Batch arrival and batch service queueing systems are of importance in the context of telecommunication networks. None of the work reported so far consider the dependence of consecutive arrival and service batches. Batch Markovian Arrival Process($ BMAP $) and Batch Markovian Service Process ($ BMSP $) take care of the dependence between successive inter-arrival and service times, respectively. However in real life situations dependence between consecutive arrival and service batch sizes also play an important role. This is to regulate the workload of the server in the context of service and to restrict the arrival batch size when the flow is from the same source. In this paper we study a queueing system with Markov dependent arrival and service batch sizes. The arrival and service batch sizes are assumed to be finite. Further, successive inter-arrival and service time durations are also assumed to be correlated. Specifically, we consider a $ BMAP/BMSP/1 $ queue with Markov dependent arrival and Markov dependent service batch sizes. The stability of the system is investigated. The steady state probability vectors of the system state and some important performance measures are computed. The Laplace-Stieltjes transform of waiting time and idle time of the server are obtained. Some numerical examples are provided.

中文翻译：

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一种 \begin{document}$ {BMAP/BMSP/1} $\end{document} 具有马尔可夫相关到达和马尔可夫相关服务批次的队列

批量到达和批量服务排队系统在电信网络环境中很重要。迄今为止报告的工作都没有考虑连续到达和服务批次的依赖性。Batch Markovian Arrival Process（$ BMAP $）和Batch Markovian Service Process（$ BMSP $）分别处理连续到达间隔和服务时间之间的依赖关系。然而，在现实生活中，连续到达和服务批次大小之间的依赖性也起着重要作用。这是为了在服务的上下文中调节服务器的工作负载，并在流来自同一源时限制到达批次大小。在本文中，我们研究了具有马尔可夫相关到达和服务批量大小的排队系统。假设到达和服务批次大小是有限的。更多，还假定连续的到达间隔和服务持续时间是相关的。具体来说，我们考虑具有马尔可夫依赖到达和马尔可夫依赖服务批量大小的 $ BMAP/BMSP/1 $ 队列。研究了系统的稳定性。计算系统状态的稳态概率向量和一些重要的性能度量。得到服务器等待时间和空闲时间的Laplace-Stieltjes变换。提供了一些数值示例。得到服务器等待时间和空闲时间的Laplace-Stieltjes变换。提供了一些数值示例。得到服务器等待时间和空闲时间的Laplace-Stieltjes变换。提供了一些数值示例。