Analysis of some service systems with low loads

The paper is concerned with stationary distribution of a semi-Markov process $K(t)$ in the presence of a small parameter. The set $F$ of the process states is digital and partially ordered. The minimal states of $F$ are most probable for $K(t)$. The case where these states with $\mu=0$ is one ergodic class a simple rule is obtained for analytical determination of the order of stationary probabilities with respect to $\mu$. A procedure for finding the associated factors is described. This approach is used in analysis of tactics for control of transit fluxes in communication networks with low loads. Asymptotic expressions are given for stationary failure probabilities in Markov models of these networks.