Large deviation asymptotics for busy periods
Large deviation asymptotics for busy periods
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繁忙期大偏差渐近
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
S. Meyn
中科院分区：
文献类型：
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作者：
K. Duffy;S. Meyn
队列的忙碌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。包括非i.i.d.增量，大偏差渐近的忙碌周期B的解决，假设增量满足标准条件下，包括负漂移。主要的结论提供了一个大的忙碌时期的概率，以及这种情况发生的方式洞察。大忙碌周期的比例概率有渐近线，对于任何B > 0，limn→∞（1/n）logP（B≥bn）=− K B，其中K=2− K 0λ*Λ（θ）dθ，λ*=sup{θ：Λ（θ）≤0}，Λ表示增量过程的比例累积量母函数。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，由下式给出：λ *（t）=−Λ（λ*（1−t））/λ*。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。虽然这两条最有可能的路径有着明显不同的形式，但它们的衍生物会导致...
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, limn→∞(1/n)logP(B≥bn)=−Kb, where K=2−∫0λ*Λ(θ)dθ, with λ*=sup{θ:Λ(θ)≤0}, and with Λ 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 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 distinctly different forms, their derivatives coin...