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Ergodicity aspects of multidimensional Markov chains with application to computer communication system analysis

  • Wojciech Szpankowski
Stationarity And Ergodicity 2
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 60)

Abstract

Ergodicity and nonergodicity of multidimensional Markov chains are discussed. We investigate ergodic and nonergodic Markov chains by the means of Lyapunov function method and so called comparison tests method. In the former case two theorems being a generalization of Pakes and Kaplan theorems are presented. Moreover, for multidimensional Markov chains we propose an explicite form for a simple Lyapunov function. On the other hand, comparison tests investigate ergodicity/nonergodicity of M-dimensional Markov chains by the means of appropriate ergodic/nonergodic K-dimensional Markov chains, where K is smaller than M. These results are applied to analysis of some computer communication systems.

Keywords

Markov Chain Lyapunov Function Queue Length Stochastic Order Packet Switching 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [CHU]
    Chung, K., Markov chains with stationary transition probabilities, Springer-Verlag, 1960.Google Scholar
  2. [FAL]
    Falin, G., About ergodicity of a multiaccess system, Tehnischeskaja Kibernetika, 4, 1981, pp.126–131 / in Russian /Google Scholar
  3. [FAY]
    Fayolle, G., Gelenbe, E., Labetoulle, J., Stability and optimal control of the packet switching broadcast channel, J. ACM, vol.24, No 3, July 1977, pp.375–386Google Scholar
  4. [FOS]
    Foster, F., On the stochastic matrices associated with certain queueing processes, Ann. Math. Statist., 24, 1953, pp. 355–360Google Scholar
  5. [JAC]
    Jacson, J., Networks of waiting times, Oper. Res., No 5, 1957 pp. 518–521Google Scholar
  6. [KAL]
    Kalashnikov, W., Qualitative analysis of complex systems by Lyapunov functions, Moscow 1978 / in Russian /Google Scholar
  7. [KAM]
    Kamae, T., Krengel, U., O'Brien, G., Stochastic inequalities on partially ordered spaces, Ann. Probab., 5, No 6 December 1977, pp.899–912Google Scholar
  8. [KAP]
    Kaplan, M., A sufficient condition for nonergodicity of a Markov chain, IEEE Trans. on Information Theory, vol.IT-25, No 4, July 1979, pp.470–471Google Scholar
  9. [KEM]
    Kemmeny, J., Snell, J., Knapp, A., Denumerable Markov chains, D. Van Nostrand Company, 1966Google Scholar
  10. [KLE]
    Kleinrock, L., Lam, S., Packet switching in a multiaccess broadcast channel: Performance evaluation, IEEE Trans. on Commun., COM-23, No 4, April 1975, pp.410–422Google Scholar
  11. [KAM]
    Kamal, S., Mahmoud, S., A study of user's buffer variations in a random access satellite channels, IEEE Trans. on Commun., vol. COM-27, No 6, June 1976, pp.857–868Google Scholar
  12. [MAL]
    Malyshev, W., A classification of two-dimensional Markov chains and pice-linear martingals, Doklady Akademii Nauk USSR, 3, 1972, pp.526–528 / in Russian /Google Scholar
  13. [OLD]
    Older, B., Qualitative analysis of congestions — Sensitive routing, Int. Symp. on Flow Control in Comp. Networks, Versailles 1979, pp.131–154Google Scholar
  14. [PAK]
    Pakes, A., Some conditions for ergodicity and recurrence of Markov chains, Oper. Res., 17, 1969, pp.1058–1061Google Scholar
  15. [SAA]
    Saadawi, T., Ephremides, A., Analysis, stability and optimization of slotted ALOHA with a finite number of buffered users, IEEE Trans. on Autom. Control, vol.AC-26, No 3, June 1981, pp.680–689Google Scholar
  16. [STO]
    Stoyan, D., Qualitative Eigenschaften und Abschatzungen Stochastischer Modelle, Akademie-Verlag-Berlin 1977Google Scholar
  17. [SZP1]
    Szpankowski, W., Analysis and stability considerations in a reservation system, IEEE Trans. on Commun, vol.COM-31, No 5, May 1983Google Scholar
  18. [SZP2]
    Szpankowski, W., Some sufficient conditions for nonergodicity of a Markov chain, to be publishedGoogle Scholar
  19. [TOB]
    Tobagi, F., Multiaccess protocols in packet communication systems, IEEE Trans on Commun., vol.COM-28, No 4, April 1980, pp. 486–488Google Scholar
  20. [TSY]
    Tsybakov, B., Mikhaikov, W., Ergodicity of slotted ALOHA system, Problemy Peredachii Informatsii, vol.15, No 4, 1979, pp.73–87 / in Russian /Google Scholar
  21. [WIT]
    Whitt, W., Comparing counting processes and queues, Adv. Appl. Prob., 13, 1981, pp.207–220Google Scholar
  22. [WON]
    Wong, E., Stochastic processes in information and dynamical systems, McGraw-Hill 1971Google Scholar
  23. [YEM]
    Yemini, Y., Kleinrock, L., On general rule for access control or, silence is golden..., Proc. of Int. Symp. on Flow Control in Comp. Networks, Versailles 1979, pp.335–347Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Wojciech Szpankowski
    • 1
  1. 1.Technical University of GdanskGdanskPoland

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