Uniform Large Deviations for Heavy-Tailed Single-Server Queues under Heavy Traffic

We provide a complete large and moderate deviations asymptotic for the steady-state waiting time of a class of subexponential M/G/1 queues under heavy traffic. The asymptotic is uniform over the whole positive axis, and reduces to Kingman’s asymptotic and heavy-tail asymptotic on two ends, both of which are known to be valid only in limited regimes. On the link between these two well-known asymptotics is a transition term that is expressible as a convolution-type integral of negative binomial sum. The class of service times that we consider includes regularly varying and Weibull tails in particular. Steady-state analysis of the waiting time ofM/G/1 first-come-first-serve queue is a classical topic in queueing theory. Despite its apparent tractability, asymptotics that have been proposed in the literature are only provably valid in restricted regimes. Among them are the well-known Kingman’s asymptotic (see Kingman (1961, 1962)) and heavy-tail asymptotic (see for example Embrechts and Veraverbeke (1982)). More precisely, Kingman’s asymptotic approximates the probability that the steady-state distribution exceeds level of order 1/(1 − ρ) under heavy traffic i.e. when the traffic intensity ρ is close to 1. On the other hand, the heavy-tail asymptotic assumes fixed traffic intensity while the level goes to ∞. It states that for service time with stationary excess distribution B0(x) lying in class S of subexponential distribution (see, for example, Embrechts et. al. (1997) and Asmussen (2000)), the probability that the steady-state waiting time is larger than x is asymptotically (ρ/(1− ρ))B̄0(x). In this paper we provide a full asymptotic description of the steady-state waiting time distribution for heavy-tailed M/G/1 under heavy traffic. In particular, we are most interested in the case when heavy traffic is present but the exceedance level is moderate, which is covered by neither Kingman’s or heavy-tail asymptotic. Such approximation is useful in the practical scenario where the system is engineered to be efficiency-driven yet the waiting time is moderately large. The complete asymptotic we propose combines Kingman’s asymptotic on one end and heavy-tail asymptotic on the other, and provide a link in between via a transition term that depends crucially on the service time distribution. Moreover, the asymptotic holds uniformly over the whole positive axis. Our assumption on the service time distribution follows closely from the one given in Rozovskii (1989, 1993), who consider asymptotics for i.i.d. sum, in addition to the class S assumption (see Embrechts et. al. (2003)). This extends and unifies the results of Olvera-Cravioto et. al. (2009) and Olvera-Cravioto and Glynn (2009). In these two papers, the authors studied first the regularly varying M/G/1 queue and showed that Kingman’s and heavy-tail asymptotics remain valid on regimes that are respectively smaller and larger than an explicitly identified transition point. Then, a separate argument is given in Olvera-Cravioto and Glynn (2009) in order to deal with Weibull type distributions. Our framework here provides means to develop a unified theory of transitions from heavy-traffic to heavy-tailed asymptotics that covers both regularly varying and Weibullian tails at once. In addition, and in contrast to Olvera-Cravioto et. al. (2009), for regularly varying