Waiting Times in Polling Systems with Markovian Server Routing

  • O. J. Boxma
  • J. A. Weststrate
Part of the Informatik-Fachberichte book series (INFORMATIK, volume 218)

Abstract

This study is devoted to a queueing analysis of polling systems with a probabilistic server routing mechanism. A single server serves a number of queues, switching between the queues according to a discrete time parameter Markov chain. The switchover times between queues are nonneghgible. It is observed that the total amount of work in this Markovian polling system can be decomposed into two independent parts, viz., (i) the total amount of work in the corresponding system without switchover times and (ii) the amount of work in the system at some epoch covered by a switching interval. This work decomposition leads to a pseudoconservation law for mean waiting times, i.e., an exact expression for a weighted sum of the mean waiting times at all queues. The results generalize known results for polling systems with strictly cyclic service.

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References

  1. 1.
    Boxma, O.J. (1989). Workloads and waiting times in single-server systems with multiple customer classes. To appear in Queueing Systems.Google Scholar
  2. 2.
    Boxma, O. J., Groenendijk, W.P. (1987). Pseudo-conservation laws in cyclic-service systems. J. Appl. Prob. 24, 949–964.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Boxma, O.J., Groenendijk, W.P., Weststrate, J.A. (1988). A pseudoconservation law for service systems with a polling table. Report Centre for Mathematics and Computer Science, Amsterdam; to appear in IEEE Trans. Commun.Google Scholar
  4. 4.
    Boxma, O.J., Meister, B. (1987). Waiting-time approximations for cyclic-service systems with switchover times, Performance Evaluation 7, 299–308.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chung, K.L. (1967). Markov Chains with Stationary Transition Probabilities (Springer, Berlin; 2nd ed.).MATHGoogle Scholar
  6. 6.
    Cinlar, E. (1975). Introduction to Stochastic Processes (Prentice Hall, Englewood Cliffs, NJ).MATHGoogle Scholar
  7. 7.
    Cohen, J.W. (1982). The Single Server Queue (North-Holland, Amsterdam; 2nd ed.).MATHGoogle Scholar
  8. 8.
    Groenendijk, W.P. (1988). Waiting-time approximations for cyclic-service systems with mixed service strategies, in: M. Bonatti (ed.), Proceedings ITC-12 (North-Holland, Amsterdam).Google Scholar
  9. 9.
    Keilson, J., Servi, L.D. (1986). Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules. J. Appl. Prob. 23, 790–802.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kelly, F.P. (1979). Reversibility and Stochastic Networks (Wiley, New York).MATHGoogle Scholar
  11. 11.
    Kleinrock, L., Levy, H. (1988). The analysis of random polling systems. Oper. Res. 36, 716–732.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Levy, H. (1984). Non-Uniform Structures and Synchronization Patterns in Shared-Channel Communication Networks. CSD-840049, Computer Science Department, University of California, Los Angeles, Ph.D. Dissertation.Google Scholar
  13. 13.
    Levy, H., Sidi, M. (1988). Correlated arrivals in polling systems. Report Department of Computer Science, Tel Aviv University.Google Scholar
  14. 14.
    Mitrani, L, Adams, J.L., Falconer, R.M. (1986). A modelling study of the Orwell ring protocol. In: Teletraffic Analysis and Computer Performance Evaluation, eds. O.J. Boxma, J.W. Cohen and H.C. Tijms (North-Holland, Amsterdam), pp. 429–438.Google Scholar
  15. 15.
    E. Seneta (1981). Non-negative Matrices and Markov Chains (Springer, New York; 2nd ed.).MATHGoogle Scholar
  16. 16.
    Takagi, H. (1986). Analysis of Polling Systems (The MIT Press, Cambridge, MA).Google Scholar
  17. 17.
    Takagi, H. (1988). Queuing analysis of polling models. ACM Comput. Surveys 20, 5–28.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Tedijanto (1988). Exact results for the cyclic-service queue with a Bernoulli schedule. Report Electrical Engineering Department and Systems Research Center, University of Maryland.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • O. J. Boxma
    • 1
    • 2
  • J. A. Weststrate
    • 2
  1. 1.Centre for Mathematics and Computer ScienceAmsterdamThe Netherlands
  2. 2.Faculty of EconomicsTilburg UniversityTilburgThe Netherlands

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