In this paper, we analyze the delay performance of queueing systems in which the service rate varies with time and the number of service states may be infinite. Except in some simple special cases, in general, the queueing model with varying service rate is mathematically intractable. Motivated by the P-K formula for M/G/1 queue, we developed a limiting analysis approach based on the connection between the fluctuation of service rate and the mean queue length. Considering the two extreme service rates, we provide a lower bound and upper bound of mean queue length. Furthermore, an approximate mean queue length formula is derived from the convex combination of these two bounds. The accuracy of our approximation has been confirmed by extensive simulation studies with different system parameters. We also verified that all limiting cases of the system behavior are consistent with the predictions made by our formula.

中文翻译：

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服务速率不同的排队系统的近似平均队列长度公式

在本文中，我们分析了排队系统的延迟性能，其中服务速率随时间变化并且服务状态的数量可能是无限的。除了在某些简单的特殊情况下，通常，具有变化的服务速率的排队模型在数学上是难解的。基于M / G / 1队列的PK公式，我们基于服务速率波动与平均队列长度之间的联系开发了一种限制分析方法。考虑到两个极端服务率，我们提供了平均队列长度的下限和上限。此外，从这两个界限的凸组合得出一个近似的平均队列长度公式。我们的近似值的准确性已通过对不同系统参数的广泛仿真研究得到证实。