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The joint distribution of sojourn times in finite Markov processes

Part of: Markov processes Operations research and management science

Published online by Cambridge University Press:  01 July 2016

Attila Csenki
Attila Csenki*
Affiliation:
Aston University
*
∗Postal address: Department of Computer Science and Applied Mathematics, Aston University, Aston Triangle, Birmingham B4 7ET, UK.
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Abstract

Rubino and Sericola (1989c) derived expressions for the mth sojourn time distribution associated with a subset of the state space of a homogeneous irreducible Markov chain for both the discrete- and continuous-parameter cases. In the present paper, it is shown that a suitable probabilistic reasoning using absorbing Markov chains can be used to obtain respectively the probability mass function and the cumulative distribution function of the joint distribution of the first m sojourn times. A concise derivation of the continuous-time result is achieved by deducing it from the discrete-time formulation by time discretization. Generalizing some further recent results by Rubino and Sericola (1991), the joint distribution of sojourn times for absorbing Markov chains is also derived. As a numerical example, the model of a fault-tolerant multiprocessor system is considered.

Keywords

MARKOV CHAINMARKOV REWARD PROCESSSOJOURN TIMETRANSIENT ANALYSISRELIABILITY MODELLING

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 90B25: Reliability, availability, maintenance, inspection
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 1 , March 1992 , pp. 141 - 160
Copyright
Copyright © Applied Probability Trust 1992 

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References

Bhat, U. N. (1984) Elements of Applied Stochastic Processes. Wiley, New York.Google Scholar
Csenki, A. (1990a) The joint distribution of sojourn times in finite Markov processes. Technical Report, Aston University, March 1990.Google Scholar
Csenki, A. (1990b) Some further results in the theory of sojourn times in finite Markov processes. Technical Report, Aston University, June 1990.Google Scholar
Darroch, J. N. and Morris, K. W. (1967) Some passage-time generating functions for discrete-time and continuous-time finite Markov chains. J Appl. Prob. 4, 496–507.CrossRefGoogle Scholar
Hill, D. R. (1988) Experiments in Computational Matrix Algebra. The Random House/Birkhauser Mathematics Series, Random House, New York.Google Scholar
Kemeny, J. G. and Snell, J. L. (1976) Finite Markov Chains. Springer-Verlag, New York.Google Scholar
Kohlas, J. (1982) Stochastic Methods of Operations Research. Cambridge University Press, Cambridge.Google Scholar
Matlab, (1989) MATLAB for Macintosh Computers User's Guide , MathWorks, South Natick, MA.Google Scholar
Moler, C. B. and Van Loan, C. F. (1979) Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev. 20, 801–836.CrossRefGoogle Scholar
Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Rubino, G. and Sericola, B. (1989a) Distribution of operational times in fault-tolerant systems modeled by semi-Markov reward processes. Technical Report 996, INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France, March 1989.Google Scholar
Rubino, G. and Sericola, B. (1989b) Accumulated reward over the n first operational periods in fault-tolerant computing systems. Technical Report 1028, INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France, May 1989.Google Scholar
Rubino, G. and Sericola, B. (1989C) Sojourn times in finite Markov processes. J. Appl. Prob. 26, 744–756.CrossRefGoogle Scholar
Rubino, G. and Sericola, B. (1991) Successive operational periods as measures of dependability. In Proc. 1st IFIP WG 10.4 International Working Conference on Dependable Computing for Critical Applications, University of California, Santa Barbara, Ca., 23–25 August 1989 , ed. Avizienis, A. and Laprie, J.-C. Dependable Computing and Fault-Tolerant Systems 4, 239–254. Springer-Verlag, New York.Google Scholar
Trivedi, K., Reibman, A. and Smith, R. (1988) Transient analysis of Markov and Markov reward models. In Computer Performance and Reliability , ed. Iazeolla, G., Courtois, P. J. and Boxma, O. J., pp. 535–545. Elsevier, Amsterdam.Google Scholar