Hostname: page-component-89b8bd64d-ktprf Total loading time: 0 Render date: 2026-05-08T03:19:58.471Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximum Queue Length and Waiting Time Revisited: Multserver G/G/c Queue

Published online by Cambridge University Press:  27 July 2009

John S. Sadowsky  and
Wojciech Szpankowski
John S. Sadowsky
Affiliation:
School of Electrical Engineering
Wojciech Szpankowski
Affiliation:
Department of Computer Science PurdueUniversity West Lafayette, Indiana 47907
Get access Rights & Permissions [Opens in a new window]

Abstract

We characterize the probabilistic nature of the maximum queue length and the maximum waiting time in a multiserver G/G/c queue. We assume a general i.i.d. interarrival process and a general i.i.d. service time process for each server with the possibility of having different service time distributions for different servers. Under a weak additional condition, we will prove that the maximum queue length and waiting time grow asymptotically in probability as logωn−1 and log n1/θ, respectively, where ω < 1 and θ > 0 are parameters of the queuing system. Furthermore, it is shown that the maximum waiting time – when appropriately normalized – converges in distribution to the extreme distribution Λ(x) = exp(−e−x). The maximum queue length exhibits similar behavior, except that some oscillation caused by the discrete nature of the queue length must be taken into account. The first results of this type were obtained for the G/M/l queue by Heyde and for the G/G/l queue by Iglehart. Our analysis is similar to that of Heyde and Iglehart. The generalization to c > 1 servers is made possible due to the recent characterization of the tail of the stationary queue length and waiting time in a G/G/c queue (cf. Sadowsky and Szpankowski [17]).

Information

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 6 , Issue 2 , April 1992 , pp. 157 - 170
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aldous, D., Hofri, M., & Szpankowski, W. (1992). Maximum size of a dynamic data structure: Hashing with lazy deletion revisited. SIAM Journal of Computing 21, 4, 1992.CrossRefGoogle Scholar
Anderson, C.W. (1970). Extreme value theory for a class of discrete distributions with applications to some stochastic processes. Journal of Applied Probability 7: 99–113.CrossRefGoogle Scholar
Berman, S. (1962). Limiting distribution of the maximum term in sequences of dependent random variables. Annals of Mathematical Statistics 33: 894–908.CrossRefGoogle Scholar
Feller, W. (1971). An introduction to probability and its applications, Vol. II. New York: John Wiley & Sons.Google Scholar
Galambos, J. (1978). The asymptotic theory of extreme order statistics. New York: John Wiley & Sons.Google Scholar
Gniedenko, B.V. (1943). Sur la distribution limite du terme maximum d'une série abéatorie. Annals of Mathematics 44: 423–453.CrossRefGoogle Scholar
Heyde, C.C. (1971). On the growth of the maximum queue length in a stable queue. Operations Research 44: 423–452.Google Scholar
Iglehart, D. (1972). Extreme values in the Gl/GI/l queue. Annals of Mathematical Statistics 43: 627–635.CrossRefGoogle Scholar
Kiefer, J. & Wolfowitz, J. (1955). On the theory of queues with many servers. Transactions of the American Mathematical Society 78: 1–18.CrossRefGoogle Scholar
Kiefer, J. & Wolfowitz, J. (1956). On the characteristics of the general queueing process, with applications to random walk. Annals of Mathematical Statistics 27: 147–161.CrossRefGoogle Scholar
Loynes, R. (1962). The stability of a queue with non-independent inter-arrival and service times. Proceedings of the Cambridge Philosophical Society, 58: 497–520.CrossRefGoogle Scholar
Neuts, M. & Takahashi, Y. (1981). Asymptotic behavior of the stationary distribution in the Gl/PM/c queue with heterogeneous servers. Zeitschrift für Wahrscheinlichkeitsiheorie und Verwunde Gebiete 57: 441–452.CrossRefGoogle Scholar
Nummelin, E. (1987). General irreducible Markov chains and non-negtive operators. Cambridge, UK: Cambridge University Press.Google Scholar
Postnikov, A.G. (1980). Tauberian theory and its applications. Proceedings of the Steklov Institute of Mathematics. Providence, RI: American Mathematical Society.Google Scholar
Sadowsky, J. (1991). Large deviation theory and efficient simulation of excessive backlogs in a Gl/G1/c queue. Transactions on Automatic Control 36: 1383–1394.CrossRefGoogle Scholar
Sadowsky, J. & Szpankowski, W. (1990). On the analysis of the tail queue length and waiting time distributions of a G/G/c queue. Proceedings of PERFORMANCE 90. Edinburgh: North Holland, pp. 93–107.Google Scholar
Sadowsky, J. & Szpankowski, W. (1990). The probability of large queue lengths and waiting times in a heterogeneous Gl/GI/c queue (submitted to a journal).Google Scholar
Serfozo, R. (1988). Extreme values of queue length in M/G/l and Gl/M/l systems. Mathematics of Operations Research 13: 349–357.CrossRefGoogle Scholar
Szpankowski, W. (1989). On the maximum queue length with applications to data structures analysis. Proceedings of the Allerton Conference. Monticello, IL: University of Illinois at Urbana-Champaign, pp. 263–272.Google Scholar
Whitt, W. (1972). Embedded renewal processes in the G1/G/s queue. Journal of Applied Probability 9: 650–658.CrossRefGoogle Scholar
Wolff, W. (1989). Stochastic modeling and the theory of queues. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar