Asymptotic expansions for a stochastic model of queue storage

Abstract

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.