Tail asymptotics of the M/G/∞ model

This paper considers the so-called M/G/infinity model: jobs arrive according to a Poisson process with rate lambda, and each of them stays in the system during a random amount of time, distributed as a non-negative random variable B. With N(t) denoting the number of jobs in the system, the random process A(t) records the load imposed on the system in [0, t], i.e., A(t) := int_0^t N(s)ds. The main result concerns the tail asymptotics of A(t)/t: we find an explicit function f(·) such that f(t) ~ P(A(t)/t> rho*(1 + epsilon)), for t large; here rho := lambda EB. A crucial issue is that A(t) cannot be written as the sum of i.i.d. increments, which makes application of the classical Bahadur-Rao result impossible; instead an adaptation of this result is required. We compare the asymptotics found with the (known) asymptotics for rho large (and t fixed).