A queueing model with independent arrivals, and its fluid and diffusion limits
A queueing model with independent arrivals, and its fluid and diffusion limits
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DOI：
10.1007/s11134-014-9428-4
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发表时间：
2012-06
期刊：
影响因子：
1.2
通讯作者：
Harsha Honnappa;R. Jain;Amy R. Ward
中科院分区：
文献类型：
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作者：
Harsha Honnappa;R. Jain;Amy R. Ward
本文研究了一个到达者为有序的排队模型，称之为排队。在这里，来自一个固定的，有限的，人口的客户独立的采样时间到达一些给定的分布，并进入队列中的顺序采样到达时间。因此，到达时间是顺序统计量，到达间隔时间是连续顺序统计量的差。它们由单个服务器提供服务，具有独立和相同分布的服务时间，具有一般服务分布。离散事件模型在分析上是难以处理的。因此，我们开发流体和扩散限制的性能指标的队列。观察到的队列长度的流体极限是一个“流体净投放”过程的反映，而扩散极限是一个布朗运动和布朗桥过程或“扩散净投放”过程的函数。扩散极限可以被视为通过流体净输出过程的Skorokhod调节器在扩散净输出过程的方向上的方向导数来反映。我们还观察到什么可能被解释为样本路径利特尔定律。样本路径分析揭示了各种操作制度，其中扩散限制之间的自由扩散，反射扩散过程，和零过程，与可能的不连续性，在政权切换。在拓扑中建立了弱收敛结果。
We study a queueing model with ordered arrivals, which can be called thequeue. Here, customers from a fixed, finite, population independently sample a time to arrive from some given distribution, and enter the queue in order of the sampled arrival times. Thus, the arrival times are order statistics, and the inter-arrival times are differences of consecutive order statistics. They are served by a single server with independent and identically distributed service times, with general service distribution. The discrete event model is analytically intractable. Thus, we develop fluid and diffusion limits for the performance metrics of the queue. The fluid limit of the queue length is observed to be a reflection of a ‘fluid netput’ process, while the diffusion limit is observed to be a function of a Brownian motion and a Brownian bridge process or ‘diffusion netput’ process. The diffusion limit can be seen as being reflected through the directional derivative of the Skorokhod regulator of the fluid netput process in the direction of the diffusion netput process. We also observe what may be interpreted as a sample path Little’s Law. Sample path analysis reveals various operating regimes where the diffusion limit switches between a free diffusion, a reflected diffusion process, and the zero process, with possible discontinuities during regime switches. The weak convergence results are established in thetopology.