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

For the GI/G/1 queueing model with traffic load a < 1, service time distribution B(t) and interarrival time distribution A(t), whenever, for t → ∞, 1 – B(t) ∼ (c/(t/β)^ν) + O(e^–δt), c > 0, 1 < ν < 2, δ > 0, and ∫^∞0 t^μdA(t) < ∞ for μ > ν, (1 – a)^1/ν–1w converges in distribution for a ↑ 1. Here w is distributed as the stationary waiting time distribution. The Laplace –Stieltjes transform of the limiting distribution is derived and an asymptotic series for its tail probabilities is obtained. The theorem actually proved in the text concerns a slightly more general asymptotic behaviour of 1 – B(t), t → ∞, than mentioned above.