This article gives closed-form analytic expressions as well as a computational analysis of the stationary system-length distribution for the renewal-input, bulk-arrival, and multi-server continuous-time queueing model. The service times are equal to the constant $D$ for any customer. The queueing model may be denoted as $G{I}^X/D/c$ queue. Using the steady-state equations, the system-length probability generating function is derived. Subsequently, by inverting this probability generating function the stationary system-length distribution is obtained using the roots of a characteristic equation. Next, a similar analysis for the corresponding multi-server queueing model with batch Markovian arrival process $\left(BMAP\right)$ is carried out using the roots of a characteristic equation associated with the vector generating function of the system-length distribution. The distribution function of the stationary actual waiting-time for the first customer of an arrival batch in a $BMAP/D/c$ queue is also derived. Some numerical implementation of the procedure for the $G{I}^X/D/c$ and $BMAP/D/c$ queues is performed. Numerical values for the expected system length and waiting time are also obtained.

本文给出了更新输入、批量到达和多服务器连续时间排队模型的固定系统长度分布的闭式解析表达式和计算分析。服务时间等于常数$D$对于任何客户。排队模型可以表示为$G{\mathrm{一世}}^X/D/C$队列。使用稳态方程，导出系统长度概率生成函数。随后，通过反转该概率生成函数，使用特征方程的根获得平稳系统长度分布。接下来对相应的多服务器排队模型与批量马尔可夫到达过程进行类似分析$\left(乙米\mathrm{一种}磷\right)$使用与系统长度分布的向量生成函数相关联的特征方程的根进行。一个到达批次的第一个顾客的平稳实际等待时间的分布函数$乙米\mathrm{一种}磷/D/C$队列也是派生的。该程序的一些数值实现$G{\mathrm{一世}}^X/D/C$ 和 $乙米\mathrm{一种}磷/D/C$队列被执行。还获得了预期系统长度和等待时间的数值。