Abstract
For a discrete-time closed cyclic network of single server queues whose service rates are non-decreasing in the queue length, we compute the queue-length distribution at each node in terms of throughputs of related networks. For the asymptotic analysis, we consider sequences of networks where the number of nodes grows to infinity, service rates are taken only from a fixed finite set of non-decreasing sequences, the ratio of customers to nodes has a limit, and the proportion of nodes for each possible service-rate sequence has a limit. Under these assumptions, the asymptotic throughput exists and is calculated explicitly. Furthermore, the asymptotic queue-length distribution at any node can be obtained in terms of the asymptotic throughput. The asymptotic throughput, regarded as a function of the limiting customer-to-node ratio, is strictly increasing for ratios up to a threshold value (possibly infinite) and is constant thereafter. For ratios less than the threshold, the asymptotic queue-length distribution at each node has finite moments of all orders. However, at or above the threshold, bottlenecks (nodes with asymptotically-infinite mean queue length) do occur, and we completely characterize such nodes.
