Skip to main content Accessibility help
Hostname: page-component-564cf476b6-dr4jh Total loading time: 0.254 Render date: 2021-06-20T14:21:42.293Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }


Published online by Cambridge University Press:  16 February 2009

E. M. M. Winands
Department of Mathematics, VU University, 1081 HV Amsterdam, The Netherlands E-mail:
I. J. B. F. Adan
Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, 5600 MB Eindhoven, The Netherlands E-mail:
G. J. van Houtum
Department of Industrial Engineering and Innovation Sciences, Technische Universiteit Eindhoven, 5600 MB Eindhoven, The Netherlands E-mail:
D. G. Down
Department of Computing and Software, McMaster University, Hamilton, Ontario L8S 4L7, Canada E-mail:


We consider a two-queue model with state-dependent setups, in which a single server alternately serves the two queues. The high-priority queue is served exhaustively, whereas the low-priority queue is served according to the k-limited strategy. A setup at a queue is incurred only if there are customers waiting at the polled queue. We obtain the transforms of the queue length and sojourn time distributions under the assumption of Poisson arrivals, generally distributed service times, and generally distributed setup times. The interest for this model is fueled by an application in the field of logistics. It is shown how the results of this analysis can be applied in the evaluation of a stochastic two-item single-capacity production system. From these results we can conclude that significant cost reductions are possible by bounding the production runs of the low-priority item, which indicates the potential of the k-limited service discipline as priority rule in production environments.

Research Article
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below.


1.Abate, J. & Whitt, W. (1992). Numerical inversion of probability generating functions. Operations Research Letters 12(4): 245251.CrossRefGoogle Scholar
2.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): 355360.CrossRefGoogle Scholar
3.Altman, E., Blanc, H., Khamisy, A. & Yechiali, U. (1994). Gated-type polling systems with walking and switch-in times. Stochastic Models 10: 741764.CrossRefGoogle Scholar
4.Blanc, J.P.C. (1992). An algorithmic solution of polling models with limited service disciplines. IEEE Transactions on Communications 40(7): 11521155.CrossRefGoogle 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): 602612.CrossRefGoogle Scholar
6.Boxma, O.J. & Down, D.G. (1997). Dynamic server assignment in a two-queue model European Journal of Operational Research 103: 101115.CrossRefGoogle Scholar
7.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. 719728.Google Scholar
8.Chaudhry, M.L. (1994). QROOT software package. Kingston: A&A Publications, Ontario, Canada.Google Scholar
9.Cohen, J.W. (1987). A two-queue, one-server model with priority for the longer queue. Queueing Systems 2(3): 261283.CrossRefGoogle Scholar
10.Dai, J.G. (1995). On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Annals of Applied Probability 5: 4977.CrossRefGoogle Scholar
11.Dai, J.G. (1999). Stability of fluid and stochastic processing networks. Publication No. 9, Centre for Mathematical Physics and Stochastics, University of Aarhus, Denmark, Available from Scholar
12.Dai, J.G. & Meyn, S.P. (1995). Stability and convergence of moments for multiclass queueing networks via fluid models. IEEE Transactions on Automatic Control 40: 18891904.CrossRefGoogle Scholar
13.Down, D. (1998). On the stability of polling models with multiple servers. Journal of Applied Probability 35: 925935.CrossRefGoogle Scholar
14.Federgruen, A. & Katalan, Z. (1996). The stochastic economic lot scheduling problem: cyclical base-stock policies with idle times. Management Science 42(6): 783796.CrossRefGoogle Scholar
15.Federgruen, A. & Katalan, Z. (1998). Determining production schedules under base-stock policies in single facility multi-item production systems. Operations Research 46(6): 883898.CrossRefGoogle Scholar
16.Flatto, L. (1989). The longer queue model. Probability in the Engineering and Informational Sciences 3: 537559.CrossRefGoogle Scholar
17.Fuhrmann, S.W. (1981). Performance analysis of a class of cyclic schedules. Bell Laboratories Technical Memorandum 81-59531-1.Google Scholar
18.Günalay, Y. & Gupta, D. (1997). A polling system with a patient server and state-dependent setup times. IIE Transactions 29: 469480.CrossRefGoogle Scholar
19.Gupta, D. & Srinivasan, M.M. (1996). Polling systems with state-dependent setup times. Queueing Systems 22: 403423.CrossRefGoogle Scholar
20.Ibe, O.C. & Cheng, X. (1988). Stability conditions for multiqueue systems with cyclic service. IEEE Transactions on Automatic Control 33(1): 102103.CrossRefGoogle Scholar
21.Keilson, J. & Servi, L.D. (1990). The distributional form of Little's law and the Fuhrmann–Cooper decomposition. Operations Research Letters 9(4): 239247.CrossRefGoogle Scholar
22.Lee, D.-S. (1996). A two-queue model with exhaustive and limited service disciplines. Stochastic Models 12(2): 285305.CrossRefGoogle Scholar
23.Levy, H. & Sidi, M. (1990). Polling systems: applications, modeling and optimization. IEEE Transactions on Communications 38(10): 17501760.CrossRefGoogle Scholar
24.Ozawa, T. (1990). Alternating service queues with mixed exhaustive and K-limited services. Performance Evaluation 11: 165175.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 the International Conference on the Performance and Management of Complex Communication Networks, Tsukuba, pp. 231250.Google Scholar
26.Resing, J.A.C. (1993). Polling systems and multitype branching processes. Queueing Systems 13: 409426.CrossRefGoogle Scholar
27.Singh, M.P. & Srinivasan, M.M. (2002). Exact analysis of the state dependent polling model. Queueing Systems 41: 371399.CrossRefGoogle Scholar
28.Sox, C.R., Jackson, P.L., Bowman, A. & Muckstadt, J.A. (1999). A review of the stochastic lot scheduling problem. International Journal of Production Economics 62: 181200.CrossRefGoogle Scholar
29.Takács, L. (1968). Two queues attended by a single server. Operations Research 16(3): 639650.CrossRefGoogle Scholar
30.Takagi, H. (1990). Queueing analysis of polling models: an update. In Takagi, H. (ed.) Stochastic analysis of computer and Communication systems. Amsterdam: North-Holland, pp. 267318.Google Scholar
31.Takagi, H. (1997). Queueing analysis of polling models: progress in 1990–1994. In Dshalalow, J.H., (ed.) Frontiers in Queueing: Models, Methods and Problems. Boca Raton, FL: CRC Press, pp. 119146.Google Scholar
32.Takagi, H., (2000). Analysis and application of polling models. In Haring, G.Lindemann, C. and Reiser, M. (eds.), Performance evaluation: Origins and directions. Lecture Notes in Computer Science Vol. 1769. Berlin: Springer, pp. 423442.CrossRefGoogle Scholar
33.Tijms, H.C. (1994). Stochastic models: An algorithmic approach. New York: Wiley.Google Scholar
34.Vuuren, M. van & Winands, E.M.M. (2006). Iterative approximation of k-limited polling systems. Queueing Systems 55(3): 161178.CrossRefGoogle Scholar
35.Winands, E.M.M., Adan, I.J.B.F. & van Houtum, G.J. (2005). The stochastic economic lot scheduling problem: a survey. Report BETA WP-133, Beta Research School for Operations Management and Logistics, Eindhoven.Google Scholar
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *