Annals of Operations Research

, Volume 35, Issue 3, pp 155–186 | Cite as

Performance evaluation of polling systems by means of the power-series algorithm

  • J. P. C. Blanc
Section III Polling Systems


Polling systems are widely used to model communication networks with several classes of messages, a single transmission channel and a collision-free access prolocol. However, they can only be analysed exactly for some special service disciplines. The power-series algorithm provides a means for the numerical analysis of polling systems with a moderate number of stations, for a wide variety of access protocols. This paper contains a general description of the power-series algorithm, with emphasis on the application to a general class of polling systems with Poisson arrival streams, with Coxian service and switching time distributions, with infinite buffers, with a fixed periodic visit order, and with a Bernoulli schedule for each visit to a station. The applicability and the complexity of the algorithm are discussed for several more service disciplines for polling systems.


Time Distribution Special Service Switching Time Polling System Moderate Number 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J.E. Baker and I. Rubin, Polling with a general-service order table, IEEE Trans. Commun. COM. 35(1987)283–288.Google Scholar
  2. [2]
    J.P.C. Blanc, A note on waiting times in systems with queues in parallel, J. Appl. Probab. 24(1987)540–546.Google Scholar
  3. [3]
    J.P.C. Blanc, On a numerical method for calculating state probabilities for qucucing systems with more than one waiting line, J. Comput. Appl. Math. 20(1987)119–125.Google Scholar
  4. [4]
    J.P.C. Blanc, A numerical study of a coupled processor model, in:Computer Performance and Reliability, ed. G. Iazcolla, P.J. Courtois and O.J. Boxma (North-Holland, Amsterdam, 1988), pp. 289–303.Google Scholar
  5. [5]
    J.P.C. Blanc, A numerical approach to cyclic-service queueing models, Queucing Systems 6(1990)173–188.Google Scholar
  6. [6]
    J.P.C. Blanc, The power-series algorithm applied to the shortest-queue model, Report FEW 379, Department of Economics, Tilburg University (1989), Oper. Res. (1992), to appear.Google Scholar
  7. [7]
    J.P.C. Blanc, Cyclic polling systems: Limited service versus Bernoulli schedules, Report FEW 422, Department of Economics, Tilburg University (1990).Google Scholar
  8. [8]
    J.P.C. Blanc, The power-series algorithm applied to cyclic polling systems, Report FEW 445, Department of Economics, Tilburg University (1990), Stoch. Models 7(1991), to appear.Google Scholar
  9. [9]
    O.J. Boxma, Workloads and waiting times in single-server systems with multiple customer classes, Qucucing Systems 5(1989)185–214.Google Scholar
  10. [10]
    O.J. Boxma, W.P. Groenendijk and J.A. Weststrate, A pseudo-conservation law for service systems with a polling table, IEEE Trans. Commun. COM-38(1990)1865–1870.Google Scholar
  11. [11]
    O.J. Boxma and J.A. Weststrate, Waiting times in polling systems with Markovian server routing. in:Messung, Modellierung und Bewertung von Rechensystemen und Netzen, ed. G. Stiege and J.S. Lie (Springer, Berlin, 1989), pp. 89–104.Google Scholar
  12. [12]
    C. Brezinski,Accélération de la Convergence en Analyse Numérique, Lecture Notes in Mathematics 584 (Springer, Heidelberg, 1977).Google Scholar
  13. [13]
    J.W. Cohen, A two-queue model with semi-exhaustive alternating service, in:Performance '87, ed. P.-J. Courtois and G. Latouche (North-Holland, Amsterdam, 1988), pp. 19–37.Google Scholar
  14. [14]
    J.W. Cohen, A two-queue, one-server model with priority for the longer queue, Queueing Systems 2(1987)261–283.Google Scholar
  15. [15]
    R.B. Cooper, Queues served in cyclic order: Waiting times, Bell. Syst. Tech. J. 49(1970)399–413.Google Scholar
  16. [16]
    M. Eisenberg, Queues with periodic service and changcover times, Oper. Res. 20(1972)440–451.Google Scholar
  17. [17]
    G. Hooghiemstra, M. Keane and S. van de Ree, Power series for stationary distributions of coupled processors models, SIAM J. Appl. Math. 48(1988)1159–1166.Google Scholar
  18. [18]
    N.K. Jaiswal,Priority Queues (Academic Press, New York, 1968).Google Scholar
  19. [19]
    L. Kleinrock and H. Levy, The analysis of random polling systems, Oper. Res. 36(1988)716–732.Google Scholar
  20. [20]
    G.P. Klimov and G.K. Mishkoy,Priority Service Systems with Orientation (Moscow University Press, Moscow, 1979), in Russian.Google Scholar
  21. [21]
    P.J. Kühn, Multi-queue systems with non-exhaustive cyclic service, Bell Syst. Tech. J. 58(1979)671–698.Google Scholar
  22. [22]
    H. Levy and M. Sidi, Polling systems: Applications, modeling, and optimization, IEEE Trans. Commun. COM-38(1990)1750–1760.Google Scholar
  23. [23]
    D. Sarkar and W.I. Zangwill, Expected waiting time for non-symmetric cyclic queueing systems — exact results and applications, Manag. Sci. 35(1989)1463–1474.Google Scholar
  24. [24]
    L.D. Servi, Average delay approximation ofM/G/I cyclic service queues with Bernoulli schedules, IEEE J. Sel. Areas Comm. SAC-4(1986)813–822.Google Scholar
  25. [25]
    H. Takagi,Analysis of Polling Systems (The MIT Press, Cambridge, MA, 1986).Google Scholar
  26. [26]
    H. Takagi, Queueing analysis of polling models: An update, in:Stochastic Analysis of Computer and Communication Systems, ed. H. Takagi (North-Holland, Amsterdam, 1990), pp. 267–318.Google Scholar
  27. [27]
    Tedijanto, Exact results for the cyclic-service queue with a Bernoulli schedule, Perf. Eval. 11(1990)107–115.Google Scholar
  28. [28]
    P. Wynn, On the convergence and stability of the epsilon algorithm, SIAM J. Numer. Anal. 3(1966)91–122.Google Scholar

Copyright information

© J.C. Baltzer AG, Scientific Publishing Company 1992

Authors and Affiliations

  • J. P. C. Blanc
    • 1
  1. 1.Faculty of EconomicsTilburg UniversityTilburgThe Netherlands

Personalised recommendations