Is fair resource sharing responsible for spreading long delays?

We show that mixing the statistically long jobs (subexponential) and short ones (exponentially bounded) using processor sharing service discipline causes long (subexponential) delays for all types of jobs in the system. Since processor sharing represents a baseline fair scheduling discipline used in designing Web servers, as well as the basic model of TCP bandwidth sharing, our finding suggests that even though fairness possesses many desirable attributes, it causes unnecessarily long delays for statistically short jobs. Hence, fairness comes with a price. We further quantify the preceding result when the long jobs follow the widely observed power law distribution x^-α, α > 0, where we discover the criticality of the <i>lognormal</i> distribution for the delay characteristics of the lighter jobs. Specifically, we find that when the shorter jobs are heavier than <i>lognormal</i>, the sojourn time <i>V</i> and the service time distribution <i>B</i> of the shorter jobs are tail equivalent P[<i>V</i> > <i>x</i>] ~ P[<i>B</i> > (1 - ρ)<i>x</i>]. However, if P[<i>B</i> > <i>x</i>] is lighter than <i>lognormal</i>, the preceding tail equivalence does not hold. Furthermore, when the shorter jobs <i>B</i> have much lighter tails <i>e</i>^{-λ<i>x</i>^&#946}, λλ > 0, β > 0, we show that the distribution of the delay <i>V</i> for these jobs satisfy, as <i>x</i> → ∞, -log P[<i>V</i> > <i>x</i>] ~ <i>c</i>(<i>x</i> log <i>x</i>) β / β+1, where <i>c</i> is explicitly computable. Note that β = 1 and β = 2 represent the exponential and Gaussian cases with the corresponding delay distributions approximately of the form <i>e</i>^{-√<i>x</i> log <i>x</i>} and <i>e</i>^{-(<i>x</i> log <i>x</i>)^2/3}, respectively. Our results are different from the existing ones in the literature that focused on the delays which are of the same form (tail equivalent) as the jobs size distribution.