Existence, uniqueness and some in variance properties of stationary distributions for general single server queues

Abstract

This note contains a modified proof of the theorem of Loynes [8] about the existence of a uniquely determined statistical equilibrium for single server system with an arbitrary stationary and ergodic distribution of the initial data and the traffic intensity β <1. The used approach permits a generalization of the theorem of LOYNES for arbitrary work conserving priority systems and waiting systems with arbitrary queueing discipline. On this basis stationary distributions of system parameters in different observation points are introduced. With the help of results from the theory of random point processes some well-known relations between these distributions are generalized respectively new proofs we given. Further some new relations are given.