Abstract
Most bounds for expected delay, E[D], in GI/GI/c queues are modifications of bounds for the GI/GI/1 case. In this paper we exploit a new delay recursion for the GI/GI/c queue to produce bounds of a different sort when the traffic intensity p = λ/μ = E[S]/E[T] is less than the integer portion of the number of servers divided by two. (S AND T denote generic service and interarrival times, respectively.) We derive two different families of new bounds for expected delay, both in terms of moments of S AND T. Our first bound is applicable when E[S2] < ∞. Our second bound for the first time does not require finite variance of S; it only involves terms of the form E[Sβ], where 1 < β < 2. We conclude by comparing our bounds to the best known bound of this type, as well as values obtained from simulation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D.J. Daley, Inequalities for moments of tails of random variables, with queueing applications, Z. Wahrsch. 41 (1977) 139–143.
D.J. Daley, Some results for the mean waiting-time and work-load in GI/GI/k queues, Preprint (1995).
D.J. Daley, A.Ya. Kreinin and C.D. Trengrove, Inequalities concerning the waiting time in single-server queues: a survey, in: Queueing and Related Models, eds. U.V. Bhat and I.V. Basawa (Oxford Univ. Press, New York, 1992) pp. 177–223.
S.I. Grossman, Calculus Part 1: The Calculus of One Variable (Academic Press, New York, 1981).
J. Kiefer and J. Wolfowitz, On the theory of queues with many servers, Trans. Amer. Math. Soc. 78 (1955) 1–18.
J. Kiefer and J. Wolfowitz, On the characterization of the general queueing process with applications to random walk, Ann. Math. Statist. 27 (1956) 147–161.
J.F.C. Kingman, Inequalities in the theory of queues, J. Roy. Statist. Soc. Ser. B 32 (1970) 102–110.
P. Meyer, Introductory Probability and Statistical Applications (Addison-Wesley, Reading, MA, 1970).
A. Scheller-Wolf, Further moment results for multiserver queues, Working paper, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA (1997).
A. Scheller-Wolf and K. Sigman, Delay moments for FIFO GI/GI/c queues, to appear in Queueing Systems (1997).
R. Wolff, An upper bound for multi-channel queues, J. Appl. Probab. 14 (1977) 884–888.
R. Wolff, Stochastic Modeling and the Theory of Queues (Prentice-Hall, Englewood Cliffs, NJ, 1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Scheller-Wolf, A., Sigman, K. New bounds for expected delay in FIFO M/G/c queues. Queueing Systems 26, 169–186 (1997). https://doi.org/10.1023/A:1019177023405
Issue Date:
DOI: https://doi.org/10.1023/A:1019177023405