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

Authors:
Sing-Kong Cheung;Hans Berg;Richard J. Boucherie
Affiliations:
Department of Applied Mathematics, Stochastic Operations Research Group, University of Twente, Enschede, The Netherlands 7500 AE;Aff2 Aff3;Department of Applied Mathematics, Stochastic Operations Research Group, University of Twente, Enschede, The Netherlands 7500 AE
Venue:
Queueing Systems: Theory and Applications
Year:
2006

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Abstract

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.