arXiv 1112.18152011-12math.PR

忙期的尾渐近性

Tail asymptotics for busy periods

Ken R. Duffy, Sean P. Meyn

备注：: 15 pages, 5 figures
发表信息：: Stochastic Systems, 4 (1), 300-319, 2014
DOI：: 10.1214/13-SSY098

摘要：

队列的繁忙期被转换为随机游走下扫描的区域，直到它第一次返回到零，$B$。包括非i.i.d.增量，$B$的大偏差渐近处理，假设增量满足标准条件，包括负漂移。主要结论提供了关于大繁忙期的概率以及发生这种情况的方式的见解：I）对于任何$b>0$，大繁忙时段的缩放概率具有渐近线，\lim_{n\to\infty} \frac{1}{\sqrt{n}} \log P（B \geq bn）= -K \sqrt{b}，\hbox{where} \quad K = 2 \sqrt{-\int_0^{λ^*} Λ(θ) dθ}，\quad \hbox{with $λ^*=\sup\{θ:Λ(θ)\leq0\}$,}，$Λ$表示增量过程的标度累积量生成函数。II）最有可能到达大扫掠面积的路径被发现是对$[0,1]$上的路径进行简单的重新缩放，由[λ ^*（t）= -Λ（λ^*（1-t））/λ^*]给出。与导致随机游走到达高水平的分段线性最可能路径相反，这通常是严格凹的。虽然这两个最可能的路径有非常不同的形式，但它们的导数在它们的轨迹开始时重合，并且在它们第一次归零时重合。这些结果部分回答了Kulick和Palmowski关于在单个服务器队列繁忙期间所做工作的尾部的一个公开问题。本文的结论与应用这些结果估计的忙期统计$(λ^*, K)$的增量的观察的基础上，提供了可能性估计的可能性，一个大的忙期提前观察。

英文摘要：

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^{λ^*} Λ(θ) dθ}, \quad \hbox{with $λ^*=\sup\{θ:Λ(θ)\leq0\}$,} and with $Λ$ 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, [ψ^*(t) = -Λ(λ^*(1-t))/λ^*.] 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 $(λ^*, K)$ based on observations of the increments, offering the possibility of estimating the likelihood of a large busy period in advance of observing one.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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