Tail asymptotics for busy periods

TL;DR

This paper analyzes the probability of large busy periods in queues by deriving large-deviation asymptotics for the area under a random walk until it returns to zero, providing insights into the likelihood and typical paths of such events.

Contribution

It introduces a large-deviations framework for busy periods with non-i.i.d. increments, deriving asymptotic probabilities and the most likely paths, addressing an open problem in queue theory.

Findings

The probability of a large busy period decays exponentially with a rate proportional to the square root of its size.

The most likely path to a large busy period is a strictly concave rescaling of the cumulant generating function.

Results enable estimation of large busy period likelihood from observed increment data.

Abstract

The busy period for a queue is cast as the area swept under the random walk until it first returns to zero, $B$. Encompassing non-i.i.d. increments, the large-deviations asymptotics of $B$ is addressed, under the assumption that the increments satisfy standard conditions, including a negative drift. The main conclusions provide insight on the probability of a large busy period, and the manner in which this occurs: I) The scaled probability of a large busy period has the asymptote, for any $b>0$, \lim_{n\to\infty} \frac{1}{\sqrt{n}} \log P(B\geq bn) = -K\sqrt{b}, \hbox{where} \quad K = 2 \sqrt{-\int_0^{\lambda^*} \Lambda(\theta) d\theta}, \quad \hbox{with $\lambda^*=\sup\{\theta:\Lambda(\theta)\leq0\}$,} and with $\Lambda$ denoting the scaled cumulant generating function of the increments process. II) The most likely path to a large swept area is found to be a simple rescaling of the path on $[0,1]$ given by, [\psi^*(t) = -\Lambda(\lambda^*(1-t))/\lambda^*.] In contrast to the piecewise linear most likely path leading the random walk to hit a high level, this is strictly concave in general. While these two most likely paths have very different forms, their derivatives coincide at the start of their trajectories, and at their first return to zero. These results partially answer an open problem of Kulick and Palmowski regarding the tail of the work done during a busy period at a single server queue. The paper concludes with applications of these results to the estimation of the busy period statistics $(\lambda^*, K)$ based on observations of the increments, offering the possibility of estimating the likelihood of a large busy period in advance of observing one.