A Large Closed Queueing Network Containing Two Types of Node and Multiple Customer Classes: One Bottleneck Station

The paper studies a closed queueing network containing two types of node. The first type (server station) is an infinite server queueing system, and the second type (client station) is a single server queueing system with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. It is assumed that there are r server stations. At the initial time moment all units are distributed in the server stations, and the ith server station contains Ni units, i = 1, 2, ..., r, where all the values Ni are large numbers of the same order. The total number of client stations is equal to k. The expected times between departures in the client stations are small values of the order O(N) (N = N1+N2+ ...+Nr). After service completion in the ith server station a unit is transmitted to the jth client station with probability pi,j (j = 1, 2, ..., k), and being served in the jth client station the unit returns to the ith server station. Under the assumption that only one of the client stations is a bottleneck node, i.e. the expected number of arrivals per time unit to the node is greater than the expected number of departures from that node, the paper derives the representation for non-stationary queue-length distributions in non-bottleneck client stations.