Performance analysis of a constrained resource sharing system

We consider a queueing system where the servers are arranged in a circle, and each arriving customer requires a pair of resources that is shared by its server with the respective neighbors on either side. If either resource is being used, the customer is denied service. Customers arrive at each server according to independent Poisson processes, and lengths of service times at each server have an exponential distribution. We derive a closed-form formula for the expected fraction of busy servers at any time in terms of the number of servers and the utilization factor (defined as the arrival rate times the mean service-time duration). This allows us to evaluate system performance when these parameters are varied, and to determine whether denying service to arrivals at alternate servers improves performance. We relate the system to Dijkstra's dining philosophers problem, which is an abstraction for resource sharing in an operating system.