In this work we look at the delay analysis of a customer in a discrete-time queueing system with one permanent server and one occasional extra server. The arrival process is assumed to be general independent, the buffer size infinite and the service times deterministically equal to one slot. The system is assumed to be in one of two different states; during the UP-state 2 servers are available and during the DOWN-state 1 server is available. State changes can only occur at slot boundaries and mark the beginnings and ends of UP-periods and DOWN-periods. The lengths of the UP-periods, expressed in their number of slots, are assumed to follow a geometric distribution, while the lengths of the DOWN-periods follow a general distribution with rational probability generating function. We provide a method to compute the tail characteristics of the delay of an arbitrary customer based on the theory of the dominant singularity. We illustrate the developed method with several numerical examples.

中文翻译：

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具有偶尔额外服务器的离散时间单服务器队列的延迟分析

在这项工作中，我们着眼于在具有一台永久服务器和一台偶尔额外服务器的离散时间排队系统中客户的延迟分析。假设到达过程是一般独立的，缓冲区大小是无限的，服务时间确定性地等于一个时隙。假设系统处于两种不同状态之一；在 UP 状态期间 2 个服务器可用，在 DOWN 状态 1 期间服务器可用。状态更改只能发生在时隙边界并标记 UP 周期和 DOWN 周期的开始和结束。UP 周期的长度，以它们的槽数表示，假设遵循几何分布，而 DOWN 周期的长度遵循具有有理概率生成函数的一般分布。我们提供了一种基于显性奇点理论计算任意客户延迟尾部特征的方法。我们用几个数值例子说明了开发的方法。