Monotonicity and correlation results for queueing network processes, generalized birth–death processes and generalized migration processes are obtained with respect to various orderings of the state space. We prove positive (e.g. association) and negative (e.g. negative association) correlations in space and positive correlations in time for different situations, in steady state as well as in the transient phase of the system. This yields exact bounds for joint probabilities in terms of their independent versions.

A highly symmetric loss network is considered, the symmetric star network previously considered by Whitt and Ziedins and Kelly. As the number of links, K, becomes large, the state space for this process also grows, so we consider a functional of the network, one which contains all information relevant to blocking probabilities within the network but which is easier to analyse. We show that this reduced process obeys a functional law of large numbers and a functional central limit theorem, the limit in this latter case being an Ornstein-Uhlenbeck diffusion process. Finally, by considering the network in equilibrium, we are able to prove that the well-known Erlang fixed point approximation for blocking probabilities is correct to within o(K-1/2) as K → ∞.

In this paper we consider the analysis of call blocking at a single resource with differing capacity requirements as well as differing arrival rates and holding times. We include in our analysis trunk reservation parameters which provide an important mechanism for tuning the relative call blockings to desired levels. We base our work on an asymptotic regime where the resource is in heavy traffic. We further derive, from our asymptotic analysis. methods for the analysis of finite systems. Empirical results suggest that these methods perform well for a wide class of examples.