Uniform Large Deviations for Heavy-tailed 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 positive axis, and reduces to heavy-traffic asymptotics and heavy-tail asymptotics on two ends, both of which are known to be valid over restricted asymptotic regimes. The link between these two well-known asymptotics is a transition term that is expressible as a convolution-type integral. The class of service times that we consider includes regularly varying and Weibull tails in particular. It is our pleasure to contribute to this special issue dedicated to the International Year of Statistics. In response to the request of the editors of this special issue we briefly overview the research topics that we have investigated recently. Our research group has pursued several themes in recent years. All of them lie under the scope of applied probability. Some of our projects deal with computational probability. In this context, our goal is to enable efficient computation in stochastic systems using (and often developing) theory of probability to inform the design of algorithm that are optimal and robust in certain sense (see Blanchet and Glynn (2008)). Most of the computations that we study relate to stochastic simulation (also known as Monte Carlo) methods (see Blanchet and Lam (2012)). Other projects that we pursue relate to classical analysis in probability, such as asymptotic approximations, large deviations, and heavy-traffic limits (Blanchet and Glynn (2006) and Lam et al (2011)). All of our research efforts are motivated by models and problems in areas such as: Finance, Insurance, Operations Research, and Statistics. Here we shall study a class of asymptotic results that lie at the intersection of large deviations and heavy-traffic limit theory. We use a classical model in queueing theory to illustrate these types of results, namely, the classical M/G/1 queue. Despite its apparent tractability, most of the asymptotics for the steadystate waiting time of the M/G/1 queue that have been proposed in the literature are only provably valid in restricted regimes. Among them are the well-known heavy-traffic or Kingman asymptotic (see Kingman (1961)) and the heavy-tail or Pakes-Veraberbeke asymptotic (see for example Embrechts and Veraverbeke (1982)). More precisely, in heavy traffic (i.e. when the long-run proportion of time the server is utilized, ρ, is close to 1) one approximates the distribution of the steady-state waiting time in spatial scales of size 1/(1−ρ) by the steady-state distribution of reflected Brownian motion (which is exponential). On the other hand, the heavy-tail asymptotic assumes fixed traffic intensity while the tail parameter 2010 Mathematics Subject Classification: 60F10.