Heavy-traffic limits for nearly deterministic queues: stationary distributions

摘要

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 Er-lang E_n/E_n/1 queue, where cyclic thinning of order n is applied to both the interar-rival times and the service times, starting from a "base" M/M/1 model. The models G_n/D/l and D/G_n/1 are special cases of G_n/G_n/l. Since waiting times before starting service in the G/D queue are equivalent to waiting times in an associated G_n/D/l model, where the interarrival times are the sum of n consecutive interarrival times in the original model, the G/D 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 p_n ↑ 1 as n → ∞, where p_n is the traffic intensity in model n.