Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T21:31:20.141Z Has data issue: false hasContentIssue false
  • English
  • Français

HEAVY-TRAFFIC ANALYSIS OF K-LIMITED POLLING SYSTEMS

Published online by Cambridge University Press:  27 June 2014

M.A.A. Boon  and
E.M.M. Winands
M.A.A. Boon
Affiliation:
Eurandom and Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The Netherlands E-mail: [email protected]
E.M.M. Winands
Affiliation:
University of Amsterdam, Korteweg-de Vries Institute for Mathematics, Science Park 904, 1098 XH Amsterdam, The Netherlands E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study a two-queue polling model with zero switchover times and k-limited service (serve at most ki customers during one visit period to queue i, i=1, 2) in each queue. The arrival processes at the two queues are Poisson, and the service times are exponentially distributed. By increasing the arrival intensities until one of the queues becomes critically loaded, we derive exact heavy-traffic limits for the joint queue-length distribution using a singular-perturbation technique. It turns out that the number of customers in the stable queue has the same distribution as the number of customers in a vacation system with Erlang-k2 distributed vacations. The queue-length distribution of the critically loaded queue, after applying an appropriate scaling, is exponentially distributed. Finally, we show that the two queue-length processes are independent in heavy traffic.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 28 , Issue 4 , October 2014 , pp. 451 - 471
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Adan, I.J.B.F., van Leeuwaarden, J.S.H. & Winands, E.M.M. (2006). On the application of Rouché’s theorem in queueing theory. Operations Research Letters 34(3): 355–360.Google Scholar
2.Bertsimas, D. & Mourtzinou, G. (1997). Multiclass queueing systems in heavy traffic: an asymptotic approach based on distributional and conservation laws. Operations Research 45(3): 470–487.Google Scholar
3.Blanc, J.P.C. (1992). An algorithmic solution of polling models with limited service disciplines. IEEE Transactions of Communications 40(7): 1152–1155.CrossRefGoogle Scholar
4.Boon, M.A.A., Adan, I.J.B.F., Winands, E.M.M. & Down, D.G. (2012). Delays at signalised intersections with exhaustive traffic control. Probability in the Engineering and Informational Sciences 26(3): 337–373.Google Scholar
5.Borst, S.C., Boxma, O.J. & Levy, H. (1995). The use of service limits for efficient operation of multistation single-medium communication systems. IEEE/ACM Transactions on Networking 3(5): 602–612.CrossRefGoogle Scholar
6.Boxma, O.J. (1985). Two symmetric queues with alternating service and switching times. In Gelenbe, E. (ed), Performance ’84, North-Holland, Amsterdam, pp. 409–431.Google Scholar
7.Boxma, O.J. & Groenendijk, W.P. (1988). Two queues with alternating service and switching times. In Boxma, O.J. & Syski, R. (eds), Queueing Theory and its Applications — Liber Amicorum for J.W. Cohen, North-Holland, Amsterdam, pp. 261–282.Google Scholar
8.Charzinski, J., Renger, T. & Tangemann, M. (1994). Simulative comparison of the waiting time distributions in cyclic polling systems with different service strategies. In Proceedings of the 14th International Teletraffic Congress, Antibes Juan-les-Pins, pp. 719–728.Google Scholar
9.Coffman, E.G. Jr., Puhalskii, A.A. & Reiman, M.I. (1995). Polling systems with zero switchover times: a heavy-traffic averaging principle. The Annals of Applied Probability 5(3): 681–719.CrossRefGoogle Scholar
10.Coffman, E.G. Jr., Puhalskii, A.A. & Reiman, M.I. (1998). Polling systems in heavy-traffic: a Bessel process limit. Mathematics of Operations Research 23: 257–304.CrossRefGoogle Scholar
11.Cohen, J.W. & Boxma, O.J. (1981). The M/G/1 queue with alternating service formulated as a Riemann-Hilbert problem. In Kylstra, F.J. (ed), Performance ’81, pp. 181–189.Google Scholar
12.Down, D.G. (1998). On the stability of polling models with multiple servers. Journal of Applied Probability, 35: 925–935.Google Scholar
13.Eisenberg, M. (1979). Two queues with alternating service. SIAM Journal on Applied Mathematics 36(2): 287–303.Google Scholar
14.Fricker, C. & Jaïbi, M.R. (1994). Monotonicity and stability of periodic polling models. Queueing Systems 15: 211–238.Google Scholar
15.Fuhrmann, S.W. (1981). Performance analysis of a class of cyclic schedules. Technical Memorandum 81-59531-1, Bell Laboratories.Google Scholar
16.Groenendijk, W.P. (1990). Conservation Laws in Polling Systems. PhD thesis, University of Utrecht.Google Scholar
17.Ibe, O.C. (1990). Analysis of polling systems with mixed service disciplines. Stochastic Models 6: 667–689.Google Scholar
18.Keilson, J. & Servi, L.D. (1990). The distributional form of Little's Law and the Fuhrmann–Cooper decomposition. Operations Research Letters 9(4): 239–247.CrossRefGoogle Scholar
19.Knessl, C. & Tier, C. (1995). Applications of singular perturbation methods in queueing. In Advances in Queueing Theory, Methods, and Open Problems, Probability and Stochastics Series. Boca Raton: CRC Press, pp. 311–336.Google Scholar
20.LaPadula, C.A. & Levy, H. (1996). Customer delay in very large multi-queue single-server systems. Performance Evaluation 26(3): 201–218.CrossRefGoogle Scholar
21.Lee, D.-S. (1996). A two-queue model with exhaustive and limited service disciplines. Stochastic Models 12(2): 285–305.Google Scholar
22.Lee, T.T. (1989). M/G/1/N queue with vacation time and limited service discipline. Performance Evaluation 9(3): 181–190.Google Scholar
23.Morrison, J.A. & Borst, S.C. (2010). Interacting queues in heavy traffic. Queueing Systems 65(2): 135–156.Google Scholar
24.Ozawa, T. (1990). Alternating service queues with mixed exhaustive and K-limited services. Performance Evaluation, 11: 165–175.CrossRefGoogle Scholar
25.Ozawa, T. (1997). Waiting time distribution in a two-queue model with mixed exhaustive and gated-type K-limited services. In Proceedings of International Conference on the Performance and Management of Complex Communication Networks, pp. 231–250.Google Scholar
26.Resing, J.A.C. (1993). Polling systems and multitype branching processes. Queueing Systems 13: 409–426.Google Scholar
27.Van der Mei, R.D. (2007). Towards a unifying theory on branching-type polling models in heavy traffic. Queueing Systems 57: 29–46.Google Scholar
28.Winands, E.M.M., Adan, I.J.B.F., van Houtum, G.J. & Down, D.G. (2009). A state-dependent polling model with k-limited service. Probability in the Engineering and Informational Sciences 23(2): 385–408.Google Scholar