This paper analyzes an infinite-buffer single-server bulk-service queueing system in which customers arrive according to a discrete-time renewal process. The customers are served under the discrete-time Markovian service process according to the general bulk-service rule. The matrix-geometric method is used to obtain the queue-length distribution at prearrival epoch. The queue-length distributions at other various time epochs are also derived based on prearrival epoch probabilities. A simple approach has been developed to compute the waiting-time distribution of an arriving customer. We also carried out closed-form analytical expression for the service batch size distribution of an arriving customer. Some numerical results are provided in the form of tables for a variety of interarrival-time distributions and model parameters to understand the system behaviour.

本文分析了一个无限缓冲的单服务器批量服务排队系统，其中客户根据离散时间更新过程到达。根据一般批量服务规则，在离散时间马尔可夫服务过程中为客户提供服务。矩阵几何方法用于获得到达前时期的队列长度分布。还基于到达前时期概率来推导其他各个不同时期的队列长度分布。已经开发出一种简单的方法来计算到达客户的等待时间分布。我们还对到达客户的服务批次大小分布进行了封闭式分析表达式。