Limits and Approximations for the Busy-period Distribution in Single-server Queues

Limit theorems are established and relatively simple closed-form approximations are developed for the busy-period distribution in single-server queues. For the M/G/1 queue, the complementary busy-period cdf is shown to be asymptotically equivalent as t → ∞ to a scaled version of the heavy-traffic limit (obtained as ρ → 1), where the scaling parameters are based on the asymptotics as t → ∞. We call this the asymptotic normal approximation, because it involves the standard normal cdf and density. The asymptotic normal approximation is asymptotically correct as t → ∞ for each fixed ρ and as ρ → 1 for each fixed t, and yields remarkably good approximations for times not too small, whereas the direct heavy-traffic (ρ → 1 ) and asymptotic (t → ∞) limits do not yield such good approximations. Indeed, even three terms of the standard asymptotic expansion does not perform well unless t is very large. As a basis for generating corresponding approximations for the busy-period distribution in more general models, we also establish a more general heavy-traffic limit theorem.