On the Asymptotic Behaviour of Queues

Abstract
The homogeneous process N(t) on the lattice of integers in continuous time is discussed. When the increment probability generating function ϵ(u) is analytic at u = 1, the transition functions γn(t) = p{N(t) = n | N(0) = 0} are Bessel‐like functions with simple asymptotic properties. The asymptotic behaviour of a variety of M/G/1, G/M/1 and related queueing systems may be deduced from these properties. An appendix is devoted to a review of saddle point methods and central limit behaviour for processes in continuous time.