Journal of Scheduling

, Volume 20, Issue 2, pp 129–145 | Cite as

Scheduling of multi-class multi-server queueing systems with abandonments

  • Urtzi Ayesta
  • Peter Jacko
  • Vladimir NovakEmail author


Many real-world situations involve queueing systems in which customers may abandon if service does not start sufficiently quickly. We study a comprehensive model of multi-class queue scheduling accounting for customer abandonment, with the objective of minimizing the total discounted or time-average sum of linear waiting costs, completion rewards, and abandonment penalties of customers in the system. We assume the service times and abandoning times are exponentially distributed. We solve analytically the case in which there is one server and there are one or two customers in the system and obtain an optimal policy. For the general case, we use the framework of restless bandits to analytically design a novel simple index rule with a natural interpretation. We show that the proposed rule achieves near-optimal or asymptotically optimal performance both in single- and multi-server cases, both in overload and underload regimes, and both in idling and non-idling systems.


Stochastic scheduling Abandonment Restless bandits Index policy Whittle index 



Research partially supported by the French “Agence Nationale de la Recherche (ANR)” through the project ANR JCJC RACON.


  1. Aksin, Z., Armony, M., & Mehrotra, V. (2007). The modern call center: A multi-disciplinary perspective on operations management research. Production and Operations Management, 16(6), 665–688.CrossRefGoogle Scholar
  2. Argon, N., Ziya, S., & Righter, R. (2010). Scheduling impatient jobs in a clearing system with insights on patient triage in mass-casualty incidents. Probability in the Engineering and Informational Sciences, 22(3), 301–332.Google Scholar
  3. Ata, B., & Tongarlak, M. H. (2013). On scheduling a multiclass queue with abandonments under general delay costs. Queueing Systems, 74(1), 65–104.CrossRefGoogle Scholar
  4. Atar, R., Giat, C., & Shimkin, N. (2010). The \(c\mu /\theta \) rule for many-server queues with abandonment. Operation Research, 58(5), 1427–1439.CrossRefGoogle Scholar
  5. Atar, R., Giat, C., & Shimkin, N. (2011). On the asymptotic optimality of the \(\text{ c }\mu /\theta \) rule under ergodic cost. Queueing Systems, 67(2), 127–144.CrossRefGoogle Scholar
  6. Ayesta, U., Jacko, P., & Novak, V. (2011). 2011 In IEEE Infocom: A nearly-optimal index rule for scheduling of users with abandonment.Google Scholar
  7. Baccelli, F., Boyer, P., & Hebuterne, G. (1984). Single-server queues with impatient customers. Advances in Applied Prabability, 16, 887–905.CrossRefGoogle Scholar
  8. Boots, N., & Tijms, H. (1999). A multiserver queueing system with impatient customers. Management Science, 45(3), 444–448.CrossRefGoogle Scholar
  9. Boxma, O., & de Waal, P. (1994). Multiserver queues with impatient customers. In In Proceedings of ITC-14 (pp. 743–756).Google Scholar
  10. Brandt, A., & Brandt, M. (2004). On the two-class M/M/1 system under preemptive resume and impatience of the prioritized customers. Queueing Systems, 47, 147–168.CrossRefGoogle Scholar
  11. Brill, P., & Posner, M. (1977). Level crossings in point processes applied to queues: Single-server case. Operations Research, 25(4), 662–674.CrossRefGoogle Scholar
  12. Buttazzo, G. C. (2011). Hard real-time computing systems: Predictable scheduling algorithms and applications (3rd ed.)., Real-time systems New York: Springer.CrossRefGoogle Scholar
  13. Buyukkoc, C., Varaya, P., & Walrand, J. (1985). The \(c\mu \) rule revisited. Advances in Applied Probability, 17, 237–238.Google Scholar
  14. Cox, D. R., & Smith, W. L. (1961). Queues. London: Methuen & Co.Google Scholar
  15. Dai, J., & He, S. (2012). Many-server queues with customer abandonment: A survey of diffusion and fluid approximations. Journal of Systems Science and Systems Engineering, 21, 1–36.CrossRefGoogle Scholar
  16. Down, D., Koole, G., & Lewis, M. (2011). Dynamic control of a single server system with abandonments. Queueing Systems, 67, 63–90.CrossRefGoogle Scholar
  17. Fife, D. (1965). Scheduling with random arrivals and linear loss functions. Management Science, 11(3), 429–437.CrossRefGoogle Scholar
  18. Gans, N., Koole, G., & Mandelbaum, A. (2003). Telephone call centers: Tutorial, review, and research prospects. Manufacturing & Service Operations Management, 5(2), 79–141.CrossRefGoogle Scholar
  19. Gittins, J. (1989). Multi-armed bandit allocation indices. Chichester: Wiley.Google Scholar
  20. Gittins, J., & Jones, D. (1974). A dynamic allocation index for the sequential design of experiments. In J. Gani (Ed.), Progress in statistics (pp. 241–266). Amsterdam: North-Holland.Google Scholar
  21. Glazebrook, K., Ansell, P., Dunn, R., & Lumley, R. (2004). On the optimal allocation of service to impatient tasks. Journal of Applied Probability, 41(1), 51–72.CrossRefGoogle Scholar
  22. Graves, S. (1984). The application of queueing theory to continuous perishable inventory systems. Management Science, 28, 401–406.Google Scholar
  23. Harrison, J. M., & Zeevi, A. (2004). Dynamic scheduling of a multiclass queue in the Halfin-Whitt heavy traffic regime. Operations Research, 52(2), 243–257.CrossRefGoogle Scholar
  24. Hasenbein, J., & Perry, D. (2013). Introduction: queueing systems special issue on queueing systems with abandonments. Queueing Systems, 75(2–4), 111–113.CrossRefGoogle Scholar
  25. Hassin, R., & Haviv, M. (2003). To queue or not to queue: Equilibrium behavior in queueing systems. Boston: Kluwer Academic Publishers.CrossRefGoogle Scholar
  26. Iravani, F., & Balcioğlu, B. (2008). On priority queues with impatient customers. Queueing Systems, 58, 239–260.CrossRefGoogle Scholar
  27. Jacko, P. (2009). Adaptive greedy rules for dynamic and stochastic resource capacity allocation problems. Medium for Econometric Applications, 17(4), 10–16.Google Scholar
  28. Jouini, O., Pot, A., Koole, G., & Dallery, Y. (2010). Online scheduling policies for multiclass call centers with impatient customers. European Journal of Operational Research, 207(1), 258–268.CrossRefGoogle Scholar
  29. Meilijson, I., & Weiss, G. (1977). Multiple feedback at a single server station. Stochastic Processes and Applications, 5, 195–205.CrossRefGoogle Scholar
  30. Niño-Mora, J. (2007). Dynamic priority allocation via restless bandit marginal productivity indices. TOP, 15(2), 161–198.CrossRefGoogle Scholar
  31. Papadimitriou, C., & Tsitsiklis, J. (1999). The complexity of optimal queueing network. Mathematics of Operations Research, 24(2), 293–305.CrossRefGoogle Scholar
  32. Puterman, M. L. (2005). Markov decision processes: Discrete stochastic dynamic programming. New York: Wiley.Google Scholar
  33. Sevcik, K. (1974). Scheduling for minimum total loss using service time distributions. Journal of the ACM, 21, 66–75.CrossRefGoogle Scholar
  34. Smith, W. (1956). Various optimizers for single stage production. Naval Research Logistics Quarterly, 3, 59–66.CrossRefGoogle Scholar
  35. Weber, R., & Weiss, G. (1990). On an index policy for restless bandits. Journal of Applied Probability, 27, 637–648.CrossRefGoogle Scholar
  36. Whitt, W. (2004). Efficiency-driven heavy-traffic approximations for many-server queues with abandonments. Management Science, 50, 1449–1461.CrossRefGoogle Scholar
  37. Whittle, P. (1981). Arm-acquiring bandits. Annals of Probability, 9(2), 284–292.CrossRefGoogle Scholar
  38. Whittle, P. (1988). Restless bandits: Activity allocation in a changing world. Journal of Applied Probability, 25, 287–298.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.CNRS, LAASToulouseFrance
  2. 2.IKERBASQUE — Basque Foundation for ScienceBilbaoSpain
  3. 3.Univ. de Toulouse, LAASToulouseFrance
  4. 4.UPV/EHUUniversity of the Basque CountryDonostiaSpain
  5. 5.Department of Management ScienceLancaster UniversityLancasterUK
  6. 6.CERGE-EI, Politickych veznu 7PragueCzech Republic

Personalised recommendations