We analyze a discrete-time two-class queueing system with a single server which is alternately available for only one customer class. The server is each time allocated to a customer class for a geometrically distributed amount of time. Service times of the customers are deterministically equal to 1 time slot each. During each time slot, both classes can have at most one arrival. The bivariate process of the number of customers of both classes can be considered as a two-dimensional nearest-neighbor random walk. The generating function of this random walk has to be obtained from a functional equation. This type of functional equation is known to be difficult to solve. In this paper, we obtain closed-form expressions for the joint probability distribution for the number of customers of both classes, in steady state.

中文翻译：

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带伯努利到达的离散时间两类随机交替服务模型分析

我们分析了一个具有单个服务器的离散时间两类排队系统，该系统仅对一个客户类别交替可用。服务器每次都被分配给一个客户类别，持续几何分布的时间。每个客户的服务时间确定性地等于 1 个时隙。在每个时段内，两个班级最多只能有一个到达。两类客户数量的二元过程可以看作是二维最近邻随机游走。这种随机游走的生成函数必须从函数方程中获得。众所周知，这种类型的函数方程很难求解。在本文中，我们获得了稳态下两类客户数量的联合概率分布的闭式表达式。