A Marginal Allocation Approach to Resource Management for a System of Multiclass Multiserver Queues Using Abandonment and CVaR QoS Measures

  • Per Enqvist
  • Göran SvenssonEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 966)


A class of resource allocation problems is considered where some quality of service measure is set against the agent related costs. Three multiobjective minimization problems are posed, one for a system of Erlang-C queues and two for systems of Erlang-A queues.

In the case of the Erlang-C systems we introduce a quality of service measure based on the Conditional Value-at-Risk with waiting time as the loss function. This is a risk coherent measure and is well established in the field of finance. An algebraic proof ensures that this quality of service measure is integer convex in the number of servers.

In the case of the Erlang-A systems we introduce two different quality of service measures. The first is a weighted sum of fractions of abandoning customers and the second is Conditional Value-at-Risk, with the waiting time in queue for a customer conditioned on eventually receiving service. Finally, numerical experiments on the two system types with the given quality of service measures, are presented and the optimal solutions are compared.


Queueing Queueing networks Marginal allocation Conditional Value-at-Risk Abandonments 



The authors would like to thank Teleopti AB, Stockholm Sweden, for their support of our work.


  1. 1.
    Armony, M., Plambeck, E., Seshadri, S.: Sensitivity of optimal capacity to customer impatience in an unobservable M/M/S queue (why you shouldn’t shout at the DMV). Manufact. Serv. Oper. Manage. 11(1), 19–32 (2009). Scholar
  2. 2.
    Artzner, P., Delbaen, F., Eber, J., Heath, D.: Coherent measures of risk. Math. Finan. 9(3), 203–228 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Baccelli, F., Hebuterne, G.: On queues with impatient customers. Performance 81 (1981)Google Scholar
  4. 4.
    Dyer, M., Proll, L.: On the validity of marginal analysis for allocating servers in “M/M/c” queues. Manage. Sci. 23(9) (1977)Google Scholar
  5. 5.
    Enqvist, P., Svensson, G.: Multi-server marginal allocation - with CVaR and abandonment based QoS measures. In: Proceedings of the 7th International Conference on Operations Research and Enterprise Systems, ICORES, vol. 1, pp. 297–303. INSTICC, SciTePress (2018).
  6. 6.
    Fox, B.: Discrete optimization via marginal analysis. Manage. Sci. 13(3), 210–216 (1966)CrossRefGoogle Scholar
  7. 7.
    Gans, N., Koole, G., Mandelbaum, A.: Telephone call centers: tutorial, review, and research prospects. Manuf. Serv. Oper. Manage. 5(2), 79–141 (2003). Scholar
  8. 8.
    Garnett, O., Mandelbaum, A., Reiman, M.: Designing a call center with impatient customers. Manuf. Serv. Oper. Manage. 4, 208–227 (2002)CrossRefGoogle Scholar
  9. 9.
    Kleinrock, L.: Queueing Systems: Problems and Solutions. Wiley, New York (1996)zbMATHGoogle Scholar
  10. 10.
    Koole, G., Pot, A.: A note on profit maximization and monotonicity for inbound call centers. Oper. Res. 59(5), 1304–1308 (2011). Technical noteCrossRefGoogle Scholar
  11. 11.
    Mandelbaum, A., Zeltyn, S.: Service engineering in action: the palm/Erlang-A queue, with applications to call centers. In: Spath, D., Fähnrich, K.P. (eds) Advances in Services Innovations, pp. 17–45. Springer, Heidelberg (2007).
  12. 12.
    Palm, C.: Research on telephone traffic carried by full availability groups. Tele 1, 107 (1957)Google Scholar
  13. 13.
    Parlar, M., Sharafali, M.: Optimal design of multi server markovian queues with polynomial waiting and service costs. Appl. Stoch. Models Bus. Ind. 30(4), 429–443 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rockafellar, R., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)CrossRefGoogle Scholar
  15. 15.
    Rockafellar, R., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finan. 26, 1443–1471 (2002)CrossRefGoogle Scholar
  16. 16.
    Rolfe, A.J.: A note on marginal allocation in multiple-server service systems. Manage. Sci. 17(9), 656–658 (1971)CrossRefGoogle Scholar
  17. 17.
    Svanberg, K.: On marginal allocation. Department of Mathematics, KTH, Stockholm (2009)Google Scholar
  18. 18.
    Weber, R.: On the marginal benefit of adding servers to “G/GI/m” queues. Manage. Sci. 26(9), 946–951 (1980)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zeng, G.: Two common properties of the Erlang-B function, Erlang-C function, and Engset blocking function. Math. Comput. Model. 37(12), 1287–1296 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.KTH Royal Institute of TechnologyStockholmSweden
  2. 2.Teleopti ABStockholmSweden

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