This paper presents the distribution of the number of customers served during a busy period for special cases of the Geo/G/1 queue when initiated with m customers. We analyze the system under the assumptions of a late arrival system with delayed access and early arrival system policies. It is not easy to invert the functional equation for the number of customers served during a busy period except for the simple case Geo/Geo/1 queue, as stated by several researchers. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We find the distribution of the number of customers served during a busy period for various service-time distributions such as geometric, deterministic, binomial, negative binomial, uniform, Delaporte, discrete phase-type and interrupted Bernoulli process. We compute the mean and variance of these distributions and also give numerical results. Due to the clarity of the expressions, the computations are very fast and robust. We also show that in the limiting case, the results tend to the analogous continuous-time counterparts.

中文翻译：

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针对 GEO/G/1 队列的特殊情况，在繁忙期间服务的客户数量分布的分析封闭式解决方案

本文给出了在繁忙时段服务的客户数量的分布，用于特殊情况地理/G/1 队列启动时使用米顾客。我们在延迟访问和提前到达系统策略的延迟到达系统的假设下分析系统。除了简单的情况外，在繁忙时段服务的客户数量的函数方程并不容易反转地理/地理/1 队列，正如几位研究人员所说。使用拉格朗日反演定理，我们给出了这个方程的优雅解。我们找到了在繁忙时期服务的客户数量的分布，包括几何、确定性、二项式、负二项式、均匀、德拉波特、离散相型和中断伯努利过程等各种服务时间分布。我们计算这些分布的均值和方差，并给出数值结果。由于表达式的清晰性，计算非常快速且稳健。我们还表明，在极限情况下，结果倾向于类似的连续时间对应物。