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Simple formulae for counting processes in reliability models

Part of: Operations research and management science Markov processes

Published online by Cambridge University Press:  01 July 2016

James Ledoux  and
Gerardo Rubino
James Ledoux*
Affiliation:
INSA
Gerardo Rubino*
Affiliation:
ENST
*
*Postal address: INSA, Campus de Beaulieu 35043 Rennes Cédex, France. Email: ledoux@{univrennes1}{irisa}.fr
**Postal address: ENST, rue de la Châtaigneraie, 35512 Cesson-Sevigné Cédex, France. Email: rubino@{rennes.enst-bretagne}{irisa}.fr
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Abstract

Dependability evaluation is a basic component in the assessment of the quality of repairable systems. We develop a model taking simultaneously into account the occurrence of failures and repairs, together with the observation of user-defined success events. The model is built from a Markovian description of the behavior of the system. We obtain the distribution function of the joint number of observed failures and of delivered services on a fixed mission period of the system. In particular, the marginal distribution of the number of failures can be directly related to the distribution of the Markovian arrival process extensively used in queueing theory. We give both the analytical expressions of the considered distributions and the algorithmic solutions for their evaluation. An asymptotic analysis is also provided.

Keywords

COUNTING PROCESSESMARKOV CHAINSUNIFORMIZATION

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 90B25: Reliability, availability, maintenance, inspection
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 4 , December 1997 , pp. 1018 - 1038
Copyright
Copyright © Probability Trust 1997 

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References

[1] Asmussen, S. and Bladt, M. (1996) Renewal theory and queueing algorithms for matrix-exponential distributions. In Matrix Analytic Methods in Stochastic Models. ed. Alfa, S. and Alfa, A. S. Marcel Dekker, New York.Google Scholar
[2] Bowerman, P. N., Nolty, R. G. and Scheuer, E. M. (1990) Calculation of the Poisson cumulative distribution function. IEEE Trans. Reliab. 39, 158–161.Google Scholar
[3] Cheung, R. C. (1980) A user-oriented software reliability model. IEEE Trans. Software Eng. 6, 118–125.CrossRefGoogle Scholar
[4] Fischer, W. and Meier-Hellstern, K. (1993) The Markov-modulated Poisson process (MMPP) cookbook. Perf. Eval. 18, 149–171.CrossRefGoogle Scholar
[5] Gross, D. and Miller, D. R. (1984) The randomization technique as a modeling tool and solution procedure for transient Markov processes. Operat. Res. 32, 343–361.Google Scholar
[6] Iosifescu, M. (1980) Finite Markov Processes and Their Applications. Wiley, New York.Google Scholar
[7] Kao, E. P. C. (1988) Computing the phase-type renewal and related functions. Technometrics 30, 87–93.Google Scholar
[8] Laprie, J. C. (1992) Dependability: Basic Concepts and Terminology. Springer, Vienna.Google Scholar
[9] Ledoux, J. and Rubino, G. (1997) A counting model for software reliability analysis. J. Modell. Simul. 17. In press.CrossRefGoogle Scholar
[10] Littlewood, B. (1975) A reliability model for systems with Markov structure. Appl. Statist. 24, 172–177.Google Scholar
[11] Lucantoni, D. M. (1993) The BMAP/G/1 queue: a tutorial. In Performance Evaluation of Computer and Communication Systems. (Lecture Notes in Computer Science 729.) ed. Donatiello, L. and Nelson, R. Springer, Berlin. pp. 330–358.Google Scholar
[12] Narayana, S. and Neuts, M. F. (1992) The first two moment matrices of the counts for the Markovian arrival process. Commun. Statist. — Stoch. Models 8, 459–477.CrossRefGoogle Scholar
[13] Neuts, M. F. (1989) Structured Stochastic Matrices of M/G/1 Type and Their Applications. Marcel Dekker, New York.Google Scholar
[14] Neuts, M. F. (1992) Models based on the Markovian arrival process. IEICE Trans. Commun. E75–B, 1255–1265.Google Scholar
[15] Siegrist, K. (1988) Reliability of systems with Markov transfer of control, II. IEEE Trans. Software Eng. 14, 1478–1480.CrossRefGoogle Scholar