The Fluid Limit of an Overloaded Processor Sharing Queue

Abstract

, http://www.math.ucsd.edu/˜williams/ This paper primarily concerns strictly supercritical fluid models, which arise as functional law of large numbers approximations for overloaded processor sharing queues. Analogous results for critical fluid models associated with heavily loaded processor sharing queues are contained in Gromoll et al. [9] and Puha and Williams [15]. An important distinction between critical and strictly supercritical fluid models is that the total mass for a solution that starts from zero grows with time for the latter, but it is identically equal to zero for the former. For strictly supercritical fluid models, this paper contains descriptions of each of the following: the distribution of the mass as it builds up from zero, the set of stationary solutions, and the limiting behavior of an arbitrary solution as time tends to infinity. In addition, a fluid limit result is proved that justifies strictly supercritical fluid models as first order approximations to overloaded processor sharing queues.

Large deviations of sojourn times in processor sharing queues

“…The fact that we twisted the load from ρ to 1 + /2 > 1 (and not 1) in the proof of Theorem 3.1 is useful for two reasons. First, it allows us to apply general theorems for transient PS queues, as derived in [15,24]. Secondly, we believe that directly twisting to a rate 1 leads to more restrictive assumptions.…”

“…Moreover, (Q p (u)) u∈ [0,x] and B 0 are independent. We now invoke a crucial result about the fluid limit of the number of customers Q(u) at time u for transient PS queues in overload, see Theorem 3.11 of Puha et al [24] -for a similar result, see Jean-Marie & Robert [15]. It entails that, if Q 0 = 0, there exists a constantα such that…”

“…Preliminary results are given in Section 2. Section 3 contains the main result and its proof, based on the change-of-measure mentioned above, and in addition a powerful fluid limit result for PS queues in overload, obtained recently by Puha et al [24]. Section 4 treats a number of ramifications.…”

Bandwidth-sharing in overloaded networks

“…Their analysis was extended by Altman et al [1] to the discriminatory processor-sharing discipline, which corresponds to a single-node network with a weighted α-fair strategy. Puha et al [25] studied a single-server overloaded processor-sharing system in terms of measure-valued processes.…”

A note on the event horizon for a processor sharing queue

“…Examples of this research include two papers of Gromoll, Puha and Williams [3] and Puha and Williams [7]. These two papers provide a general framework for studying the fluid limits of GI/GI/1/P S processor sharing queues via a measure-valued state descriptor, where the queue length and the residual service time process are modeled as measure-valued processes.…”