Bottlenecks in Markovian Queueing Networks

We will consider the problem of identifying regions of congestion in closed queueing networks with state-dependent service rates. A particular queue will be called a bottleneck if the number of customers in that queue grows without bound as the total number of customers in the network becomes large. We will review methods for identifying potential bottlenecks, with a view to controlling congestion. We will see that the problem of identifying bottlenecks can be reduced to one of finding them in an isolated subnetwork with suitably modified routing intensities. Several special cases will be studied, illustrating a range of behaviour. For example, it is possible for a subnetwork to be congested, yet each queue in that subnetwork is not strictly a bottleneck.