一个重试超市模型的混沌传播
The Chaos of Propagation in a Retrial Supermarket Model
摘要：
通过将总轨道分解为与每个$N$服务器相关的$N$子轨道（或简称轨道），并通过比较这些轨道上的顾客数，我们引入了一个由$N$相同服务器组成的重试超市模型.值得注意的是，这两个选择数的设计目的可以有效地提高这个重试超市模型的性能指标。本文分析了一个简单而基本的N个相同服务员的重试超市模型，即泊松到达、指数服务和重试次数。为此，我们首先提供了一个详细的概率计算，以建立一个无限维系统的微分方程（或平均场方程）所满足的期望分数向量。然后，当N趋于无穷大时，我们应用算子半群来获得表示该重试超市模型的状态的马尔可夫过程序列的平均场极限（或传播的混沌）。具体地说，通过任意轨道上队长的前两个矩，建立了平均场极限和Lipschitz条件的一些简单而基本的条件。最后，我们证明了该不动点满足一个非线性方程组，该方程组是M/M/1重试排队尾方程的有趣网络推广，并利用该不动点通过数值计算对该重试超市模型进行了性能分析.
英文摘要：
When decomposing the total orbit into $N$ sub-orbits (or simply orbits) related to each of $N$ servers and through comparing the numbers of customers in these orbits, we introduce a retrial supermarket model of $N$ identical servers, where two probing-server choice numbers are respectively designed for dynamically allocating each primary arrival and each retrial arrival into these orbits when the chosen servers are all busy. Note that the designed purpose of the two choice numbers can effectively improve performance measures of this retrial supermarket model. This paper analyzes a simple and basic retrial supermarket model of N identical servers, that is, Poisson arrivals, exponential service and retrial times. To this end, we first provide a detailed probability computation to set up an infinite-dimensional system of differential equations (or mean-field equations) satisfied by the expected fraction vector. Then, as N goes to infinity, we apply the operator semigroup to obtaining the mean-field limit (or chaos of propagation) for the sequence of Markov processes which express the state of this retrial supermarket model. Specifically, some simple and basic conditions for the mean-field limit as well as for the Lipschitz condition are established through the first two moments of the queue length in any orbit. Finally, we show that the fixed point satisfies a system of nonlinear equations which is an interesting networking generalization of the tail equations given in the M/M/1 retrial queue, and also use the fixed point to give performance analysis of this retrial supermarket model through numerical computation.
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