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Monotonicity and stability of periodic polling models

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Abstract

This paper deals with the stability of periodic polling models with a mixture of service policies. Customers arrive according to independent Poisson processes. The service times and the switchover times are independent with general distributions. The necessary and sufficient condition for the stability of such polling systems is established. The proof is based on the stochastic monotonicity of the state process at the polling instants. The stability of only a subset of the queues is also analyzed and, in case of heavy traffic, the order of explosion of the queues is given. The results are valid for a model with set-up times, and also when there is a local priority rule at the queues.

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References

  1. E. Altman, P. Konstantopoulos and Z. Liu, Stability, monotonicity and invariant quantities in general polling systems, Queueing Systems 11 (1992) 35–58.

    Google Scholar 

  2. S. Asmussen,Applied Probability and Queues (Wiley, New York, 1987).

    Google Scholar 

  3. P. Billingsley,Ergodic Theory and Information (Wiley, New York, 1965).

    Google Scholar 

  4. O.J. Boxma and W.P. Groenendijk, Pseudo-conservation laws in cyclic queues, J. Appl. Prob. 24 (1990) 949–964.

    Google Scholar 

  5. M. Eisenberg, Queues with periodic service and changeover time, Oper. Res. 20 (1972) 440–451.

    Google Scholar 

  6. L. Fournier and Z. Rosberg, Expected waiting times in polling systems under priority disciplines, Queueing Systems 9 (1991) 419–440.

    Google Scholar 

  7. C. Fricker and M.R. Jaïbi, Stability of polling models with Markovian routing, Techn. Report (1993).

  8. S.W. Fuhrmann and R.B. Cooper, Stochastic decomposition in theM/G/1 queue with generalized vacations, Oper. Res. 33 (1985) 1117–1129.

    Google Scholar 

  9. L. Georgiadis and W. Szpankowski, Stability of token passing rings, Queueing Systems 11 (1992) 7–34.

    Google Scholar 

  10. P.J. Kuehn, Multiqueue systems with nonexhaustive cyclic service, Bell Syst. Tech. J. 58 (1979) 671–698.

    Google Scholar 

  11. H. Levy, M. Sidi and O.J. Boxma, Dominance relations in polling systems, Queueing Systems 6 (1990) 155–171.

    Google Scholar 

  12. J. Neveu, Construction de files d'attente stationnaires, Lect. Notes on Control and Information Sci. 60 (1983) 31–41.

    Google Scholar 

  13. D. Stoyan,Comparison Methods for Queues and Other Stochastic Models (Wiley, New York, 1983).

    Google Scholar 

  14. H. Takagi, Queueing analysis of polling systems, in:Stochastic Analysis of Computer and Communication Systems, ed. H. Takagi (North-Holland, Amsterdam, 1990) pp. 267–318.

    Google Scholar 

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Authors and Affiliations

  1. Domaine de Voluceau, INRIA, BP. 105, 78153, Le Chesnay Cedex, France

    C. Fricker

  2. Tilburg University, P.O. Box 90153, 5000, LE Tilburg, The Netherlands

    M. R. Jaibi

Authors
  1. C. Fricker
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  2. M. R. Jaibi
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Additional information

This work was supported in part by a Fellowship of the Netherlands Organization for Scientific Research NWO-ECOZOEK.

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Fricker, C., Jaibi, M.R. Monotonicity and stability of periodic polling models. Queueing Syst 15, 211–238 (1994). https://doi.org/10.1007/BF01189238

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  • DOI: https://doi.org/10.1007/BF01189238

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