On the existence of fixed points for the ./GI/1 queue
Résumé
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 a weak moment condition, then there is an unbounded set S such that for each s in S there exists a unique ergodic fixed point with mean inter-arrival time equal to s. We conjecture that the result is true for all means.