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2014 Large deviation asymptotics for busy periods
Ken R. Duffy, Sean P. Meyn
Stoch. Syst. 4(1): 300-319 (2014). DOI: 10.1214/13-SSY098

## Abstract

The busy period for a queue is cast as the area swept under the random walk until it first returns to zero. Encompassing non-i.i.d. increments, the large-deviations asymptotics of the busy period $B$ 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. The scaled probability of a large busy period has the asymptote, for any $b>0$, \begin{aligned}&\lim_{n\to\infty}\frac{1}{\sqrt{n}}\log P(B\geq bn)=-K\sqrt{b},\\[3pt]\hbox{where}\quad K=2&\sqrt{-\int_{0}^{\lambda^{*}}\Lambda(\theta)\,d\theta},\quad \hbox{with }\lambda^{*}=\sup\{\theta:\Lambda(\theta)\leq0\},\end{aligned} and with $\Lambda$ denoting the scaled cumulant generating function of the increments process. 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 distinctly 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.

## Citation

Ken R. Duffy. Sean P. Meyn. "Large deviation asymptotics for busy periods." Stoch. Syst. 4 (1) 300 - 319, 2014. https://doi.org/10.1214/13-SSY098

## Information

Published: 2014
First available in Project Euclid: 18 September 2014

zbMATH: 1310.60130
MathSciNet: MR3353220
Digital Object Identifier: 10.1214/13-SSY098

Subjects:
Primary: 60K25
Secondary: 60F10  