Tail asymptotics of the stationary distribution for a two-node generalized Jackson networkTail asymptotics of the stationary distribution for a two-node generalized Jackson network
Tail asymptotics of the stationary distribution for a two-node generalized Jackson networkTail asymptotics of the stationary distribution for a two-node generalized Jackson network
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两节点广义 Jackson 网络的平稳分布的尾部渐进性两节点广义 Jackson 网络的平稳分布的尾部渐进性
DOI：
10.1145/2667522.2667545
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Masakiyo Miyazawa
中科院分区：
文献类型：
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作者：
J. Mori;Y. Kajikawa;H. Kashima;and I. Sakata;Masakiyo Miyazawa
本文的主要目的是在相型条件下，研究两节点广义杰克逊网络平稳分布的尾渐近性，并给出其稳定性条件。这里，相位类型设置意味着到达过程是所谓的马尔可夫到达过程，并且服务时间分布是相位类型。我们考虑了两种类型的尾渐近性，即任意方向上的边际平稳分布的尾衰减率和坐标方向上的联合平稳概率的尾衰减率，这两种尾渐近性对两节点广义杰克逊网络特别有意义，主要有两个原因.在讨论它们之前，我们回顾一下什么是广义杰克逊网络以及它是如何被研究的。当各节点的外生到达和服务时间独立但分布一般时，完成服务的顾客在各节点按给定的概率独立地路由，其服务时间为iid的排队网络称为广义杰克逊网络.这里，假设每个节点处的服务规则是先到先服务的。如果外生到达过程是时齐Poisson过程，服务时间服从指数分布，则该网络成为著名的杰克逊网络。因此，广义杰克逊网络是杰克逊网络的自然推广。在本文中，我们假设每个节点都有一个服务器。
A primary interest of this paper is to find tail asymptotics of the stationary distribution of a generalized Jackson network with two nodes under phase-type setting, provided its stability holds. Here, the phase type setting is meant that the arrival processes are the so called Markov arrival processes, and the service time distributions are of phase type. We consider two types of the tail asymptotics, the tail decay rate of the marginal stationary distribution in an arbitrary direction and those for the joint stationary probabilities in the coordinate directions.There are two major reasons why those tail asymptotics are interesting particularly for the two-node generalized Jackson network. Before discussing them, we recall what is the generalized Jackson network and how it has been studied. A queueing network in which at each node customers finishing service are independently routed according to given probabilities and their service times are iid is called a generalized Jackson network when exogenous arrivals and service times at each node are independent but their distributions are general. Here, service discipline at each node is assumed to be first-come and first-served. If the exogenous arrival processes are time-homogeneous Poisson and the service times are exponentially distributed, then this network becomes the well known Jackson network. Thus, the generalized Jackson network is a natural generalization of the Jackson network. In this paper, we assume that each node has a single server.