Abstract
Scheller-Wolf [12] established necessary and sufficient conditions for finite stationary delay moments in stable FIFO GI/GI/s queues that incorporate the interaction between service time distribution, traffic intensity (ρ) and the number of servers in the queue. These conditions can be used to show that when the service time has finite first but infinite αth moment, s slow servers can give lower delays than one fast server. In this paper, we derive an alternative derivation of these moment results: Both upper bounds, that serve as sufficient conditions, and lower bounds, that serve as necessary conditions are presented. In addition, we extend the class of service time distributions for which the necessary conditions are valid. Our new derivations provide a structural interpretation of the moment bounds, giving intuition into their origin: We show that FIFO GI/GI/s delay can be represented as the minimum of (s − k) i.i.d. GI/GI/1 delays, when ρ satisfies k < ρ < k+1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Asmussen, Applied Probability and Queues (Springer Verlag, New York, 2003).
D.J. Daley and C.M. Goldie, The moment index of minima (II). Statist. Probab. Lett. 76(8) (2006) 831–837.
S. Foss and D. Korshunov, Heavy tails in multi-server queues. Queueing Systems 52 (2006) 31–48.
M. Harchol-Balter, The effect of heavy-tailed job size. Distributions on Computer System Design. In: Proceedings of ASA-IMS Conference on Applications of Heavy Tailed Distributions in Economics, Engineering and Statistics (Washington, DC, June 1999).
M. Harchol-Balter, Task assignment with unknown duration, Journal of the ACM 49(2) (2002) 260–288.
J. Kiefer and J. Wolfowitz, On the theory of queues with many servers. Transactions of the American Mathematical Society 78 (1955) 1–18.
J. Kiefer and J. Wolfowitz, On the characteristics of the general queueing process with applications to random walk. Annals of Mathematical Statistics 27 (1956) 47–161.
D.S. Mitrinović, Analytic Inequalities (Springer Verlag, Berlin, 1970).
S. Ross, A First Course in Probability, 3rd edition (Macmillan Publishing Company, New York, 1988).
A. Scheller-Wolf and K. Sigman, Delay moments for FIFO GI/GI/c queues. Queueing Systems 25 (1997) 77–95.
A. Scheller-Wolf, Further delay moment results for FIFO multiserver queues. Queueing Systems 34 (2000) 387–400.
A. Scheller-Wolf, Necessary and sufficient conditions for delay moments in FIFO multiserver queues: Why s slow servers are better than one fast server for heavy-tailed systems. Operations Research 51 (2003) 748–758.
K. Sigman and D. Yao, Finite moments for inventory processes. Annals of Applied Probability 4(3) (1994) 765–778.
W. Whitt, The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution. Queueing Systems 36 (2000) 71–87.
R. Wolff, Stochastic Modeling and the Theory of Queues (Prentice-Hall, New Jersey, 1989).
Author information
Authors and Affiliations
Corresponding author
Additional information
AMS Subject Classification 60K25
Rights and permissions
About this article
Cite this article
Scheller-Wolf, A., Vesilo, R. Structural interpretation and derivation of necessary and sufficient conditions for delay moments in FIFO multiserver queues. Queueing Syst 54, 221–232 (2006). https://doi.org/10.1007/s11134-006-0068-1
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11134-006-0068-1