Tamrakar G K, Banerjee A
Indian Institute of Technology (BHU), Varanasi, India.
Int J Appl Comput Math. 2021;7(6):252. doi: 10.1007/s40819-021-01194-0. Epub 2021 Nov 22.
Queueing models with vacations have drawn the attention of researchers over several decades as a handy tool for tackling real-life congestion problems. Keeping this in mind, we pay attention to an infinite buffer single server batch-size dependent batch service queue with queue size (queue length) dependent vacation. The arrival pattern of the customers in the system follows the Poisson process where they get the service in packets/group following the general batch service (GBS) rule. An embedded Markov chain technique is used for the mathematical analysis where service (vacation) completion epochs have been taken as an embedded Markov point. We obtain the bivariate generating functions of the queue size and vacation type (queue size at vacation initiation epoch) at vacation termination epoch, and the bivariate generating function of the queue size and batch size with the server at service completion epoch, and then we successfully extract the steady-state joint probabilities of the queue size and batch size with the server and the joint probabilities of the queue size and vacation type at various epochs. Finally, various performance measures are presented. Also, the behavior of the considered model is presented by the graphs and tables.
几十年来，带休假的排队模型作为解决现实生活中拥堵问题的便捷工具，一直吸引着研究人员的关注。基于此，我们关注一个具有无限缓冲区的单服务器批量服务队列，该队列的批量大小取决于队列长度，且休假取决于队列大小（队列长度）。系统中客户的到达模式遵循泊松过程，他们按照一般批量服务（GBS）规则以分组/批次的形式接受服务。采用嵌入式马尔可夫链技术进行数学分析，其中将服务（休假）完成时刻作为嵌入式马尔可夫点。我们得到了休假终止时刻队列大小和休假类型（休假开始时刻的队列大小）的二元生成函数，以及服务完成时刻服务器的队列大小和批量大小的二元生成函数，然后成功提取了服务器的队列大小和批量大小的稳态联合概率以及不同时刻队列大小和休假类型的联合概率。最后，给出了各种性能指标。此外，通过图表展示了所考虑模型的行为。
