Skip to main content
SpringerLink
Log in

Waiting Times in Polling Systems with Markovian Server Routing

  • Conference paper
Messung, Modellierung und Bewertung von Rechensystemen und Netzen

Part of the book series: Informatik-Fachberichte ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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. Boxma, O. J., Groenendijk, W.P. (1987). Pseudo-conservation laws in cyclic-service systems. J. Appl. Prob. 24, 949–964.

    Article  MathSciNet  MATH  Google Scholar 

  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. Boxma, O.J., Meister, B. (1987). Waiting-time approximations for cyclic-service systems with switchover times, Performance Evaluation 7, 299–308.

    Article  MathSciNet  Google Scholar 

  5. Chung, K.L. (1967). Markov Chains with Stationary Transition Probabilities (Springer, Berlin; 2nd ed.).

    MATH  Google Scholar 

  6. Cinlar, E. (1975). Introduction to Stochastic Processes (Prentice Hall, Englewood Cliffs, NJ).

    MATH  Google Scholar 

  7. Cohen, J.W. (1982). The Single Server Queue (North-Holland, Amsterdam; 2nd ed.).

    MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  10. Kelly, F.P. (1979). Reversibility and Stochastic Networks (Wiley, New York).

    MATH  Google Scholar 

  11. Kleinrock, L., Levy, H. (1988). The analysis of random polling systems. Oper. Res. 36, 716–732.

    Article  MathSciNet  MATH  Google Scholar 

  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. Levy, H., Sidi, M. (1988). Correlated arrivals in polling systems. Report Department of Computer Science, Tel Aviv University.

    Google Scholar 

  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. E. Seneta (1981). Non-negative Matrices and Markov Chains (Springer, New York; 2nd ed.).

    MATH  Google Scholar 

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

    Google Scholar 

  17. Takagi, H. (1988). Queuing analysis of polling models. ACM Comput. Surveys 20, 5–28.

    Article  MathSciNet  MATH  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. Centre for Mathematics and Computer Science, P.O. Box 4079, 1009 AB, Amsterdam, The Netherlands

    O. J. Boxma

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

    O. J. Boxma & J. A. Weststrate

Authors
  1. O. J. Boxma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. Weststrate
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Betriebssysteme und Rechnerverbund, TU Braunschweig, Postfach 3329, D-3300, Braunschweig, Deutschland

    G. Stiege  & J. S. Lie  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Boxma, O.J., Weststrate, J.A. (1989). Waiting Times in Polling Systems with Markovian Server Routing. In: Stiege, G., Lie, J.S. (eds) Messung, Modellierung und Bewertung von Rechensystemen und Netzen. Informatik-Fachberichte, vol 218. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75079-3_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-75079-3_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51713-9

  • Online ISBN: 978-3-642-75079-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation