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

Abstract

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.

Insensitive Bounds for the Moments of the Sojourn Times in M/GI Systems Under State-Dependent Processor Sharing

“…For the general M/M/SDPS system, see [6]. The aim of this paper is to derive for the M/GI/SDPS system tight insensitive upper bounds for the moments of V (τ ), which generalize corresponding results, recently given for the M/GI/1-PS system by [9] and for the M/M/SDPS system by the authors [6], as well as expressions for the Laplace-Stieltjes transforms (LSTs) and moments of V (τ ) and V , respectively, which are useful for deriving explicit and numerically treatable representations in some special cases; see Example 2.1, below, and, in particular, [8]. The tight insensitive upper bounds for the moments of V (τ ) are summarized in the following theorem, which will be proved in Section 3.…”

“…Note that the right-hand side of (1.8) is just the kth moment of the instantaneous sojourn timeV (τ ), i.e. E[V k (τ )] is for all τ ∈ R + an upper bound for E[V k (τ )], and, thus, Theorem 1.1 generalizes the result of [9] to the general M/GI/SDPS system. In the case of a stable M/GI/1-PS system, assumption (1.7) is satisfied for all k ∈ N, in view of p(n) = (1 − ) n , n ∈ Z + , and < 1, and, hence, all moments of V (τ ) are finite.…”

“…for ϕ 1 (n) = 1/n, n ∈ N, Theorem 1.1 was proved in [9] (see their Theorems 4.1 and 5.11) using stochastic ordering theory and particular results known for M/GI/1-PS systems. Analogously to [9] for the M/GI/1-PS system, the right-hand side of (1.8) can be interpreted as follows. Consider a tagged τ -request finding at its arrival n requests in the system.…”

“…is approximately the sojourn time of a τ -request, and, therefore,V (τ ) is called in [9] the instantaneous sojourn time as τ ↓ 0. Note that the right-hand side of (1.8) is just the kth moment of the instantaneous sojourn timeV (τ ), i.e.…”

Monotonicity Properties for Multi-Class Queueing Systems

“…In this paper we will consider a special case of stochastic ordering: We use a sample-path approach to compare two stochastic networks, that is for both networks we consider the same realizations of the arrival processes and service requirements (see [11,23] for more details). A related research direction is to obtain bounds for the stochastic process of interest [5,41,8]. In a recent paper [5] the authors consider a network of processor sharing queues with independent Poisson arrival processes.…”