Queueing Systems

, Volume 67, Issue 1, pp 63–90 | Cite as

Dynamic control of a single-server system with abandonments

  • Douglas G. Down
  • Ger Koole
  • Mark E. LewisEmail author


In this paper, we discuss the dynamic server control in a two-class service system with abandonments. Two models are considered. In the first case, rewards are received upon service completion, and there are no abandonment costs (other than the lost opportunity to gain rewards). In the second, holding costs per customer per unit time are accrued, and each abandonment involves a fixed cost. Both cases are considered under the discounted or average reward/cost criterion. These are extensions of the classic scheduling question (without abandonments) where it is well known that simple priority rules hold.

The contributions in this paper are twofold. First, we show that the classic cμ rule does not hold in general. An added condition on the ordering of the abandonment rates is sufficient to recover the priority rule. Counterexamples show that this condition is not necessary, but when it is violated, significant loss can occur. In the reward case, we show that the decision involves an intuitive tradeoff between getting more rewards and avoiding idling. Secondly, we note that traditional solution techniques are not directly applicable. Since customers may leave in between services, an interchange argument cannot be applied. Since the abandonment rates are unbounded we cannot apply uniformization—and thus cannot use the usual discrete-time Markov decision process techniques. After formulating the problem as a continuous-time Markov decision process (CTMDP), we use sample path arguments in the reward case and a savvy use of truncation in the holding cost case to yield the results. As far as we know, this is the first time that either have been used in conjunction with the CTMDP to show structure in a queueing control problem. The insights made in each model are supported by a detailed numerical study.


Priority rules Dynamic programming Control of queues 

Mathematics Subject Classification (2000)

90B36 60K25 90C40 


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  1. 1.
    Aksin, Z., Armony, M., Mehrotra, V.: The modern call-center: a multi-disciplinary perspective on operations management research. Prod. Oper. Manag. 16, 665–668 (2007) Google Scholar
  2. 2.
    Argon, N.T., Ziya, S., Righter, R.: Scheduling impatient jobs in a clearing system with insights on patient triage in mass casualty incidents. Probab. Eng. Inf. Sci. 22(3), 301–332 (2008) CrossRefGoogle Scholar
  3. 3.
    Armony, M., Maglaras, C.: Contact centers with a call-back option and real-time delay information. Oper. Res. 52(4), 527–545 (2004) CrossRefGoogle Scholar
  4. 4.
    Armony, M., Maglaras, C.: On customer contact centers with a call-back option: customer decisions, routing rules and system design. Oper. Res. 52(2), 271–292 (2004) CrossRefGoogle Scholar
  5. 5.
    Armony, M., Shimkin, N., Whitt, W.: The impact of delay announcements in many-server queues with abandonment. Oper. Res. 57, 66–81 (2009) CrossRefGoogle Scholar
  6. 6.
    Atar, R., Giat, C., Shimkin, N.: The c μ/θ rule for many-server queues with abandonment. Preprint (2010) Google Scholar
  7. 7.
    Buyukkoc, C., Varaiya, P., Walrand, J.: The c μ-rule revisited. Adv. Appl. Probab. 17(1), 237–238 (1985) CrossRefGoogle Scholar
  8. 8.
    Gans, N., Koole, G., Mandelbaum, A.: Telephone call centers: tutorial, review and research prospects. Manuf. Serv. Oper. Manag. 5(2), 79–141 (2003) CrossRefGoogle Scholar
  9. 9.
    Gayon, J.-P., Benjaafar, S., de Véricourt, F.: Using imperfect advance demand information in production-inventory systems with multiple customer classes. Manuf. Serv. Oper. Manag. 11(1), 128–143 (2009) CrossRefGoogle Scholar
  10. 10.
    Ghamami, S., Ward, A.: Dynamic scheduling of an N-system with reneging. Preprint (2010) Google Scholar
  11. 11.
    Guo, X., Hernández-Lerma, O.: Continuous-time controlled Markov chains. Ann. Appl. Probab. 13(1), 363–388 (2003) CrossRefGoogle Scholar
  12. 12.
    Guo, X., Hernández-Lerma, O., Prieto-Rumeau, T.: A survey of recent results on continuous-time Markov decision processes. Top 14(2), 177–257 (2006) CrossRefGoogle Scholar
  13. 13.
    Harrison, J., Zeevi, A.: Dynamic scheduling of a multiclass queue in the Halfin and Whitt heavy traffic regime. Oper. Res. 52, 243–257 (2004) CrossRefGoogle Scholar
  14. 14.
    Iravani, F., Balcıog̃lu, B.: On priority queues with impatient customers. Queueing Syst. 58, 239–260 (2008) CrossRefGoogle Scholar
  15. 15.
    Koçag̃a, Y., Ward, A.: Admission control for a multi-server queue with abandonment. Preprint (2010) Google Scholar
  16. 16.
    Lippman, S.: Applying a new device in the optimization of exponential queueing systems. Oper. Res. 23(4), 687–710 (1975) CrossRefGoogle Scholar
  17. 17.
    Nain, P.: Interchange arguments for classical scheduling problems in queues. Syst. Control Lett. 12, 177–184 (1989) CrossRefGoogle Scholar
  18. 18.
    Prieto-Rumeau, T., Hernández-Lerma, O.: A unified approach to continuous-time discounted Markov control processes. Morfismos 10(1), 1–40 (2006) Google Scholar
  19. 19.
    Puterman, M.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1994) Google Scholar
  20. 20.
    Serfozo, R.: An equivalence between continuous and discrete time Markov decision processes. Oper. Res. 27(3), 616–620 (1978) CrossRefGoogle Scholar
  21. 21.
    Tezcan, T., Dai, J.: Dynamic control of N-systems with many servers. asymptotic optimality of a static priority policy in heavy traffic. Preprint (2010) Google Scholar
  22. 22.
    Ward, A.R., Glynn, P.W.: A diffusion approximation for a Markovian queue with reneging. Queueing Syst. 43(1), 103–128 (2003) CrossRefGoogle Scholar
  23. 23.
    Ward, A.R., Glynn, P.W.: A diffusion approximation for a GI/GI/1 queue with balking or reneging. Queueing Syst. 50(4), 371–400 (2005) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Computing and SoftwareMcMaster UniversityHamiltonCanada
  2. 2.Faculty of Sciences, Department of MathematicsVU UniversityAmsterdamThe Netherlands
  3. 3.School of Operations Research and Information EngineeringCornell UniversityIthacaUSA

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