Moments and tails in monotone-separable stochastic networks
Moments and tails in monotone-separable stochastic networks
复制标题
单调可分离随机网络中的矩和尾
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
S. Foss
中科院分区：
文献类型：
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作者：
F. Baccelli;S. Foss
如果网络的状态变量是到达过程的齐次单调函数，则网络属于单调可分类。该框架最初用于导出具有平稳遍历驱动序列的随机网络的稳定区域。它包含几个经典的排队网络模型，包括广义Jackson网络、max-plus网络、轮询系统、多服务器队列和各种类型的随机Petri网。我们的目的是分析稳态变量的尾部在特定情况下的i.i.d驱动序列。为此，我们建立了该类网络与GI/GI/1/的一般比较关系\\排队。我们首先用它证明了GI/GI/1/的渐近理论的两个经典结果\\Infty队列可以直接扩展到这个框架。第一个是关于稳态变量的矩的存在性。我们为所有人确立了这一点 \\α\\geq1, (\\服务时间的α +1)矩条件是存在的充分必要条件 \\在这类网络中，静止最大时间点（通常是停止进一步到达时清空网络的时间）的α矩。第二个是GI/GI/1/中平稳等待时间的veraverbekeka尾渐近的直接推广\\排队。
A network belongs to the monotone separable class if its state variables are homogeneous and monotone functions of the epochs of the arrival process. This framework, which was first introduced to derive the stability region for stochastic networks with stationary and ergodic driving sequences, is revisited. It contains several classical queueing network models, including generalized Jackson networks, max-plus networks, polling systems, multiserver queues, and various classes of stochastic Petri nets. Our purpose is the analysis of the tails of the stationary state variables in the particular case of i.i.d. driving sequences. For this, we establish general comparison relationships between networks of this class and the GI/GI/1/\\infty queue. We first use this to show that two classical results of the asymptotic theory for GI/GI/1/\\infty queues can be directly extended to this framework. The first one concerns the existence of moments for the stationary state variables. We establish that for all \\alpha\\geq 1, the (\\alpha+1)-moment condition for service times is necessary and sufficient for the existence of the \\alpha-moment for the stationary maximal dater (typically the time to empty the network when stopping further arrivals) in any network of this class. The second one is a direct extension of Veraverbeke\'s tail asymptotic for the stationary waiting times in the GI/GI/1/\\infty queue.