This paper presents an explicit and straightforward method for finding the sojourn-time distribution of a random customer in an \(M/G^a/1\) queue with a fixed-size batch service. The exhibited process is much more straightforward than the approach discussed by Yu and Tang (Methodology and Computing in Applied Probability 20(4):1503–1514, 2018). We obtain two closed-form expressions for probability density functions by using the inside and outside roots of the underlying characteristic equation. Applying partial fractions and residue theorem, we determine an explicit form of sojourn-time distribution and evaluate the distribution function for any specific time. In illustrative examples, we compare the results obtained by both methods and find that the results match excellently.

本文提出了一种明确且直接的方法，用于在具有固定大小批量服务的\(M/G^a/1\)队列中找到随机客户的逗留时间分布。所展示的过程比 Yu 和 Tang 讨论的方法更直接（Methodology and Computing in Applied Probability 20(4):1503–1514, 2018）。我们通过使用基础特征方程的内根和外根获得概率密度函数的两个封闭形式的表达式。应用部分分数和剩余定理，我们确定了逗留时间分布的显式形式并评估任何特定时间的分布函数。在说明性示例中，我们比较了两种方法获得的结果，发现结果非常匹配。