Skip to main content Accessibility help


  • Bruno Gaujal (a1), Arie Hordijk (a2) and Dinard van der Laan (a3)


In this article, we consider deterministic (both fluid and discrete) polling systems with N queues with infinite buffers and we show how to compute the best polling sequence (minimizing the average total workload). With two queues, we show that the best polling sequence is always periodic when the system is stable and forms a regular sequence. The fraction of time spent by the server in the first queue is highly noncontinuous in the parameters of the system (arrival rate and service rate) and shows a fractal behavior. Moreover, convexity properties are shown and are used in a generalization of the computation of the optimal control policy (in open loop) for the stochastic exponential case.



Hide All


Altman, E., Gaujal, B., & Hordijk, A. (2000). Optimal open-loop control of vacations, polling and service assignment. Queueing Systems 36: 303325.
Altman, E., Gaujal, B., & Hordijk, A. (2003). Discrete-event control of stochastic networks: Multimodularity and regularity. New York: Springer-Verlag.
Arian, Y. & Levy, Y. (1992). Algorithms for generalized round robin routing. Operations Research Letters 12: 313319.
Borst, S. (1994). Polling systems. PhD thesis, Katholieke Universiteit Brabant, Tilburg.
Boxma, O.J., Levy, H., & Weststrate, J.A. (1991). Efficient vist frequencies for polling tables: Minimization of waiting cost. Queueing Systems 9: 133162.
Boxma, O.J. & Tagaki, H. (eds.) (1992). Queueing Systems [Special issue on polling systems].
Coffman, E.G., Puhalskii, A.A., & Reiman, M.I. (1995). Polling systems with zero switchover times: A heavy-traffic averaging principle. Annals of Applied Probability 5: 681719.
Combé, M.B. & Boxma, O.J. (1994). Optimization of static traffic allocation policies. Theoretical Computer Science 125: 1743.
Gaujal, B. & Hyon, E. (2000). Optimal routing policy in two deterministic queues. Calculateurs Parallèles 13: 601634.
Gaujal, B., Hyon, E., & Jean-Marie, A. (2004). Optimal open-loop routing in two parallel queues with exponential service times. In IEEE, Wodes, Reims, 2004. Long version available from
Hardy, G.H. & Wright, E.M. (1960). An introduction to the theory of numbers, 4th ed. Oxford: Oxford University Press.
Hordijk, A. & Loeve, J.A. (1997). Optimal noncyclic server allocation in a polling model. In IEEE, Conference on Decision and Control, vol. 36, pp. 29412945.
Hordijk, A. & van der Laan, D.A. (2003). Note on the convexity of the stationary waiting time as a function of the density. Probability in the Engineering and Information Sciences 17: 503508.
Hordijk, A. & van der Laan, D.A. (2005). On the average waiting time for regular routing to deterministic queues. Mathematics of Operations Research 30: 521544.
Kruskal, J.B. (1969). Work-scheduling algorithms: A non-probabilistic queueing study. Bell System Technical Journal 48: 29632974.
Lothaire, M. (2002). Algebraic combinatorics on words. Cambridge: Cambridge University Press.
Mack, C. (1957). The efficiency of n machines uni-directionally controlled by one operative when walking time and repair times are variable. Journal of the Royal Statistical Society Press, Series B 19(1): 173178.
Mack, C., Murphy, T., & Webb, N.L. (1957). The efficiency of n machines uni-directionally controlled by one operative when walking time and repair times are constants. Journal of the Royal Statistical Society Press, Series B 19(1): 166172.
Neuts, M.F. (1989). Structured stochastic matrices of M/G/1 type and their applications. New York: Marcel Dekker.
Perron, O. (1954). Die Lehre von den Kettenbrüchen. Stuttgart: Teubner.
Sinai, Y.G. (1976). Introduction to ergodic theory. Princeton, NJ: Princeton University Press.
Tagaki, H. (1986). Analysis of polling systems. Cambridge, MA: MIT Press.
Tijdeman, R. (2000). Fraenkel's conjecture for six sequences. Discrete Mathematics 222: 223234.
van der Laan, D.A. (2003). The structure and performance of optimal routing sequences. PhD thesis, Leiden University.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed