The existence of fixed points for the ·/GI/1 queue

A celebrated theorem of Burke's asserts that the Poisson process is a fixed point for a stable exponential single server queue; that is, when the arrival process is Poisson, the equilibrium departure process is Poisson of the same rate. This paper considers the following question: Do fixed points exist for queues which dispense i.i.d. services of finite mean, but otherwise of arbitrary distribution (i.e., the so-called ·/GI/1/∞/FCFS queues)? We show that if the service time S is nonconstant and satisfies ∫\!\! P ≥ u\1/2\,du <∞, then there is an unbounded set ⊂(E[S],∞) such that for each α∈ there exists a unique ergodic fixed point with mean inter-arrival time equal to α. We conjecture that in fact =(E[S],∞).