On the heavy-traffic limit theorem for GI/G/∞ queues
On the heavy-traffic limit theorem for GI/G/∞ queues
复制标题
关于GI/G/∞队列的大流量极限定理
DOI：
10.2307/1426738
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发表时间：
1982
影响因子：
1.2
通讯作者：
W. Whitt
中科院分区：
文献类型：
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作者：
W. Whitt
为GI/G/s队列的Iglehart (1965), (1973) -Borovkov（1967）大流量极限定理提供了一个有启发性的替代证明。这种大流量是通过考虑一系列GI/G/s系统而得到的，这些系统的服务器数量和到达率趋于∞，而服务时间分布保持不变。该定理建立了收敛于高斯过程，通常不是马尔可夫过程，对于表示在任意时间服务的客户数量的随机过程序列的适当规范化。新证明的关键思想是考虑服务时间分布是指数阶段的随机停止和，然后用离散时间向量值马尔可夫链表示到达时刻服务的每个阶段的顾客数量。然后，通过应用Stroock和Varadhan（1979）中的简单准则，很容易证明该马尔可夫链序列收敛于多元O-U （Ornstein-Uhlenbeck）扩散过程。这些特殊服役时间分布的Iglehart-Borovkov极限是这个多元O-U过程各分量的和。利用随机阶性质，建立了GI/M/s队列在相同条件下的稳态分布的大流量收敛性。
A revealing alternate proof is provided for the Iglehart (1965), (1973)–Borovkov (1967) heavy-traffic limit theorem for GI/G/s queues. This kind of heavy traffic is obtained by considering a sequence of GI/G/s systems with the numbers of servers and the arrival rates going to ∞ while the service-time distributions are held fixed. The theorem establishes convergence to a Gaussian process, which in general is not Markov, for an appropriate normalization of the sequence of stochastic processes representing the number of customers in service at arbitrary times. The key idea in the new proof is to consider service-time distributions that are randomly stopped sums of exponential phases, and then work with the discrete-time vector-valued Markov chain representing the number of customers in each phase of service at arrival epochs. It is then easy to show that this sequence of Markov chains converges to a multivariate O–U (Ornstein–Uhlenbeck) diffusion process by applying simple criteria in Stroock and Varadhan (1979). The Iglehart–Borovkov limit for these special service-time distributions is the sum of the components of this multivariate O–U process. Heavy-traffic convergence is also established for the steady-state distributions of GI/M/s queues under the same conditions by exploiting stochastic-order properties.