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Generalized product-form stationary distributions for Markov chains in random environments with queueing applications

Part of: Special processes Probabilistic methods, simulation and stochastic differential equations Markov processes

Published online by Cambridge University Press:  01 July 2016

Antonis Economou
Antonis Economou*
Affiliation:
University of Athens
*
∗ Postal address: Department of Mathematics, University of Athens, Panepistemiopolis, Athens 15784, Greece. Email address: [email protected]
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Abstract

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Consider a continuous-time Markov chain evolving in a random environment. We study certain forms of interaction between the process of interest and the environmental process, under which the stationary joint distribution is tractable. Moreover, we obtain necessary and sufficient conditions for a product-form stationary distribution. A number of examples that illustrate the applicability of our results in queueing and population growth models are also included.

Keywords

Continuous-time Markov chainrandom environmentqueueingproduct-form stationary distributionpartial balancepotentialresolventESTA property

MSC classification

Primary: 60J22: Computational methods in Markov chains 65C40: Computational Markov chains
Secondary: 60K37: Processes in random environments 60J27: Continuous-time Markov processes on discrete state spaces
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 1 , March 2005 , pp. 185 - 211
Copyright
Copyright © Applied Probability Trust 2005 

References

Anisimov, V. and Sztrik, J. (1989). Asymptotic analysis of some complex renewable systems operating in random environments. Europ. J. Operat. Res. 41, 162–168.Google Scholar
Bourgin, R. D. and Cogburn, R. (1981). On determining absorption probabilities for Markov chains in random environments. Adv. Appl. Prob. 13, 369–387.Google Scholar
Chang, C.-S. and Nelson, R. (1993). Perturbation analysis of the M/M/1 queue in a Markovian environment via the matrix-geometric method. Commun. Statist. Stoch. Models 9, 233–246.Google Scholar
Chao, X., Pinedo, M. and Miyazawa, M. (1999). Queueing Networks: Negative Customers, Signals and Product Form. John Wiley, New York.Google Scholar
Çinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Cogburn, R. (1991). On the central limit theorem for Markov chains in random environments. Ann. Prob. 19, 587–604.Google Scholar
Cogburn, R. and Torrez, W. C. (1981). Birth and death processes with random environments in continuous time. J. Appl. Prob. 18, 19–30.CrossRefGoogle ScholarPubMed
El-Taha, M. and Stidham, S. Jr. (1999). Sample-Path Analysis of Queueing Systems. Kluwer, Boston, MA.CrossRefGoogle Scholar
Fakinos, D. (1982). The generalized M/G/k blocking system with heterogeneous servers. J. Operat. Res. Soc. 33, 801–809.Google Scholar
Fakinos, D. (1991). Insensitivity of generalized semi-Markov processes evolving in a random environment. J. Operat. Res. Soc. 42, 1111–1115.Google Scholar
Fakinos, D. and Economou, A. (1998). Overall station balance and decomposability for non-Markovian queueing networks. Adv. Appl. Prob. 30, 870–887.Google Scholar
Falin, G. (1996). A heterogeneous blocking system in a random environment. J. Appl. Prob. 33, 211–216.CrossRefGoogle Scholar
Gaver, D. P., Jacobs, P. A. and Latouche, G. (1984). Finite birth-and-death models in randomly changing environments. Adv. Appl. Prob. 16, 715–731.Google Scholar
Gupta, P. L. and Gupta, R. D. (1990). A bivariate random environmental stress model. Adv. Appl. Prob. 22, 501–503.Google Scholar
Hambly, B. (1992). On the limiting distribution of a supercritical branching process in a random environment. J. Appl. Prob. 29, 499–518.Google Scholar
Helm, W. E. and Waldmann, K.-H. (1984). Optimal control of arrivals to multiserver queues in a random environment. J. Appl. Prob. 21, 602–615.Google Scholar
Karlin, S. and McGregor, J. L. (1965). Ehrenfest urn models. J. Appl. Prob. 2, 352–376.Google Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, New York.Google Scholar
Lefèvre, C. and Milhaud, X. (1990). On the association of the lifelengths of components subjected to a stochastic environment. Adv. Appl. Prob. 22, 961–964.Google Scholar
Melamed, B. and Yao, D. D. (1995). The ASTA property. In Advances in Queueing (Prob. Stoch. Ser.), ed. Dshalalow, J. H., CRC, Boca Raton, FL, pp. 195–224.Google Scholar
Mitrani, I. and Chakka, R. (1995). Spectral expansion solution for a class of Markov models: application and comparison with the matrix-geometric method. Perf. Eval. 23, 241–260.Google Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach (Johns Hopkins Ser. Math. Sci. 2). Johns Hopkins University Press, Baltimore, MD.Google Scholar
Norris, J. R. (1997). Markov Chains. Cambridge University Press.CrossRefGoogle Scholar
Núñez-Queija, R. (1997). Steady-state analysis of a queue with varying service rate. Res. Rep. PNA-R9712, CWI.Google Scholar
O' Cinneide, C. A. and Purdue, P. (1986). The M/M/∞ queue in a random environment. J. Appl. Prob. 23, 175–184.Google Scholar
Posner, M. J. M. and Zuckerman, D. (1990). Optimal R & D programs in a random environment. J. Appl. Prob. 27, 343–350.Google Scholar
Serfozo, R. (1999). Introduction to Stochastic Networks. Springer, New York.Google Scholar
Stidham, S. Jr. and El-Taha, M. (1995). Sample-path techniques in queueing theory. In Advances in Queueing (Prob. Stoch. Ser.), ed. Dshalalow, J. H., CRC, Boca Raton, FL, pp. 119–166.Google Scholar
Sztrik, J. (1987). On the heterogeneous M/G/n blocking system in a random environment. J. Operat. Res. Soc. 38, 57–63.Google Scholar
Tsitsiashvili, G. Sh., Osipova, M. A., Koliev, N. V. and Baum, D. (2002). A product theorem for Markov chains with application to PF-queueing networks. Ann. Operat. Res. 113, 141–154.Google Scholar
Van Assche, W., Parthasarathy, P. R. and Lenin, R. B. (1999). Spectral representation of four finite birth and death processes. Math. Scientist 24, 105–112.Google Scholar
Van Dijk, N. M. (1993). Queueing Networks and Product Forms: A System Approach. John Wiley, Chichester.Google Scholar
Yamazaki, G. and Miyazawa, M. (1995). Decomposability in queues with background states. Queueing Systems 20, 453–469.Google Scholar
Zhu, Y. (1991). A Markov-modulated M/M/1 queue with group arrivals. Queueing Systems 8, 255–263.Google Scholar
Zhu, Y. (1994). Markovian queueing networks in a random environment. Operat. Res. Lett. 15, 11–17.Google Scholar