For a class of discrete-time queueing systems, we present a new exact method of computing both the expectation and the distribution of the queue length. This class of systems includes the bulk-service queue and the fixed-cycle traffic-light (FCTL) queue, which is a basic model in traffic-control research and can be seen as a non-exhaustive time-limited polling system. Our method avoids finding the roots of the characteristic equation, which enhances both the reliability and the speed of the computations compared to the classical root-finding approach. We represent the queue-length expectation in an exact closed-form expression using a contour integral. We also introduce several realistic modifications of the FCTL model. For the FCTL model for a turning flow, we prove a decomposition result. This allows us to derive a bound on the difference between the bulk-service and FCTL expected queue lengths, which turns out to be small in most of the realistic cases.

中文翻译：

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一类离散时间排队系统期望队列长度的精确无根方法

对于一类离散时间排队系统，我们提出了一种新的精确方法来计算队列长度的期望和分布。此类系统包括批量服务队列和固定周期红绿灯（FCTL）队列，这是交通控制研究中的基本模型，可以看作是一个非穷举的限时轮询系统。我们的方法避免了寻找特征方程的根，与经典的求根方法相比，这提高了计算的可靠性和速度。我们使用轮廓积分以精确的封闭形式表达队列长度期望。我们还介绍了 FCTL 模型的几个现实修改。对于转向流的 FCTL 模型，我们证明了分解结果。