Waiting time with critical load for priority systems with semi-Markov service (Q1074225)

The paper contains limit distributions of actual and virtual waiting times for the following single-server queueing system. There are N Poisson arrival flows. The server has L states and changes them when beginning a new service act in accordance with a Markov matrix P. The d.f. of service time \(\beta_ k\) for the k-th customer is equal to \(Q^ k_{ij}(x)=P(\beta_ K\leq x| i\to j)\) given that the server is in the state i and its next state will be j. Hence, \(P=(Q^ 1_{ij}(\infty))=(Q^ 2_{ij}(\infty))=...=(Q^ N_{ij}(\infty))\quad.\) The author obtained the limit (under heavy traffic) joint distribution of waiting time \(w_ k\) for customers of the k-th flow and the current state of the server. A similar distribution is obtained for the virtual waiting time and the state of the server.