The mean-field computation in a supermarket model with server multiple vacations
The mean-field computation in a supermarket model with server multiple vacations
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DOI：
10.1007/s10626-013-0171-5
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发表时间：
2013-11
期刊：
影响因子：
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通讯作者：
Quanlin Li;Guirong Dai;John C.S. Lui;Yang Wang
中科院分区：
文献类型：
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作者：
Quanlin Li;Guirong Dai;John C.S. Lui;Yang Wang
虽然休假过程被认为是服务器的普通行为，但对具有服务器休假的休假网络的研究是有限的，有趣的和具有挑战性的。本文对一类服务员多重休假的超市模型的研究提供了一种统一有效的泛函分析方法。首先，我们分析了具有多重休假的N个相同服务员的超市模型，利用尾方程技巧建立了期望分数向量所满足的无穷维微分（或平均场）方程组.其次，当N → ∞时，我们利用算子半群为Markov过程序列提供了一个平均场极限，该平均场极限渐近地逼近由无穷维极限微分方程组的唯一全局解所确定的一条轨迹.第三，给出了一个计算无限维极限微分方程组不动点的有效算法，并利用该不动点给出了该超市模型的性能分析，包括任意服务台的稳态队长均值和任意到达顾客在该系统中的期望逗留时间。最后，我们用数值例子分析了超市模型中的一些关键因素对绩效指标的影响。值得注意的是，本文的方法将是有用的和有效的性能分析的复杂超市模型的资源管理方面的实际领域，如计算机网络，制造系统和运输网络。
While vacation processes are considered to be ordinary behavior for servers, the study of queueing networks with server vacations is limited, interesting, and challenging. In this paper, we provide a unified and effective method of functional analysis for the study of a supermarket model with server multiple vacations. Firstly, we analyze a supermarket model ofNidentical servers with server multiple vacations, and set up an infinite-dimensional system of differential (or mean-field) equations, which is satisfied by the expected fraction vector, in terms of a technique of tailed equations. Secondly, asN→ ∞ we use the operator semigroup to provide a mean-field limit for the sequence of Markov processes, which asymptotically approaches a single trajectory identified by the unique and global solution to the infinite-dimensional system of limiting differential equations. Thirdly, we provide an effective algorithm for computing the fixed point of the infinite-dimensional system of limiting differential equations, and use the fixed point to give performance analysis of this supermarket model, including the mean of stationary queue length in any server and the expected sojourn time that any arriving customer spends in this system. Finally, we use some numerical examples to analyze how the performance measures depend on some crucial factors of this supermarket model. Note that the method of this paper will be useful and effective for performance analysis of complicated supermarket models with respect to resource management in practical areas such as computer networks, manufacturing systems and transportation networks.