马尔可夫队列多服务器大流量限制的鞅证明*

IF 1.3 Q2 STATISTICS & PROBABILITY Probability Surveys Pub Date : 2007-12-27 DOI:10.1214/06-PS091
G. Pang, Rishi Talreja, W. Whitt
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引用次数: 251

摘要

这是一篇说明性的综述文章,说明了“martin- gale方法”用于证明多服务器大流量队列模型的随机过程约束,支持扩散过程近似。本文对一个基本模型——经典的无限服务器模型M/M/1进行了详细的讨论,但也对有限多服务器和客户放弃的模型进行了讨论。表示系统中顾客数量的马尔可夫随机过程用率- 1泊松过程以两种方式构造:(i)通过随机时间变化和(ii)通过随机变薄。通过分别应用:(i)可选停止定理(其中随机时间变化是停止时间)和(ii)与计数过程随机细化相关的积分定理,获得了这些结构的相关鞅表示。应用连续映射定理,得到了适当序列的伸缩排队过程收敛到扩散过程极限的问题。该框架中的关键FCLT和关键FWLLN分别在使用和不使用鞅的情况下建立。
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Martingale proofs of many-server heavy-traffic limits for Markovian queues ∗
This is an expository review paper illustrating the "martin- gale method" for proving many-server heavy-traffic stochastic-process lim- its for queueing models, supporting diffusion-process approximations. Care- ful treatment is given to an elementary model - the classical infinite-server model M/M/1, but models with finitely many servers and customer aban- donment are also treated. The Markovian stochastic process representing the number of customers in the system is constructed in terms of rate- 1 Poisson processes in two ways: (i) through random time changes and (ii) through random thinnings. Associated martingale representations are obtained for these constructions by applying, respectively: (i) optional stop- ping theorems where the random time changes are the stopping times and (ii) the integration theorem associated with random thinning of a counting process. Convergence to the diffusion process limit for the appropriate se- quence of scaled queueing processes is obtained by applying the continuous mapping theorem. A key FCLT and a key FWLLN in this framework are established both with and without applying martingales.
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来源期刊
Probability Surveys
Probability Surveys STATISTICS & PROBABILITY-
CiteScore
4.70
自引率
0.00%
发文量
9
期刊最新文献
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