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 non-constant and satisfies: ∫ P{S ≥ u}du < ∞, then there is an unbounded set S ⊂ (E[S],∞) such that for each α ∈ S there exists a unique ergodic fixed point with mean inter-arrival time equal to α. We conjecture that in fact S = (E[S],∞).