On the stability of open networks: A unified approach by stochastic dominance

Abstract

Using stochastic dominance, in this paper we provide a new characterization of point processes. This characterization leads to a unified proof for various stability results of open Jackson networks where service times are i.i.d. with a general distribution, external interarrivai times are i.i.d. with a general distribution and the routing is Bernoulli. We show that if the traffic condition is satisfied, i.e., the input rate is smaller than the service rate at each queue, then the queue length process (the number of customers at each queue) is tight. Under the traffic condition, the pth moment of the queue length process is bounded for all t if the p+lth moment of the service times at all queues are finite. If, furthermore, the moment generating functions of the service times at all queues exist, then all the moments of the queue length process are bounded for all t. When the interarrivai times are unbounded and non-lattice (resp. spreadout), the queue lengths and the remaining service times converge in distribution (resp. in total variation) to a steady state. Also, the moments converge if the corresponding moment conditions are satisfied. © 1994 J.C. Baltzer AG, Science Publishers.