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

摘要

The paper studies a closed queueing network containing two types of node. Thefirst type (server station) is an infinite server queueing system, and thesecond type (client station) is a single server queueing system with autonomousservice, i.e. every client station serves customers (units) only at randominstants generated by strictly stationary and ergodic sequence of randomvariables. It is assumed that there are $r$ server stations. At the initialtime moment all units are distributed in the server stations, and the $i$thserver station contains $N_i$ units, $i=1,2,...,r$, where all the values $N_i$are large numbers of the same order. The total number of client stations isequal to $k$. The expected times between departures in the client stations aresmall values of the order $O(N^{-1})$ ~ $(N=N_1+N_2+...+N_r)$. After servicecompletion in the $i$th server station a unit is transmitted to the $j$thclient station with probability $p_{i,j}$ ~ ($j=1,2,...,k$), and being servedin the $j$th client station the unit returns to the $i$th server station. Underthe 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 theexpected number of departures from that node, the paper derives therepresentation for non-stationary queue-length distributions in non-bottleneckclient stations.