A Limit Result Respecting Graph Structure for a Fully Connected Loss Network with Alternative Routing

Abstract

Recently there has been a considerable amount of work on the transient behaviour of loss networks in two different limiting regimes. The first of these, which we do not consider here, is when link capacities and offered traffics become large but the number of links remains finite. The second is the diverse routing limit when the number of links increases with the offered load to each link held constant. Thus far, however, all results of this latter type have been for simplified models with exchangeable links. In adopting such a simplified model one loses the inherent graph structure of the original loss network, which appears to be a serious drawback. In this paper we consider a loss network with graph structure and show that, subject to natural constraints on the initial configuration, the model behaves asymptotically exactly like one with exchangeable links. Our result is proved by combining the techniques of Gibbens, Hunt and Kelly with those of Hajek for a problem in random graph theory.