Recently multiserver queues with setup times have been extensively studied because they have applications in power-saving data centers. The most challenging model is the M/M/$c$/Setup queue where a server is turned off when it is idle and is turned on if there are some waiting jobs. Recently, Gandhi et al.~(SIGMETRICS 2013, QUESTA 2014) present the recursive renewal reward approach as a new mathematical tool to analyze the model. In this paper, we derive exact solutions for the same model using two alternative methodologies: generating function approach and matrix analytic method. The former yields several theoretical insights into the systems while the latter provides an exact recursive algorithm to calculate the joint stationary distribution and then some performance measures so as to give new application insights.
翻译：最近对安装时间的多服务器队列进行了广泛研究,因为它们在节能数据中心有应用程序。 最具挑战性的模式是 M/M/$/ $c$/ Setup 队列, 服务器闲置时即关闭, 如果有些等待工作则打开。 最近, 甘地等人 ~ (SIGMETRICS 2013, QUESTA 2014) 介绍了循环更新奖励方法, 作为分析模型的新数学工具。 在本文中, 我们用两种替代方法为同一模型找到精确的解决方案: 生成函数法和矩阵分析法。 前者在系统中产生一些理论洞察力, 而后者则提供精确的循环算法, 以计算联合固定分布, 然后提供一些业绩计量, 以便提供新的应用洞察力。
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