Abstract
Motivated by service levels in terms of the waiting-time distribution seen, for instance, in call centers, we consider two models for systems with a service discipline that depends on the waiting time. The first model deals with a single server that continuously adapts its service rate based on the waiting time of the first customer in line. In the second model, one queue is served by a primary server which is supplemented by a secondary server when the waiting of the first customer in line exceeds a threshold. Using level crossings for the waiting-time process of the first customer in line, we derive steady-state waiting-time distributions for both models. The results are illustrated with numerical examples.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Armony, M.: Dynamic routing in large-scale service systems with heterogeneous servers. Queueing Syst. 51, 287–329 (2005)
Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)
Barth, W., Manitz, M., Stolletz, R.: Analysis of two-level support systems with time-dependent overflow—a banking application. Prod. Oper. Manag. 19, 757–768 (2010)
Bekker, R., Borst, S.C., Boxma, O.J., Kella, O.: Queues with workload-dependent arrival and service rates. Queueing Syst. 46, 537–556 (2004)
Boxma, O.J., Vlasiou, M.: On queues with service and interarrival times depending on waiting times. Queueing Syst. 56, 121–132 (2007)
Boxma, O., Kaspi, H., Kella, O., Perry, D.: On/off storage systems with state dependent input, output and switching rates. Probab. Eng. Inf. Sci. 19, 1–14 (2005)
Brill, P.H., Posner, M.J.M.: A two server queue with nonwaiting customers receiving specialized service. Manag. Sci. 27, 914–925 (1981)
Brill, P.H., Posner, M.J.M.: The system point method in exponential queues: a level crossing approach. Math. Oper. Res. 6, 31–49 (1981)
Browne, S., Sigman, K.: Work-modulated queues with applications to storage processes. J. Appl. Probab. 29, 699–712 (1992)
Cohen, J.W., Rubinovitch, M.: On level crossings and cycles in dam processes. Math. Oper. Res. 2, 297–310 (1977)
Franx, G.J., Koole, G.M., Pot, S.A.: Approximating multi-skill blocking systems by hyperexponential decomposition. Perform. Eval. 63, 799–824 (2006)
Gans, N., Zhou, Y.-P.: A call-routing problem with service-level constraints. Oper. Res. 51, 255–271 (2003)
Gans, N., Zhou, Y.-P.: Call-routing schemes for call-center outsourcing. Manuf. Serv. Oper. Manag. 9, 33–50 (2007)
Gans, N., Koole, G.M., Mandelbaum, A.: Telephone call centers: tutorial, review, and research prospects. Manuf. Serv. Oper. Manag. 5, 79–141 (2003)
Gurvich, I., Armony, M., Mandelbaum, A.: Service-level differentiation in call centers with fully flexible servers. Manag. Sci. 54, 279–294 (2008)
Harrison, J.M., Resnick, S.I.: The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Oper. Res. 1, 347–358 (1976)
Jackson, J.R.: Some problems in queueing with dynamic priorities. Nav. Res. Logist. Q. 7, 235–249 (1960)
Koole, G.M.: A simple proof of the optimality of a threshold policy in a two-server queueing system. Syst. Control Lett. 26, 301–303 (1995)
Lin, W., Kumar, P.R.: Optimal control of a queueing system with two heterogeneous servers. IEEE Trans. Autom. Control 29, 696–703 (1984)
Liu, L., Parlar, M., Zhu, S.: Pricing and lead time decisions in decentralized supply chains. Manag. Sci. 53, 713–725 (2007)
Lucent Technologies CentreVu Release 8 Advocate User Guide. P.O. Box 4100, Crawfordsville, IN 47933, USA: Lucent Technologies (1999).
Perry, D., Benny, L.: Continuous production/inventory model with analogy to certain queueing and Dam models. Adv. Appl. Probab. 21, 123–141 (1989)
Posner, M.J.M.: Single-server queues with service time dependent on waiting time. Oper. Res. 21, 610–616 (1973)
Rubinovitch, M.: The slow server problem. J. Appl. Probab. 22, 205–213 (1985)
Scheinhardt, W.R.W., van Foreest, N., Mandjes, M.: Continuous feedback fluid queues. Oper. Res. Lett. 33, 551–559 (2005)
Stockbridge, R.H.: A martingale approach to the slow server problem. J. Appl. Probab. 28, 480–486 (1991)
Whitt, W.: Queues with service times and interarrival times depending linearly and randomly upon waiting times. Queueing Syst. 6, 335–351 (1990)
Zabreiko, P.P., Koshelev, A.I., Krasnosel’skii, M.A.: Integral Equations: A Reference Text. Monographs and Textbooks on Pure and Applied Mathematics. Noordhoff, Leiden (1975). Transl. and ed. by T.O. Shaposhnikova, R.S. Anderssen and S.G. Mikhlin
Author information
Authors and Affiliations
Corresponding author
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.
About this article
Cite this article
Bekker, R., Koole, G.M., Nielsen, B.F. et al. Queues with waiting time dependent service. Queueing Syst 68, 61–78 (2011). https://doi.org/10.1007/s11134-011-9225-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-011-9225-2
Keywords
- Waiting-time distribution
- Adaptive service rate
- Call centers
- Contact centers
- Queues
- Deterministic threshold
- Overflow
- Level crossing