ABSTRACT

The current investigation of our paper is concentrated on an infinite-buffer single-server queueing model with discrete-time batch Markovian arrival process and Markovian service process. The steady-state behaviour of this discrete-time queueing system at various significant time epochs are presented. We first analyze the system-length distribution at outside observer’s epoch using the $RG$-factorization technique combined with the spectral expansion method to compute the stochastic matrix $\mathbf{G}$. For the purpose of comparative study, a comprehensive analysis of the system-length distribution at outside observer’s epoch is also carried out using the roots method. Next, we obtain the explicit closed-form expressions of the system-length distributions at pre-arrival, intermediate, post-departure and random epochs by developing the relations among them in equilibrium state. The analysis of waiting-time distribution in the queue measured in slots of an arbitrary customer in an arriving batch is also discussed. Computational experiences with adequate numerical results are conferred in the structure of tables and graphs. Finally, we suggest that our queueing model seems to be fitted for outpatient service of any specialized hospital in a smart city.

摘要

本文的当前研究集中在具有离散时间批马尔可夫到达过程和马尔可夫服务过程的无限缓冲单服务器排队模型。给出了该离散时间排队系统在各个重要时间段的稳态行为。我们首先使用$\mathrm{[R}G$分解技术结合频谱扩展方法计算随机矩阵 $\mathbf{G}$。为了进行比较研究，还使用根法对外部观察者时代的系统长度分布进行了综合分析。接下来，通过在平衡状态下建立它们之间的关系，我们获得了到达前，中间，出发后和随机时期的系统长度分布的显式封闭形式表达式。还讨论了在到达批次中任意客户的插槽中测量的队列中等待时间分布的分析。在表格和图表的结构中赋予了具有足够数值结果的计算经验。最后，我们建议我们的排队模型似乎适合智能城市中任何专业医院的门诊服务。