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Stochastic Recursive Equations with Applications to Queues with Dependent Vacations

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Abstract

We focus on a special class of nonlinear multidimensional stochastic recursive equations in which the coefficients are stationary ergodic (not necessarily independent). Under appropriate conditions, an explicit ergodic stationary solution for these equations is obtained and the convergence to this stationary regime is established. We use these results to analyze several queueing models with vacations. We obtain explicit solutions for several performance measures for the case of general non-independent vacation processes. We finally extend some of these results to polling systems with general vacations.

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References

  1. E. Altman, K. Avratchenkov and C. Barakat, A stochastic model of TCP/IP with stationary random losses, in: Proc. of ACM SIGCOMM, Stockholm, Sweden (August 2000). (See also INRIA Research Report RR-3824.)

  2. E. Altman, K. Avratchenkov, C. Barakat and R. Núñez-Queija, TCP modeling in the presence of nonlinear window growth, in: Proceedings of the 17th International Teletraffic Congress, Salvador da Bahia, Brazil (December 2001).

  3. E. Altman and A. Hordijk, Applications of Borovkov's renovation theory to non-stationary stochastic recursive sequences and their control, Adv. in Appl. Probab. 29 (1997) 388-413.

    Google Scholar 

  4. E. Altman, A. Khamisy and U. Yechiali, On elevator polling with globally gated regime, Queueing Systems 11 (1992) 85-90.

    Google Scholar 

  5. E. Altman, P. Konstantopoulos and Z. Liu, Stability, monotonicity and invariant quantities in general polling systems, Queueing Systems 11 (1992) 35-57.

    Google Scholar 

  6. E. Altman and F. Spieksma, Polling systems: moment stability of station times and central limit theorems, Stochastic Models 12(2) (1996) 307-328.

    Google Scholar 

  7. F. Baccelli and P. Brémaud, Elements of Queueing theory (Springer, 1994).

  8. A.A. Borovkov, Asymptotic Methods in Queueing Theory (Wiley, 1984) (translated from Russian).

  9. A.A. Borovkov and S.G. Foss, Stochastically recursive sequences and their generalizations, Siberian Advances in Mathematics 2(1) (1992) 16-81.

    Google Scholar 

  10. A.A. Borovkov and S.G. Foss, Two ergodicity criteria for stochastically recursive sequences, Acta Appl. Math. 34 (1994) 125-134.

    Google Scholar 

  11. O.J. Boxma, H. Levy and U. Yechiali, Cyclic reservation schemes for efficient operation of multiplequeue single-server systems, Ann. Oper. Res. 35 (1992) 187-208.

    Google Scholar 

  12. A. Brandt, The stochastic equation Y n+1 = A n Y n + B n with stationary coefficients, Adv. in Appl. Probab. 18 (1986).

  13. A. Brandt, P. Franken and B. Lisek, Stationary Stochastic Models (Akademie, Berlin, 1992).

    Google Scholar 

  14. S. Foss and N. Chernova, Ergodic properties of polling systems, Research Report 6/1995, Institute of Mathematics, Novosibirsk (1995).

    Google Scholar 

  15. P. Glasserman and D.D. Yao, Stochastic vector difference equations with stationary coefficients, J. Appl. Prob. 32 (1995) 851-866.

    Google Scholar 

  16. A. Khamisy, E. Altman and M. Sidi, Polling systems with synchronization constraints, Ann. Oper. Res. 35 (1992) 231-267.

    Google Scholar 

  17. H. Levy, M. Sidi and O.J. Boxma, Dominance relations in polling systems, Queueing Systems 6 (1990) 155-172.

    Google Scholar 

  18. R. Loynes, The stability of a queue with non-independent inter-arrival and service times, Proc. Cambridge Philos. Soc. 58(3) (1962) 497-520.

    Google Scholar 

  19. H. Takagi, Analysis of Polling Systems (The MIT Press, Cambridge, MA, 1986).

    Google Scholar 

  20. H. Takagi, Queueing analysis of polling models: an update, in: Stochastic Analysis of Computer and Communications Systems, ed. H. Takagi (Elsevier Science, Amsterdam, 1990) pp. 267-318.

    Google Scholar 

  21. Tedijanto, Exact results for the cyclic service queue with a Bernoulli schedule, Performance Evaluation 11 (1990) 107-115.

    Google Scholar 

  22. U. Yechiali, Analysis and control of polling systems, in: Performance Evaluation of Computer and Communications Systems, eds. L. Donantiello and R. Nelson (Springer, Berlin, 1993) pp. 630-650.

    Google Scholar 

  23. W. Vervaat, On stochastic difference equation and a representation of non-negative infinitely random variables, Adv. in Appl. Probab. 11 (1979) 750-783.

    Google Scholar 

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Authors and Affiliations

  1. INRIA, BP93, 06902, Sophia Antipolis, France

    Eitan Altman

  2. Facultad de Ingenieria, C.E.S.I.M.O., Universidad de Los Andes, Mérida, Venezuela

    Eitan Altman

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  1. Eitan Altman
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Altman, E. Stochastic Recursive Equations with Applications to Queues with Dependent Vacations. Annals of Operations Research 112, 43–61 (2002). https://doi.org/10.1023/A:1020972803727

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