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May 2000 Asymptotic expansions for a stochastic model of queue storage
Charles Knessl
Ann. Appl. Probab. 10(2): 592-615 (May 2000). DOI: 10.1214/aoap/1019487357


We consider an $M/M/infty$ queue with servers ranked as ${1, 2, 3,\dots}$. The Poisson arrival stream has rate $\lambda$ and each server works at rate $\mu$. A new arrival takes the lowest ranked available server.We let $S$ be the set of occupied servers and $|S|$ is the number of elements of $S$.We study the distribution of max $(S)$ in the asymptotic limit of $\rho = \lambda/\mu \to \infty$.Setting $P(m) = \Pr[max(S) > m]$ we find that the asymptotic structure of the problem is different according as $m = O(1)$ or $m \to \infty$, at the same rate as $\rho$. For the latter it is furthermore necessary to distinguish the cases $m/\rho < 1, m/\rho \approx 1$ and $m/\rho > 1$.We also estimate the average amount of wasted stor- age space, which is defined by $E(max(S)) - \rho$. This is the average number of idle servers that are ranked below the maximum occupied one.We also relate our results to those obtained by probabilistic approaches. Numerical studies demonstrate the accuracy of the asymptotic results.


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Charles Knessl. "Asymptotic expansions for a stochastic model of queue storage." Ann. Appl. Probab. 10 (2) 592 - 615, May 2000.


Published: May 2000
First available in Project Euclid: 22 April 2002

zbMATH: 1056.60095
MathSciNet: MR1768221
Digital Object Identifier: 10.1214/aoap/1019487357

Primary: 34E20 , 60K30

Keywords: asymptotics , first-fit allocation , queue storage

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.10 • No. 2 • May 2000
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