Asymptotic expansions for a stochastic model of queue storage

We consider an M/M/infinity queue with servers ranked as {1, 2, 3,...}. 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 ISI is the number of elements of S. We study the di stribution of max(S) in the asymptotic limit of rho = lambda/mu --> infinit y. 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 --> infinity, at the sa me rate as rho. For the latter it is furthermore necessary to distinguish t he cases m/rho < 1, m/<rho> approximate to 1 and m/rho > 1. We also estimat e the average amount of wasted storage space, which is defined by E (max(S) )- rho. This is the average number of idle servers that are ranked below th e maximum occupied one. We also relate our results to those obtained by pro babilistic approaches. Numerical studies demonstrate the accuracy of the as ymptotic results.