Weak Convergence Limits for Sojourn Times in Cyclic Queues Under Heavy Traffic Conditions

Abstract

We consider sequences of closed cycles of exponential single-server nodes with a single bottleneck. We study the cycle time and the successive sojourn times of a customer when the population sizes go to infinity. Starting from old results on the mean cycle times under heavy traffic conditions, we prove a central limit theorem for the cycle time distribution. This result is then utilised to prove a weak convergence characteristic of the vector of a customer's successive sojourn times during a cycle for a sequence of networks with population sizes going to infinity. The limiting picture is a composition of a central limit theorem for the bottleneck node and an exponential limit for the unscaled sequences of sojourn times for the nonbottleneck nodes.

Weak Convergence Limits for Sojourn Times in Cyclic Queues Under Heavy Traffic Conditions

“…An early survey on both problems is [13,Section 3], which dealt with closed systems of the repairman type: approximating the equilibrium distribution and the processes as a whole. More recent book sections on diffusion approximations for general closed networks via functional central [19]. As will be seen, our results are related to both paths, because on the one hand we only consider systems in steady state, but on the other hand we observe the systems' behaviour over time.…”