Abstract In this paper, we consider the number of departures seen by a tagged customer while in service in a classical processor sharing queue. By exploiting the underlying orthogonal structure of this queuing system revealed in an earlier study, we compute the probability mass function of the random variable under consideration. We moreover derive the asymptotic behavior of this probability mass function. The knowledge of the number of departures seen by a tagged customer allows us to test the validity of an approximation, which consists of assuming that the tagged customer is randomly served among those customers in the residual busy period of the queue following the arrival of the tagged customer. A numerical study shows that this approximation is reasonable for moderate values of the number of departures, given that the asymptotic behaviors of the probability mass functions are very different even if the exponential decay rates are equal.

中文翻译：

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M/M/1 处理器共享队列中任意客户的逗留

摘要在本文中，我们考虑了在经典处理器共享队列中服务时被标记的客户看到的离开次数。通过利用早期研究中揭示的这个排队系统的基本正交结构，我们计算了所考虑的随机变量的概率质量函数。此外，我们推导出这个概率质量函数的渐近行为。对标记客户所看到的离开次数的了解使我们能够测试近似值的有效性，该近似值包括假设在标记客户到达后队列的剩余繁忙时段内，在这些客户中随机为标记客户提供服务顾客。数值研究表明，这种近似值对于离场次数的中等值是合理的，