction.- 10.2 Definitions and Software Reliability Model.- 10.3 Software Reliability Growth Modeling.- 10.4 Imperfect Debugging Modeling.- 10.4.1 Imperfect debugging model with perfect correction rate.- 10.4.2 Imperfect debugging model for introduced faults.- 10.5 Software Availability Modeling.- 10.5.1 Model description.- 10.5.2 Software availability measures.- 10.6 Application of Software Reliability Assessment.- 10.6.1 Optimal software release problem.- 10.6.1.1 Maintenance cost model.- 10.6.1.2 Maintenance cost model with reliability requirement.- 10.6.2 Statistical software testing-progress control.- 10.6.3 Optimal testing-effort allocation problem.- 11. Reliability Models in Data Communication Systems.- 11.1 Introduction.- 11.2 SW ARQ Model with Intermittent Faults.- 11.2.1 Intermittent faults.- 11.2.2 ARQ policy.- 11.2.3 Optimal retransmission number.- 11.2.4 Numerical examples and remarks.- 11.3 SR ARQ Model with Retransmission Number.- 11.3.1 Model and analysis.- 11.3.2 Optimal policy.- 11.3.3 Numerical examples and remarks.- 11.4 Hybrid ARQ Models with Response Time.- 11.4.1 Type-I hybrid ARQ.- 11.4.2 Type-II hybrid ARQ.- 11.4.3 Comparison of type-I and type-II hybrid ARQs.- 11.4.4 Numerical examples and remarks.- 12. Quick Monte Carlo Methods in Stochastic Systems and Reliability.- 12.1 Introduction.- 12.2 The Problem with Direct Simulation.- 12.3 Importance Sampling.- 12.4 The Optimal Change of Measure.- 12.4.1 Remarks.- 12.4.2 Preliminary definitions.- 12.4.3 The recursive approach.- 12.4.4 Exact calculation of ?(x).- 12.5 Cases of Application of the Recursive Approach.- 12.6 System Model.- 12.7 Regenerative Simulation.- 12.8 Failure Biasing Methods.- 12.8.1 Simple failure biasing (SFB).- 12.8.2 Balanced failure biasing (BFB).- 12.8.3 Bias2 failure biasing.- 12.8.4 Failure distance biasing (FDB).- 12.8.5 Balanced 1 failure biasing (B1FB).- 12.8.6 Balanced 2 failure biasing (B2FB).- 12.8.7 Bounded relative error and failure biasing.- 12.9 Unreliability Estimation.- 12.9.1 One-component system.- 12.9.2 General case.- 12.9.3 Example.- 12.10 Analytical-Statistical Methods.- 12.11 Concluding Remarks.
