Skip to main content

A polling model with smart customers

Abstract

In this paper we consider a single-server, cyclic polling system with switch-over times. A distinguishing feature of the model is that the rates of the Poisson arrival processes at the various queues depend on the server location. For this model we study the joint queue length distribution at polling epochs and at the server’s departure epochs. We also study the marginal queue length distribution at arrival epochs, as well as at arbitrary epochs (which is not the same in general, since we cannot use the PASTA property). A generalised version of the distributional form of Little’s law is applied to the joint queue length distribution at customer’s departure epochs in order to find the waiting time distribution for each customer type. We also provide an alternative, more efficient way to determine the mean queue lengths and mean waiting times, using Mean Value Analysis. Furthermore, we show that under certain conditions a Pseudo-Conservation Law for the total amount of work in the system holds. Finally, typical features of the model under consideration are demonstrated in several numerical examples.

References

  1. 1.

    Boon, M.A.A.: A polling model with reneging at polling instants. Ann. Oper. Res. (2010, to appear). doi:10.1007/s10479-010-0758-2

  2. 2.

    Boon, M.A.A., Adan, I.J.B.F.: Mixed gated/exhaustive service in a polling model with priorities. Queueing Syst. 63, 383–399 (2009)

    Article  Google Scholar 

  3. 3.

    Borst, S.C.: Polling Systems. CWI Tracts, vol. 115 (1996)

  4. 4.

    Borst, S.C., Boxma, O.J.: Polling models with and without switchover times. Oper. Res. 45(4), 536–543 (1997)

    Article  Google Scholar 

  5. 5.

    Boxma, O.J.: Polling systems. In: From Universal Morphisms to Megabytes: A Baayen Space Odyssey. Liber Amicorum for P.C. Baayen, pp. 215–230. CWI, Amsterdam (1994)

    Google Scholar 

  6. 6.

    Boxma, O.J., Groenendijk, W.P.: Pseudo-conservation laws in cyclic-service systems. J. Appl. Probab. 24(4), 949–964 (1987)

    Article  Google Scholar 

  7. 7.

    Boxma, O.J., Kelbert, M.: Stochastic bounds for a polling system. Ann. Oper. Res. 48, 295–310 (1994)

    Article  Google Scholar 

  8. 8.

    Boxma, O.J., Weststrate, J.A., Yechiali, U.: A globally gated polling system with server interruptions, and applications to the repairman problem. Probab. Eng. Inf. Sci. 7, 187–208 (1993)

    Article  Google Scholar 

  9. 9.

    Boxma, O.J., van Wijk, A.C.C., Adan, I.J.B.F.: Polling systems with a gated-exhaustive discipline. In: ValueTools 2008 (Third International Conference on Performance Evaluation Methodologies and Tools, Athens, Greece, October 20–24, 2008)

  10. 10.

    Boxma, O.J., Bruin, J., Fralix, B.H.: Waiting times in polling systems with various service disciplines. Perform. Eval. 66, 621–639 (2009)

    Article  Google Scholar 

  11. 11.

    Boxma, O.J., Ivanovs, J., Kosiński, K., Mandjes, M.: Lévy-driven polling systems and continuous-state branching processes. Eurandom report 2009-026, Eurandom (2009)

  12. 12.

    Cohen, J.W.: The Single Server Queue, revised edn. North-Holland, Amsterdam (1982)

    Google Scholar 

  13. 13.

    Eisenberg, M.: Queues with periodic service and changeover time. Oper. Res. 20(2), 440–451 (1972)

    Article  Google Scholar 

  14. 14.

    Fuhrmann, S.W.: Performance analysis of a class of cyclic schedules. Technical memorandum 81-59531-1, Bell Laboratories (March 1981)

  15. 15.

    Fuhrmann, S.W., Cooper, R.B.: Stochastic decompositions in the M/G/1 queue with generalized vacations. Oper. Res. 33(5), 1117–1129 (1985)

    Article  Google Scholar 

  16. 16.

    Gong, Y., de Koster, R.: A polling-based dynamic order picking system for online retailers. IIE Trans. 40, 1070–1082 (2008)

    Article  Google Scholar 

  17. 17.

    Ibe, O.C., Trivedi, K.S.: Two queues with alternating service and server breakdown. Queueing Syst. 7, 253–268 (1990)

    Article  Google Scholar 

  18. 18.

    Jain, M., Jain, A.: Working vacations queueing model with multiple types of server breakdowns. Appl. Math. Model. 34(1), 1–13 (2010)

    Article  Google Scholar 

  19. 19.

    Keilson, J., Servi, L.D.: The distributional form of Little’s Law and the Fuhrmann–Cooper decomposition. Oper. Res. Lett. 9(4), 239–247 (1990)

    Article  Google Scholar 

  20. 20.

    Kofman, D., Yechiali, U.: Polling systems with station breakdowns. Perform. Eval. 27–28, 647–672 (1996)

    Google Scholar 

  21. 21.

    Levy, H., Sidi, M.: Polling systems: applications, modeling, and optimization. IEEE Trans. Commun. 38, 1750–1760 (1990)

    Article  Google Scholar 

  22. 22.

    Mandelbaum, A., Yechiali, U.: Optimal entering rules for a customer with wait option at an M/G/1 queue. Manag. Sci. 29(2), 174–187 (1983)

    Article  Google Scholar 

  23. 23.

    Nakdimon, O., Yechiali, U.: Polling systems with breakdowns and repairs. Eur. J. Oper. Res. 149, 588–613 (2003)

    Article  Google Scholar 

  24. 24.

    Resing, J.A.C.: Polling systems and multitype branching processes. Queueing Syst. 13, 409–426 (1993)

    Article  Google Scholar 

  25. 25.

    Shanthikumar, J.G.: On stochastic decomposition in M/G/1 type queues with generalized server vacations. Oper. Res. 36(4), 566–569 (1988)

    Article  Google Scholar 

  26. 26.

    Shogan, A.W.: A single server queue with arrival rate dependent on server breakdowns. Nav. Res. Logist. Q. 26(3), 487–497 (1979)

    Article  Google Scholar 

  27. 27.

    Takagi, H.: Queuing analysis of polling models. ACM Comput. Surv. (CSUR) 20, 5–28 (1988)

    Article  Google Scholar 

  28. 28.

    van der Mei, R.D., Resing, J.A.C.: Analysis of polling systems with two-stage gated service: fairness versus efficiency. In: Mason, L., Drwiega, T., Yan, J. (eds.) Proc. ICT-20—Managing Traffic Performance in Converged Networks: The Interplay Between Convergent and Divergent Forces, pp. 544–555. Springer, Berlin (2007)

    Google Scholar 

  29. 29.

    Vishnevskii, V.M., Semenova, O.V.: Mathematical methods to study the polling systems. Autom. Remote Control 67(2), 173–220 (2006)

    Article  Google Scholar 

  30. 30.

    Winands, E.M.M.: Polling, Production & Priorities. PhD thesis, Eindhoven University of Technology (2007)

  31. 31.

    Winands, E.M.M., Adan, I.J.B.F., van Houtum, G.-J.: Mean value analysis for polling systems. Queueing Syst. 54, 35–44 (2006)

    Article  Google Scholar 

  32. 32.

    Wolff, R.: Stochastic Modeling and the Theory of Queues. Prentice-Hall, Englewood Cliffs (1989)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to M. A. A. Boon.

Additional information

The research was done in the framework of the BSIK/BRICKS project, and of the European Network of Excellence Euro-NF.

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Boon, M.A.A., van Wijk, A.C.C., Adan, I.J.B.F. et al. A polling model with smart customers. Queueing Syst 66, 239–274 (2010). https://doi.org/10.1007/s11134-010-9191-0

Download citation

Keywords

  • Polling
  • Smart customers
  • Varying arrival rates
  • Queue lengths
  • Waiting times
  • Pseudo-conservation law

Mathematics Subject Classification (2000)

  • 60K25
  • 90B22