马尔可夫超市模型的超指数解：框架与挑战
Super-Exponential Solution in Markovian Supermarket Models: Framework and Challenge
摘要：
Marcel F. Neuts利用PH分布和马尔可夫到达过程为随机模型的数值计算打开了一扇钥匙。为了庆祝他的75岁生日，本文报道了一个更一般的马尔可夫超市模型的框架，包括一个微分方程组的分数测度和一个非线性方程组的不动点。为了更好地理解这一框架，本文对M/G/1型、GI/M/1型和多重选择三个重要的超市实例进行了详细的分析，说明了如何利用密度依赖跳马尔可夫过程导出微分方程组，并通过求解非线性方程组证明了不动点可以是简单的超指数的.由于超市模型是一类复杂的排队系统，其分析不能应用流行的排队论，因此在超市模型的研究中有必要总结出这样一个更一般的框架，使我们能够专注于重要的研究问题。在此基础上，本文发展了马尔可夫超市模型的矩阵分析方法。希望能为超市模式的性能评估开辟一条矩阵分析方法的新途径。
英文摘要：
Marcel F. Neuts opened a key door in numerical computation of stochastic models by means of phase-type (PH) distributions and Markovian arrival processes (MAPs). To celebrate his 75th birthday, this paper reports a more general framework of Markovian supermarket models, including a system of differential equations for the fraction measure and a system of nonlinear equations for the fixed point. To understand this framework heuristically, this paper gives a detailed analysis for three important supermarket examples: M/G/1 type, GI/M/1 type and multiple choices, explains how to derive the system of differential equations by means of density-dependent jump Markov processes, and shows that the fixed point may be simply super-exponential through solving the system of nonlinear equations. Note that supermarket models are a class of complicated queueing systems and their analysis can not apply popular queueing theory, it is necessary in the study of supermarket models to summarize such a more general framework which enables us to focus on important research issues. On this line, this paper develops matrix-analytical methods of Markovian supermarket models. We hope this will be able to open a new avenue in performance evaluation of supermarket models by means of matrix-analytical methods.
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