This paper studies the M/G/1 processor-sharing (PS) queue, in particular the sojourn time distribution conditioned on the initial job size. Although several expressions for the Laplace-Stieltjes transform (LST) are known, these expressions are not suitable for computational purposes. This paper derives readily applicable insensitive bounds for all moments of the conditional sojourn time distribution. The instantaneous sojourn time, i.e., the sojourn time of an infinitesimally small job, leads to insensitive upper bounds requiring only knowledge of the traffic intensity and the initial job size. Interestingly, the upper bounds involve polynomials with so-called Eulerian numbers as coefficients. In addition, stochastic ordering and moment ordering results for the sojourn time distribution are obtained. (Keywords: M/G/1 PS - Conditional sojourn time - Moments - Insensitive bounds - Instantaneous sojourn time - Euler's number triangle - Moment ordering - Permanent customers)
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机译：本文研究了M / G / 1处理器共享（PS）队列，特别是以初始作业大小为条件的停留时间分布。尽管已知用于Laplace-Stieltjes变换（LST）的几种表达式，但这些表达式不适合用于计算目的。本文推导了适用于有条件停留时间分布的所有时刻的不敏感边界。瞬时停留时间，即无限小的工作的停留时间，导致不敏感的上限，只需要了解交通强度和初始工作量即可。有趣的是，上限涉及具有所谓的欧拉数作为系数的多项式。另外，获得了停留时间分布的随机排序和矩排序结果。 （关键字：M / G / 1 PS-有条件的停留时间-时刻-不敏感的界限-瞬时的停留时间-欧拉数三角形-时刻订购-永久客户）
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