Chakravarthy Srinivas R, Rumyantsev Alexander
Departments of Industrial and Manufacturing Engineering, Mathematics, Kettering University, Flint, MI 48504 USA.
Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani Campus, Pilani, Rajasthan 333031 India.
Methodol Comput Appl Probab. 2021;23(4):1551-1579. doi: 10.1007/s11009-020-09828-4. Epub 2020 Oct 19.
In this paper we consider a single server queueing model with under general bulk service rule with infinite upper bound on the batch size which we call . The arrivals occur according to a batch Markovian point process and the services are generally distributed. The customers arriving after the service initiation cannot enter the ongoing service. The service time is independent on the batch size. First, we employ the classical embedded Markov renewal process approach to study the model. Secondly, under the assumption that the services are of phase type, we study the model as a continuous-time Markov chain whose generator has a very special structure. Using matrix-analytic methods we study the model in steady-state and discuss some special cases of the model as well as representative numerical examples covering a wide range of service time distributions such as constant, uniform, Weibull, and phase type.
在本文中，我们考虑一个单服务器排队模型，其具有一般批量服务规则，批量大小有无限上界，我们称之为 。到达过程根据批量马尔可夫点过程发生，服务时间服从一般分布。服务开始后到达的客户不能进入正在进行的服务。服务时间与批量大小无关。首先，我们采用经典的嵌入式马尔可夫更新过程方法来研究该模型。其次，在服务时间为相位型的假设下，我们将该模型作为一个连续时间马尔可夫链来研究，其生成器具有非常特殊的结构。使用矩阵分析方法，我们研究该模型的稳态，并讨论该模型的一些特殊情况以及涵盖广泛服务时间分布（如常数、均匀、威布尔和相位型）的代表性数值示例。
