On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case.

Two of the most popular approximations for the distribution of the steady-state waiting time, W∞, of the M/G/1 queue are the socalled heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic intensity,�, is close to 1 and the processing times have finite variance, the heavy-traffic approximation s that the distribution of W∞ is roughly exponential at scale O((1 �) −1 ), while the heavy tailed asymptotic describes power law decay in the tail of the distribution of W∞ for a fixed traffic intensity. In this paper, we assume a regularly varying processing time distribution and obtain a sharp threshold in terms of the tail value, or equivalently in terms of (1 �), that describes the point at which the tail behavior transitions from the heavy-traffic regime to the heavy-tailed asymptotic. We also provide new approximations that are either uniform in the traffic intensity, or uniform on the positive axis, that avoid the need to use different expressions on the two regions defined by the threshold. 1. Introduction. A substantial literature has been developed over the last forty years that recognizes the simplifications that arise in the analysis of queueing systems in the presence of “heavy traffic.” The earliest such “heavy traffic” approximation was that obtained byKingman (1961, 1962) for the steady-state waiting time W∞ for the G/G/1 queue. In particular, let Wn be the waiting time (exclusive of service) of the nth customer for a first-in first-out (FIFO) single-server queue (with an infinite capacity waiting room) fed by a renewal arrival process [with i.i.d. inter-arrival times (�n:n ≥ 1)] and an independent stream of i.i.d. processing times (Vn:n ≥ 0). If � ,