Insensitivity of the mean field limit of loss systems under SQ(d) routeing
Insensitivity of the mean field limit of loss systems under SQ(d) routeing
复制标题
SQ(d) 路由下损耗系统平均场限制的不敏感性
DOI：
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发表时间：
2019
影响因子：
1.2
通讯作者：
R. Mazumdar
中科院分区：
文献类型：
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作者：
Thirupathaiah Vasantam;Arpan Mukhopadhyay;R. Mazumdar
摘要本文研究了SQ（d）路由方案下服务时间服从一般有限均值分布的大型多服务台损失模型。以前的工作已经解决了指数服务时间的情况下，服务器的数量趋于无穷大，从而产生一个平均场模型。极限平均场方程（MFE）的不动点在模拟中被认为对服务时间分布不敏感，但没有证据。虽然损失系统的不敏感性是众所周知的，但即使具有状态依赖输入的模型也属于线性马尔可夫模型。在SQ（d）路由的上下文中，所得到的模型属于非线性马尔可夫过程（其生成器本身取决于分布的过程），传统的参数不直接适用于这类过程。因此，对一般服务时间分布不敏感仍然是一个悬而未决的问题。在这种情况下，获得的MFE构成了一个挑战，由于所得的马尔可夫描述的系统是在正正交的，而不是在指数的情况下，有限链。在本文中，我们首先得到的MFE，然后表明，MFE有一个唯一的不动点，在指数的情况下，与不动点重合，从而建立不灵敏度。该方法是通过一个测度值马尔可夫过程表示和鞅问题来建立平均场极限。
Abstract In this paper, we study a large multi-server loss model under the SQ(d) routeing scheme when the service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity, giving rise to a mean field model. The fixed point of the limiting mean field equations (MFEs) was seen to be insensitive to the service time distribution in simulations, but no proof was available. While insensitivity is well known for loss systems, the models, even with state-dependent inputs, belong to the class of linear Markov models. In the context of SQ(d) routeing, the resulting model belongs to the class of nonlinear Markov processes (processes whose generator itself depends on the distribution) for which traditional arguments do not directly apply. Showing insensitivity to the general service time distributions has thus remained an open problem. Obtaining the MFEs in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFEs and then show that the MFEs have a unique fixed point that coincides with the fixed point in the exponential case, thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean field limit.