Insensitive bounds for the moments of the sojourn time distribution in the M/G/1 processor-sharing queue

This paper studies the M/G/1 processor-sharing (PS) queue and 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 applicable for computational purposes. This paper derives readily applicable expressions for insensitive bounds of \textit{all} moments of the conditional sojourn time distribution. The \textit{instantaneous} sojourn time, the sojourn time of a very small job, leads to insensitive upper bounds with special structure requiring only knowledge of the traffic load and the initial job size. Interestingly, the special form of the upper bounds involves polynomials with so-called Eulerian numbers as coefficients. In addition, stochastic ordering and moment ordering results for the sojourn time distribution are obtained.