Heavy-traffic limits for nearly deterministic queues: stationary distributions

Abstract

We establish heavy-traffic limits for stationary waiting times and other performance measures in G _n /G _n /1 queues, where G _n indicates that an original point process is modified by cyclic thinning of order n, i.e., the thinned process contains every nth point from the original point process. The classical example is the Erlang E _n /E _n /1 queue, where cyclic thinning of order n is applied to both the interarrival times and the service times, starting from a “base” M/M/1 model. The models G _n /D/1 and D/G _n /1 are special cases of G _n /G _n /1. Since waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated G _n /D/1 model, where the interarrival times are the sum of n consecutive interarrival times in the original model, the G/D/n model is a special case as well. As n→∞, the G _n /G _n /1 models approach the deterministic D/D/1 model. We obtain revealing limits by letting ρ _n ↑1 as n→∞, where ρ _n is the traffic intensity in model n.