Markov chains and dependability theory / Gerardo Rubino and Bruno Sericola, Inria Rennes-Bretagne Atlantique, France.

Format:

Book

Author/Creator:

Rubino, Gerardo, 1955- author.

Sericola, Bruno, author.

Language:

English

Subjects (All):

Markov processes.

Physical Description:

1 online resource (viii, 278 pages) : digital, PDF file(s).

Other Title:

Markov Chains & Dependability Theory

Place of Publication:

Cambridge : Cambridge University Press, 2014.

Language Note:

English

Summary:

Dependability metrics are omnipresent in every engineering field, from simple ones through to more complex measures combining performance and dependability aspects of systems. This book presents the mathematical basis of the analysis of these metrics in the most used framework, Markov models, describing both basic results and specialised techniques. The authors first present both discrete and continuous time Markov chains before focusing on dependability measures, which necessitate the study of Markov chains on a subset of states representing different user satisfaction levels for the modelled system. Topics covered include Markovian state lumping, analysis of sojourns on subset of states of Markov chains, analysis of most dependability metrics, fundamentals of performability analysis, and bounding and simulation techniques designed to evaluate dependability measures. The book is of interest to graduate students and researchers in all areas of engineering where the concepts of lifetime, repair duration, availability, reliability and risk are important.

Contents:

Cover; Half-title; Title page; Copyright information; Table of contents; 1 Introduction; 1.1 Preliminary words; 1.2 Dependability and performability models; 1.2.1 Basic dependability metrics; 1.2.2 More complex metrics; 1.2.3 Performability; 1.2.4 Some general definitions in dependability; 1.3 Considering subsets of the state space; 1.3.1 Aggregation; 1.3.2 Lumping and performance evaluation; 1.3.3 Lumping and dependability models; 1.3.4 Using lumping to evaluate numerical procedures; 1.4 The contents of this book; 2 Discrete-time Markov chains; 2.1 Definitions and properties

2.2 Strong Markov property2.3 Recurrent and transient states; 2.4 Visits to a fixed state; 2.5 Invariant measures and irreducible Markov chains; 2.6 Aperiodic Markov chains; 2.7 Convergence to steady-state; 2.8 Ergodic theorem; 2.9 Absorbing Markov chains; 2.9.1 Application to irreducible Markov chains; 2.9.2 Computational aspects; 3 Continuous-time Markov chains; 3.1 Definitions and properties; 3.2 Transition function matrix; 3.3 Backward and forward equations; 3.4 Uniformization; 3.5 Limiting behavior; 3.6 Recurrent and transient states; 3.6.1 General case; 3.6.2 Irreducible case

3.7 Ergodic theorem3.8 Absorbing Markov chains; 4 State aggregation; 4.1 State aggregation in irreducible DTMC; 4.1.1 Introduction and notation; 4.1.2 Preliminaries; 4.1.3 Strong and weak lumpability; 4.1.4 Characterization of weak lumpability; 4.2 State aggregation in absorbing DTMC; 4.2.1 Quasi-stationary distribution; 4.2.2 Weak lumpability; 4.2.3 Link with the irreducible case; 4.3 State aggregation in CTMC; 5 Sojourn times in subsets of states; 5.1 Successive sojourn times in irreducible DTMC; 5.2 Successive sojourn times in irreducible CTMC; 5.3 Pseudo-aggregation

5.3.1 The pseudo-aggregated process5.3.2 Pseudo-aggregation and sojourn times: discrete-time case; 5.3.3 Pseudo-aggregation and sojourn times: continuous-time case; 5.4 The case of absorbing Markov chains; 5.4.1 Discrete time; 5.4.2 Continuous time; 5.4.3 An illustrative example; 6 Occupation times of subsets of states - interval availability; 6.1 The discrete-time case; 6.2 Order statistics and Poisson process; 6.3 The continuous-time case; 7 Linear combination of occupation times - performability; 7.1 Backward and forward equations; 7.2 Solution; 7.3 Examples; 7.3.1 A two-state example

7.3.2 A three-state example7.4 Algorithmic aspects; 7.4.1 Numerical examples; 8 Stationarity detection; 8.1 Point availability; 8.1.1 The classical uniformization method; 8.1.2 Stationarity detection; 8.1.3 The new algorithm; 8.2 Expected interval availability analysis; 8.2.1 Stationarity detection for the expected interval availability; 8.3 Numerical example; 8.4 Extension to performability analysis; 8.5 Conclusions; 9 Simulation techniques; 9.1 The standard Monte Carlo method; 9.1.1 Standard estimation of the MTTF; 9.1.2 Standard estimation of the reliability at t, R(t)

9.2 Estimating the MTTF of a multicomponent repairable system

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-139-98953-7

1-316-01015-5

1-139-98491-8

1-316-01239-5

1-139-05170-9

1-316-00565-8

1-316-00115-6

1-316-00789-8

1-316-00339-6

OCLC:

881165588