This paper analyzes an infinite-buffer single-server queueing system wherein customers arrive in batches of random size according to a discrete-time renewal process. The customers are served one at a time under discrete-time Markovian service process. Based on the censoring technique, the UL-type $ RG $-factorization for the Toeplitz type block-structured Markov chain is used to obtain the prearrival epoch probabilities. The random epoch probabilities are obtained with the help of classical principle based on Markov renewal theory. The system-length distributions at outside observer's, intermediate and post-departure epochs are obtained by making relations among various time epochs. The analysis of waiting-time distribution measured in slots of an arbitrary customer in an arrival batch has also been investigated. In order to unify the results of both discrete-time and its continuous-time counterpart, we give a brief demonstration to get the continuous-time results from those of the discrete-time ones. A variety of numerical results are provided to illustrate the effect of model parameters on the performance measures.

中文翻译：

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使用$ RG $因子分析$ GI ^ {[X]} / D $-$ MSP / 1 / \ infty $队列

本文分析了一个无限缓冲区单服务器排队系统，其中，客户根据离散时间更新过程以随机大小批量到达。在离散时间的马尔可夫服务过程中，一次为客户提供服务。基于检查技术，使用Toeplitz型块结构马尔可夫链的UL型$ RG $分解来获得到达前的时期概率。借助马尔可夫更新理论，借助经典原理获得了随机纪元概率。通过在各个时间纪元之间建立关系，可以获得外部观察者纪元，中间纪元和出发后纪元的系统长度分布。还研究了在到达批次中任意客户的插槽中测量的等待时间分布的分析。为了统一离散时间及其连续时间的结果，我们给出一个简短的演示，以从离散时间的结果中获取连续时间的结果。提供了各种数值结果来说明模型参数对性能指标的影响。