Weak Convergence Limits for Closed Cyclic Networks of Queues with Multiple Bottleneck Nodes

Abstract

We consider a sequence of cycles of exponential single server nodes, where the number of nodes is fixed and the number of customers grows unboundedly. We prove a central limit theorem for the cycle times distribution. We investigate the idle time structure of the bottleneck nodes and the joint sojourn time distribution that a test customer observes at the non-bottleneck nodes during a cycle. Furthermore, we study the filling behaviour of the bottleneck nodes and show that there is a different asymptotic depending on having a single bottleneck or having multiple bottlenecks.