Queueing Systems

, Volume 79, Issue 3–4, pp 251–291 | Cite as

A law of iterated logarithm for multiclass queues with preemptive priority service discipline

  • Yongjiang Guo
  • Yunan LiuEmail author


A law of iterated logarithm (LIL) is established for a multiclass queueing model, having a preemptive priority service discipline, one server and \(K\) customer classes, with each class characterized by a renewal arrival process and i.i.d. service times. The LIL limits quantify the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensated by their deterministic fluid limits. The LIL is established in three cases: underloaded, critically loaded, and overloaded, for five performance measures: queue length, workload, busy time, idle time, and number of departures. The proof of the LIL is based on a strong approximation approach, which approximates discrete performance processes with reflected Brownian motions. We conduct numerical examples to provide insights on these LIL results.


Law of iterated logarithm Multiclass queues Priority queues Preemptive-resume discipline Non-Markovian queues Strong approximation 

Mathematics Subject Classification

60K25 90B22 60F15 60J65 



We thank Prof. Ward Whitt, Prof. Junfei Huang and the anonymous referees for providing constructive comments. Both authors were supported by NSFC grant 11471053. The first author also acknowledges support from NSFC grant 11101050. The second author also acknowledges support from NSF Grant CMMI 1362310.


  1. 1.
    Bramson, M., Dai, J.G.: Heavy traffic limits for some queueing networks. Ann. Appl. Probab. 11(1), 49–90 (2001)CrossRefGoogle Scholar
  2. 2.
    Harrison, M.: A limit theorem for priority queues in heavy traffic. J. Appl. Probab. 10(4), 907–912 (1973)CrossRefGoogle Scholar
  3. 3.
    Whitt, W.: Weak convergence theorems for priority queues: preemptive-Resume discipline. J. Appl. Probab. 8(1), 74–94 (1971)CrossRefGoogle Scholar
  4. 4.
    Chen, H., Zhang, H.Q.: A sufficient condition and a necessary condition for the diffusion approximations of multiclass queueing networks under priority service disciplines. Queueing Syst. 34(1–4), 237–268 (2000)CrossRefGoogle Scholar
  5. 5.
    Peterson, W.P.: A heavy traffic limit theorem for networks of queues with multiple customer types. Math. Oper. Res. 16(1), 90–118 (1991)CrossRefGoogle Scholar
  6. 6.
    Chen, H.: Fluid approximations and stability of multiclass queueing networks I: work-conserving disciplines. Ann. Appl. Probab. 5(3), 637–665 (1995)CrossRefGoogle Scholar
  7. 7.
    Chen, H., Shen, X.: Strong approximations for multiclass feedforward queueing networks. Ann. Appl. Probab. 10(3), 828–876 (2000)CrossRefGoogle Scholar
  8. 8.
    Dai, J.G.: On the positive Harris recurrence for multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Probab. 5(1), 49–77 (1995)CrossRefGoogle Scholar
  9. 9.
    Zhang, H.Q., Hsu, G.X.: Strong approximations for priority queues: head-of-the-line-first discipline. Queueing Syst. 10(3), 213–234 (1992)CrossRefGoogle Scholar
  10. 10.
    Lévy, P.: Théorie de l’addition des variables aléatories. Gauthier-Villars, Paris (1937)Google Scholar
  11. 11.
    Lévy, P.: Procesus stochastique et mouvement Brownien. Gauthier-Villars, Paris (1948)Google Scholar
  12. 12.
    Csörgő, M., Révész, P.: How big are the increments of a Wiener process? Ann. Probab. 7(4), 731–737 (1979)CrossRefGoogle Scholar
  13. 13.
    Csörgő, M., Révész, P.: Strong Approximations in Probability and Statistics. Academic Press, New York (1981)Google Scholar
  14. 14.
    Iglehart, G.L.: Multiple channel queues in heavy traffic: IV. Law of the iterated logarithm. Z.Wahrscheinlichkeitstheorie verw. Geb. 17, 168–180 (1971)CrossRefGoogle Scholar
  15. 15.
    Minkevičius, S.: On the law of the iterated logarithm in multiserver open queueing networks. Stoch. Int. J. Probab. Stoch. Process. 86(1), 46–59 (2014)Google Scholar
  16. 16.
    Sakalauskas, L.L., Minkevičius, S.: On the law of the iterated logarithm in open queueing networks. Eur. J. Oper. Res. 120(3), 632–640 (2000)CrossRefGoogle Scholar
  17. 17.
    Minkevičius, S., Steišūnas, S.: A law of the iterated logarithm for global values of waiting time in multiphase queues. Statist. Probab. Lett. 61(4), 359–371 (2003)CrossRefGoogle Scholar
  18. 18.
    Chen, H., Yao, D.D.: Fundamentals of Queueing Networks. Springer-Verlag, New York (2001)CrossRefGoogle Scholar
  19. 19.
    Chen, H., Mandelbaum, A.: Hierarchical modeling of stochastic network, part II: strong approximations. In: Yao, D.D. (Eds.) Stochastic Modeling and Analysis of Manufacturing Systems, 107–131 (1994)Google Scholar
  20. 20.
    Strassen, V.: An invariance principle for the law of the iterated logarith. Z. Wahrscheinlichkeitstheorie verw. Geb. 3(3), 211–226 (1964)CrossRefGoogle Scholar
  21. 21.
    Whitt, W.: Stochastic-Process Limits. Springer, Berlin (2002)Google Scholar
  22. 22.
    Glynn, P.W., Whitt, W.: A central-limit-theorem version of \(L=\lambda W\). Queueing Syst. 1(2), 191–215 (1986)CrossRefGoogle Scholar
  23. 23.
    Glynn, P.W., Whitt, W.: Sufficient conditions for functional limit theorem versions of \(L=\lambda W\). Queueing Syst. 1(3), 279–287 (1987)CrossRefGoogle Scholar
  24. 24.
    Glynn, P.W., Whitt, W.: An LIL version of \(L=\lambda W\). Math. Oper. Res. 13(4), 693–710 (1988)CrossRefGoogle Scholar
  25. 25.
    Horváth, L.: Strong approximation of renewal processes. Stoch. Process. Appl. 18(1), 127–138 (1984)CrossRefGoogle Scholar
  26. 26.
    Horváth, L.: Strong approximation of extended renewal processes. Ann. Probab. 12(4), 1149–1166 (1984)CrossRefGoogle Scholar
  27. 27.
    Csörgő, M., Deheuvels, P., Horváth, L.: An approximation of stopped sums with applications in queueing theory. Adv. Appl. Probab. 19(3), 674–690 (1987)CrossRefGoogle Scholar
  28. 28.
    Csörgő, M., Horváth, L.: Weighted Approximations in Probability and Statistics. Wiley, New York (1993)Google Scholar
  29. 29.
    Glynn, P.W., Whitt, W.: A new view of the heavy-traffic limit for infinite-server queues. Adv. Appl. Probab. 23(1), 188–209 (1991)CrossRefGoogle Scholar
  30. 30.
    Zhang, H.Q., Hsu, G.X., Wang, R.X.: Strong approximations for multiple channels in heavy traffic. J. Appl. Probab. 27(3), 658–670 (1990)CrossRefGoogle Scholar
  31. 31.
    Glynn, P.W., Whitt, W.: Departures from many queues in series. Ann. Appl. Probab. 1(4), 546–572 (1991)CrossRefGoogle Scholar
  32. 32.
    Horváth, L.: Strong approximations of open oueueing networks. Math. Oper. Res. 17(2), 487–508 (1992)CrossRefGoogle Scholar
  33. 33.
    Zhang, H.Q.: Strong approximations of irreducible closed queueing networks. Adv. Appl. Probab. 29(2), 498–522 (1997)CrossRefGoogle Scholar
  34. 34.
    Mandelbaum, A., Massey, W.A.: Strong approximations for time-dependent queues. Math. Oper. Res. 20(1), 33–64 (1995)CrossRefGoogle Scholar
  35. 35.
    Mandelbaum, A., Massey, W.A., Reiman, M.: Strong approximations for Markovian service networks. Queueing Syst. 30, 149–201 (1998)CrossRefGoogle Scholar
  36. 36.
    Dai, J.G.: Stability of Fluid and Stochastic Processing Networks, vol. 9. MaPhySto Miscellanea Publication, Aarhus, Denmark (1999)Google Scholar
  37. 37.
    Chen, H.: Rate of convergence of the fluid approximation for generalized Jackson networks. J. Appl. Probab. 33(3), 804–814 (1996)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina
  2. 2.Department of Industrial and Systems EngineeringNorth Carolina State UniversityRaleighUSA

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