Annals of Operations Research

, Volume 264, Issue 1–2, pp 157–191 | Cite as

Functional law of the iterated logarithm for multi-server queues with batch arrivals and customer feedback

Original Paper
  • 66 Downloads

Abstract

A functional law of the iterated logarithm (FLIL) and its corresponding law of the iterated logarithm (LIL) are established for a multi-server queue with batch arrivals and customer feedback. The FLIL and LIL, which quantify the magnitude of asymptotic fluctuations of the stochastic processes around their mean values, are developed in three cases: underloaded, critically loaded and overloaded, for five performance measures: queue length, workload, busy time, idle time and departure process. Both FLIL and LIL are proved using an approach based on strong approximations.

Keywords

Functional law of the iterated logarithm Multi-server queue Batch arrival Customer feedback Nonexponential service times Strong approximation 

Notes

Acknowledgements

The first author acknowledge support from NSFC Grant 11471053. The second author also acknowledges support from NSF Grant CMMI 1362310.

References

  1. Borovkov, A. A. (1984). Asymptotic methods in queueing theory. New York: Wiley.Google Scholar
  2. Chen, H., & Mandelbaum, A. (1994). Hierarchical modeling of stochastic network, part II: Strong approximations. In D. D. Yao (Ed.), Stochastic modeling and analysis of manufacturing systems (pp. 107–131).Google Scholar
  3. Chen, H., & Shanthikumar, J. G. (1994). Fluid limits and diffusion approximations for networks of multi-server queues in heavy traffic. Disctete Event Dynamic Systems, 4, 269–291.CrossRefGoogle Scholar
  4. Chen, H., & Shen, X. (2000). Strong approximations for multiclass feedforward queueing networks. Annals of Applied Probability, 10(3), 828–876.CrossRefGoogle Scholar
  5. Chen, H., & Yao, D. D. (2001). Fundamentals of queueing networks. New York: Springer.CrossRefGoogle Scholar
  6. Csörgő, M., Deheuvels, P., & Horváth, L. (1987). An approximation of stopped sums with applications in queueing theory. Advances in Applied Probability, 19(3), 674–690.CrossRefGoogle Scholar
  7. Csörgő, M., & Horváth, L. (1993). Weighted approximations in probability and statistics. New York: Wiley.Google Scholar
  8. Csörgő, M., & Révész, P. (1981). Strong approximations in probability and statistics. New York: Academic.Google Scholar
  9. Dai, J. G. (1995). On the positive Harris recurrence for multiclass queueing networks: A unified approach via fluid limit models. Annals of Applied Probability, 5(1), 49–77.CrossRefGoogle Scholar
  10. Ethier, S. N., & Kurtz, T. G. (1986). Markov processes: Characterization and convergence. New York: Wiley.CrossRefGoogle Scholar
  11. Glynn, P. W., & Whitt, W. (1986). A central-limit-theorem version of \(L=\lambda W\). Queueing Systems, 1(2), 191–215.CrossRefGoogle Scholar
  12. Glynn, P. W., & Whitt, W. (1987). Sufficient conditions for functional limit theorem versions of \(L=\lambda W\). Queueing Systems, 1(3), 279–287.CrossRefGoogle Scholar
  13. Glynn, P. W., & Whitt, W. (1988). An LIL version of \(L=\lambda W\). Mathematics of Operations Research, 13(4), 693–710.CrossRefGoogle Scholar
  14. Glynn, P. W., & Whitt, W. (1991a). A new view of the heavy-traffic limit for infinite-server queues. Advances in Applied Probability, 23(1), 188–209.CrossRefGoogle Scholar
  15. Glynn, P. W., & Whitt, W. (1991b). Departures from many queues in series. Annals of Applied Probability, 1(4), 546–572.CrossRefGoogle Scholar
  16. Guo, Y., & Liu, Y. (2015). A law of iterated logarithm for multiclass queues with preemptive priority service discipline. Queueing Systems, 79, 251–291.CrossRefGoogle Scholar
  17. Harrison, J. M. (1985). Brownian motion and stochastic flow system. New York: Wiley.Google Scholar
  18. Horváth, L. (1984a). Strong approximation of renewal processes. Stochastic Processes and Their Applications, 18(1), 127–138.CrossRefGoogle Scholar
  19. Horváth, L. (1984b). Strong approximation of extended renewal processes. The Annals of Probability, 12(4), 1149–1166.CrossRefGoogle Scholar
  20. Horváth, L. (1992). Strong approximations of open queueing networks. Mathematics of Operations Research, 17(2), 487–508.CrossRefGoogle Scholar
  21. Iglehart, G. L. (1971). Multiple channel queues in heavy traffic: IV. Law of the iterated logarithm. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 17, 168–180.CrossRefGoogle Scholar
  22. Lee, H. S., & Srinivasan, M. M. (1989). Control policies for \(M^{X}/G/1\) queueing system. Management Science, 35(6), 708–721.CrossRefGoogle Scholar
  23. Lévy, P. (1937). Théorie de l’addition des variables aléatories. Paris: Gauthier-Villars.Google Scholar
  24. Lévy, P. (1948). Procesus stochastique et mouvement Brownien. Paris: Gauthier-Villars.Google Scholar
  25. Liu, Y., & Whitt, W. (2014a). Algorithms for time-varying networks of many-server fluid queues. INFORMS Journal on Computing, 26(1), 59–73.CrossRefGoogle Scholar
  26. Liu, Y., & Whitt, W. (2014b). Stabilizing performance in networks of queues with time-varying arrival rates. Probability in the Engineering and Informational Sciences, 28(4), 419–449.CrossRefGoogle Scholar
  27. Liu, Y., & Whitt, W. (2017). Stabilizing performance in a service system with time-varying arrivals and customer feedback. European Journal of Operational Research, 256(2), 473–486.CrossRefGoogle Scholar
  28. Machihara, F. (1999). A \(BMAP/SM/1\) queue with service times depending on the arrival process. Queueing Systems, 33(4), 277–291.CrossRefGoogle Scholar
  29. Mandelbaum, A., & Massey, W. A. (1995). Strong approximations for time-dependent queues. Mathematics of Operations Research, 20(1), 33–64.CrossRefGoogle Scholar
  30. Mandelbaum, A., Massey, W. A., & Reiman, M. (1998). Strong approximations for Markovian service networks. Queueing Systems, 30, 149–201.CrossRefGoogle Scholar
  31. Minkevičius, S. (2014). On the law of the iterated logarithm in multiserver open queueing networks. Stochastics, 86(1), 46–59.CrossRefGoogle Scholar
  32. Minkevičius, S., & Steišūnas, S. (2003). A law of the iterated logarithm for global values of waiting time in multiphase queues. Statistics and Probability Letters, 61(4), 359–371.CrossRefGoogle Scholar
  33. Pang, G. D., & Whitt, W. (2012). Infinite-server queues with batch arrivals and dependent service times. Probability in the Engineering and Information Sciences, 26, 197–220.CrossRefGoogle Scholar
  34. Reiman, M. I. (1984). Open queueing networks in heavy traffic. Mathematic of Operations Research, 9, 441–458.CrossRefGoogle Scholar
  35. Sakalauskas, L. L., & Minkevičius, S. (2000). On the law of the iterated logarithm in open queueing networks. European Journal of Operational Research, 120(3), 632–640.CrossRefGoogle Scholar
  36. Strassen, V. (1964). An invariance principle for the law of the iterated logarith. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 3(3), 211–226.CrossRefGoogle Scholar
  37. Van Ommeren, J. C. W. (1990). Simple approximations for the batch-arrival \(M^{X}/G/1\) queue. Operations Research, 38(4), 678–685.CrossRefGoogle Scholar
  38. Whitt, W. (1983). Comparing batch delays and customer delays. Bell System Technical Journal, 62(7), 2001–2009.CrossRefGoogle Scholar
  39. Whitt, W. (2002). Stochastic-process limits. New York: Springer.Google Scholar
  40. Yom-Tov, G., & Mandelbaum, A. (2014). Erlang R: A time-varying queue with reentrant customers, in support of healthcare staffiong. Manufacturing and Service Operations Management, 16, 283–299.CrossRefGoogle Scholar
  41. Zhang, H. (1997). Strong approximations of irreducible closed queueing networks. Advances in Applied Probability, 29(2), 498–522.CrossRefGoogle Scholar
  42. Zhang, H., & Hsu, G. X. (1992). Strong approximations for priority queues: Head-of-the-line-first discipline. Queueing Systems, 10(3), 213–234.CrossRefGoogle Scholar
  43. Zhang, H., Hsu, G. X., & Wang, R. X. (1990). Strong approximations for multiple channels in heavy traffic. Journal of Applied Probability, 27(3), 658–670.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

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

Personalised recommendations