QED limits for many-server systems under a priority policy



A multi-class many-server priority system operating in the quality-and-efficiency-driven regime is considered. Both arrival processes and service times are general. The many-server heavy-traffic diffusion asymptotic is characterized in terms of the corresponding limiting infinite-server process.


QED regime Heavy traffic Priority scheduling 

Mathematics Subject Classification

60K25 90B22 


  1. 1.
    Aghajani, R., Ramanan, K.: The limit of stationary distributions of many-server queues in the Halfin–Whitt regime. Preprint (2016)Google Scholar
  2. 2.
    Armony, M., Ward, A.: Fair dynamic routing policies in large-scale systems with heterogeneous servers. Oper. Res. 58(3), 624–637 (2010)CrossRefGoogle Scholar
  3. 3.
    Ata, B., Gurvich, I.: On optimality gaps in the Halfin–Whitt regime. Ann. Appl. Probab. 22(1), 407–455 (2012)CrossRefGoogle Scholar
  4. 4.
    Atar, R.: A diffusion model of scheduling control in queueing systems with many servers. Ann. Appl. Probab. 15(1B), 820–852 (2005)CrossRefGoogle Scholar
  5. 5.
    Atar, R.: Scheduling control for queueing systems with many servers: asymptotic optimality in heavy traffic. Ann. Appl. Probab. 15(4), 2606–2650 (2005)CrossRefGoogle Scholar
  6. 6.
    Atar, R., Giat, C., Shimkin, N.: The \(c\mu /\theta \) rule for many-server queues with abandonment. Oper. Res. 58(5), 1427–1439 (2010)CrossRefGoogle Scholar
  7. 7.
    Atar, R., Giat, C., Shimkin, N.: On the asymptotic optimality of the \(c\mu /\theta \) rule under ergodic cost. Queueing Syst. Theory Appl. 67(2), 127–144 (2011)CrossRefGoogle Scholar
  8. 8.
    Atar, R., Kaspi, H., Shimkin, N.: Fluid limits for many-server systems with reneging under a priority policy. Math. Oper. Res. 39(3), 672–696 (2013)CrossRefGoogle Scholar
  9. 9.
    Atar, R., Mandelbaum, A., Reiman, M.: Scheduling a multi class queue with many exponential servers: asymptotic optimality in heavy traffic. Ann. Appl. Probab. 14(3), 1084–1134 (2004)CrossRefGoogle Scholar
  10. 10.
    Atar, R., Shaki, Y.Y., Shwartz, A.: A blind policy for equalizing cumulative idleness. Queueing Syst. Theory Appl. 67(4), 275–293 (2011)CrossRefGoogle Scholar
  11. 11.
    Daffer, P., Taylor, R.: Laws of large numbers for \(D[0,1]\). Ann. Probab. 7(1), 85–95 (1979)CrossRefGoogle Scholar
  12. 12.
    Dai, J., He, S.: Customer abandonment in many-server queues. Math. Oper. Res. 35(2), 347–362 (2010)CrossRefGoogle Scholar
  13. 13.
    de Véricourt, F., Jennings, O.: Dimensioning large-scale membership services. Oper. Res. 56(1), 173–187 (2008)CrossRefGoogle Scholar
  14. 14.
    Erlang, A.K.: On the rational determination of the number of circuits. In: Brockmeyer, E., Halstrom, H.L., Jensen, A. (eds.) The Life and Works of A.K. Erlang, pp. 216–221. The Copenhagen Telephone Company, Copenhagen (1948)Google Scholar
  15. 15.
    Gamarnik, D., Goldberg, D.: On the rate of convergence to stationarity of the M/M/N queue in the Halfin–Whitt regime. Ann. Appl. Probab. 23(5), 1879–1912 (2013)CrossRefGoogle Scholar
  16. 16.
    Gamarnik, D., Goldberg, D.: Steady-state GI/GI/N queue in the Halfin–Whitt regime. Ann. Appl. Probab. 23(6), 2382–2419 (2013)CrossRefGoogle Scholar
  17. 17.
    Gamarnik, D., Momčilović, P.: Steady-state analysis of a multi-server queue in the Halfin–Whitt regime. Adv. Appl. Probab. 40(2), 548–577 (2008)CrossRefGoogle Scholar
  18. 18.
    Gurvich, I., Whitt, W.: Queue-and-idleness-ratio controls in many-server service systems. Math. Oper. Res. 34(2), 363–396 (2009)CrossRefGoogle Scholar
  19. 19.
    Halfin, S., Whitt, W.: Heavy-traffic limits for queues with many exponential servers. Oper. Res. 29(3), 567–588 (1981)CrossRefGoogle Scholar
  20. 20.
    Harrison, J.M., Zeevi, A.: Dynamic scheduling of a multiclass queue in the Halfin–Whitt heavy traffic regime. Oper. Res. 52(2), 243–257 (2004)CrossRefGoogle Scholar
  21. 21.
    Jagerman, D.: Some properties of the Erlang loss function. Bell Syst. Tech. J. 53(3), 525–551 (1974)CrossRefGoogle Scholar
  22. 22.
    Janssen, A.J.E.M., van Leeuwaarden, J.S.H., Zwart, B.: Refining square-root safety staffing by expanding Erlang C. Oper. Res. 59(6), 1512–1522 (2011)CrossRefGoogle Scholar
  23. 23.
    Jelenković, P., Mandelbaum, A., Momčilović, P.: Heavy traffic limits for queues with many deterministic servers. Queueing Syst. Theory Appl. 47(1–2), 53–69 (2004)CrossRefGoogle Scholar
  24. 24.
    Kaspi, H., Ramanan, K.: SPDE limits of many-server queues. Ann. Appl. Probab. 23(1), 145–229 (2013)CrossRefGoogle Scholar
  25. 25.
    Krichagina, E., Puhalskii, A.: A heavy-traffic analysis of a closed queueing system with a GI/\(\infty \) service center. Queueing Syst. Theory Appl. 25(1–4), 235–280 (1997)CrossRefGoogle Scholar
  26. 26.
    Mandelbaum, A., Massey, W., Reiman, M.: Strong approximations for Markovian service networks. Queueing Syst. Theory Appl. 30(1–2), 149–201 (1998)CrossRefGoogle Scholar
  27. 27.
    Mandelbaum, A., Momčilović, P.: Queues with many servers: the virtual waiting-time process in the QED regime. Math. Oper. Res. 33(3), 561–586 (2008)CrossRefGoogle Scholar
  28. 28.
    Mandelbaum, A., Momčilović, P.: Queues with many servers and impatient customers. Math. Oper. Res. 37(1), 41–64 (2012)CrossRefGoogle Scholar
  29. 29.
    Mandelbaum, A., Momčilović, P., Tseytlin, Y.: On fair routing from emergency departments to hospital wards: QED queues with heterogeneous servers. Manag. Sci. 58(7), 1273–1291 (2012)CrossRefGoogle Scholar
  30. 30.
    Mandelbaum, A., Stolyar, A.: Scheduling flexible servers with convex delay costs: heavy-traffic optimality of the generalized \(c \mu \)-rule. Oper. Res. 52(6), 836–855 (2004)CrossRefGoogle Scholar
  31. 31.
    Momčilović, P., Motaei, A.: An analysis of a large-scale machine repair model. Stoch. Syst. (to appear)Google Scholar
  32. 32.
    Puhalskii, A., Reed, J.: On many-server queues in heavy traffic. Ann. Appl. Probab. 20(1), 129–195 (2010)CrossRefGoogle Scholar
  33. 33.
    Puhalskii, A., Reiman, M.: The multiclass GI/PH/N queue in the Halfin–Whitt regime. Adv. Appl. Probab. 32(3), 564–595 (2000)CrossRefGoogle Scholar
  34. 34.
    Reed, J.: The G/GI/N queue in the Halfin–Whitt regime. Ann. Appl. Probab. 19(6), 2211–2269 (2009)CrossRefGoogle Scholar
  35. 35.
    Sigman, K., Whitt, W.: Heavy-traffic limits for nearly deterministic queues: stationary distributions. Queueing Syst. Theory Appl. 69(2), 145–173 (2011)CrossRefGoogle Scholar
  36. 36.
    Tezcan, T.: Optimal control of distributed parallel server systems under the Halfin and Whitt regime. Math. Oper. Res. 33(1), 51–90 (2008)CrossRefGoogle Scholar
  37. 37.
    van Leeuwaarden, J.S.H., Knessl, C.: Spectral gap of the Erlang A model in the Halfin–Whitt regime. Stoch. Syst. 2(1), 149–207 (2012)CrossRefGoogle Scholar
  38. 38.
    van Mieghem, J.: Dynamic scheduling with convex delay costs: the generalized \(c \mu \) rule. Ann. Appl. Probab. 5(3), 809–833 (1995)CrossRefGoogle Scholar
  39. 39.
    van Mieghem, J.: Due-date scheduling: asymptotic optimality of generalized longest queue and generalized largest delay rules. Oper. Res. 51(1), 113–122 (2003)CrossRefGoogle Scholar
  40. 40.
    Zhang, B., van Leeuwaarden, J.S.H., Zwart, B.: Staffing call centers with impatient customers: refinements to many-server asymptotics. Oper. Res. 60(2), 461–474 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA

Personalised recommendations