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 LaplaceStieltjes transform (LST) are known, these expressions are not applicable for computational purposes. This paper derives readily applicable expressions for insensitive bounds of all moments of the conditional sojourn time distribution. The 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.

We obtain a decomposition result for the steady state queue length distribution in egalitarian processor-sharing (PS) models. In particular, for multi-class egalitarian PS queues, we show that the marginal queue length distribution for each class equals the queue length distribution of an equivalent single class PS model with a random number of permanent customers. Similarly, the mean sojourn time (conditioned on the initial service requirement) for each class can be obtained by conditioning on the number of permanent customers. The decomposition result implies linear relations between the marginal queue length probabilities, which also hold for other PS models such as the egalitarian PS models with state-dependent system capacity that only depends on the total number of customers in the system. Based on the exact decomposition result for egalitarian PS queues, we propose a similar decomposition for discriminatory processor-sharing (DPS) models, and numerically show that the approximation is accurate for moderate differences in service weights.

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