A heavy-traffic theorem for the GI/G/1 queue with a Pareto-type service time distribution (Q1271241)

The author considers a GI/G/1 queueing model, with traffic load \(a\), in which the service time distribution \(B(t)\) has a Pareto-type tail (i.e., for \(t \rightarrow \infty, 1-B(t) \sim \frac{c}{t^\nu}+ O[e^{-\delta t}], c>0, 1<\nu<2, \delta>0\)) and the \(\mu\)th moment of the interarrival time distribution \(A(t)\) exists for \(\mu>\nu\). The author proves that \((1-a)^{1/(\nu-1)} {\mathbf w}\) converges in distribution for \(a \uparrow 1\), where \({\mathbf w}\) is distributed as the stationary waiting time distribution. He obtains the Laplace-Stieltjes transform of the limiting distribution and an asymptotic series for its tail probabilities.