Skip to main content Accessibility help
×
×
Home

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

  • Antonis Economou (a1)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

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

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

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

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

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

Copyright

Corresponding author

Postal address: Department of Mathematics, University of Athens, Panepistemiopolis, Athens 15784, Greece. Email address: aeconom@math.uoa.gr

References

Hide All
Anisimov, V. and Sztrik, J. (1989). Asymptotic analysis of some complex renewable systems operating in random environments. Europ. J. Operat. Res. 41, 162168.
Bourgin, R. D. and Cogburn, R. (1981). On determining absorption probabilities for Markov chains in random environments. Adv. Appl. Prob. 13, 369387.
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, 233246.
Chao, X., Pinedo, M. and Miyazawa, M. (1999). Queueing Networks: Negative Customers, Signals and Product Form. John Wiley, New York.
Çinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.
Cogburn, R. (1991). On the central limit theorem for Markov chains in random environments. Ann. Prob. 19, 587604.
Cogburn, R. and Torrez, W. C. (1981). Birth and death processes with random environments in continuous time. J. Appl. Prob. 18, 1930.
El-Taha, M. and Stidham, S. Jr. (1999). Sample-Path Analysis of Queueing Systems. Kluwer, Boston, MA.
Fakinos, D. (1982). The generalized M/G/k blocking system with heterogeneous servers. J. Operat. Res. Soc. 33, 801809.
Fakinos, D. (1991). Insensitivity of generalized semi-Markov processes evolving in a random environment. J. Operat. Res. Soc. 42, 11111115.
Fakinos, D. and Economou, A. (1998). Overall station balance and decomposability for non-Markovian queueing networks. Adv. Appl. Prob. 30, 870887.
Falin, G. (1996). A heterogeneous blocking system in a random environment. J. Appl. Prob. 33, 211216.
Gaver, D. P., Jacobs, P. A. and Latouche, G. (1984). Finite birth-and-death models in randomly changing environments. Adv. Appl. Prob. 16, 715731.
Gupta, P. L. and Gupta, R. D. (1990). A bivariate random environmental stress model. Adv. Appl. Prob. 22, 501503.
Hambly, B. (1992). On the limiting distribution of a supercritical branching process in a random environment. J. Appl. Prob. 29, 499518.
Helm, W. E. and Waldmann, K.-H. (1984). Optimal control of arrivals to multiserver queues in a random environment. J. Appl. Prob. 21, 602615.
Karlin, S. and McGregor, J. L. (1965). Ehrenfest urn models. J. Appl. Prob. 2, 352376.
Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, New York.
Lefèvre, C. and Milhaud, X. (1990). On the association of the lifelengths of components subjected to a stochastic environment. Adv. Appl. Prob. 22, 961964.
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. 195224.
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, 241260.
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.
Norris, J. R. (1997). Markov Chains. Cambridge University Press.
Núñez-Queija, R. (1997). Steady-state analysis of a queue with varying service rate. Res. Rep. PNA-R9712, CWI.
O' Cinneide, C. A. and Purdue, P. (1986). The M/M/∞ queue in a random environment. J. Appl. Prob. 23, 175184.
Posner, M. J. M. and Zuckerman, D. (1990). Optimal R & D programs in a random environment. J. Appl. Prob. 27, 343350.
Serfozo, R. (1999). Introduction to Stochastic Networks. Springer, New York.
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. 119166.
Sztrik, J. (1987). On the heterogeneous M/G/n blocking system in a random environment. J. Operat. Res. Soc. 38, 5763.
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, 141154.
Van Assche, W., Parthasarathy, P. R. and Lenin, R. B. (1999). Spectral representation of four finite birth and death processes. Math. Scientist 24, 105112.
Van Dijk, N. M. (1993). Queueing Networks and Product Forms: A System Approach. John Wiley, Chichester.
Yamazaki, G. and Miyazawa, M. (1995). Decomposability in queues with background states. Queueing Systems 20, 453469.
Zhu, Y. (1991). A Markov-modulated M/M/1 queue with group arrivals. Queueing Systems 8, 255263.
Zhu, Y. (1994). Markovian queueing networks in a random environment. Operat. Res. Lett. 15, 1117.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed