Abstract Queueing models with batch service have been studied frequently, for instance in the domain of telecommunications or manufacturing. Although the batch server’s capacity may be variable in practice, only a few authors have included variable capacity in their models. We analyse a batch server with multiple customer classes and a variable service capacity that depends on both the number of waiting customers and their classes. The service times are generally distributed and class-dependent. These features complicate the analysis in a non-trivial way. We tackle it by examining the system state at embedded points, and studying the resulting Markov Chain. We first establish the joint probability generating function (pgf) of the service capacity and the number of customers left behind in the queue immediately after service initiation epochs. From this joint pgf, we extract the pgf for the number of customers in the queue and in the system respectively at service initiation epochs and departure epochs, and the pgf of the actual server capacity. Combined with additional techniques, we also obtain the pgf of the queue and system content at customer arrival epochs and random slot boundaries, and the pgf of the delay of a random customer. In the numerical experiments, we focus on the impact of correlation between the classes of consecutive customers, and on the influence of different service time distributions on the system performance.

中文翻译：

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具有可变服务容量、相关客户类型和一般分布的类相关服务时间的批量服务队列分析

摘要 具有批处理服务的排队模型已被频繁研究，例如在电信或制造领域。尽管批处理服务器的容量在实践中可能是可变的，但只有少数作者在他们的模型中包含了可变容量。我们分析了具有多个客户类别和可变服务容量的批处理服务器，该服务容量取决于等待客户的数量及其类别。服务时间通常是分布式的并且依赖于类。这些特征以一种非平凡的方式使分析复杂化。我们通过检查嵌入点的系统状态并研究由此产生的马尔可夫链来解决它。我们首先建立服务能力和服务启动时期后立即留在队列中的客户数量的联合概率生成函数（pgf）。从这个联合 pgf 中，我们分别提取服务启动时期和离开时期队列中和系统中客户数量的 pgf，以及实际服务器容量的 pgf。结合其他技术，我们还获得了客户到达时期和随机时隙边界的队列和系统内容的 pgf，以及随机客户延迟的 pgf。在数值实验中，我们关注连续客户类别之间相关性的影响，以及不同服务时间分布对系统性能的影响。我们还获得了在客户到达时期和随机时隙边界的队列和系统内容的 pgf，以及随机客户延迟的 pgf。在数值实验中，我们关注连续客户类别之间相关性的影响，以及不同服务时间分布对系统性能的影响。我们还获得了在客户到达时期和随机时隙边界的队列和系统内容的 pgf，以及随机客户延迟的 pgf。在数值实验中，我们关注连续客户类别之间相关性的影响，以及不同服务时间分布对系统性能的影响。