Asymptotic behavior of a critical fluid model for a processor sharing queue via relative entropy

Abstract

In this paper, we develop a new approach to studying the asymptotic behavior of fluid model solutions for critically loaded processor sharing queues. For this, we introduce a notion of relative entropy associated with measure-valued fluid model solutions. In contrast to the approach used in [12], which does not readily generalize to networks of processor sharing queues, we expect the approach developed in this paper to be more robust. Indeed, we anticipate that similar notions involving relative entropy may be helpful for understanding the asymptotic behavior of critical fluid model solutions for stochastic networks operating under various resource sharing protocols naturally described by measure-valued processes. Introduction.In the context of multiclass queueing networks operating under head-of-the-line (HL) service disciplines, Bramson [1] and Williams [15] have developed a modular approach for establishing heavy traffic diffusion approximations to such networks. In particular, they have given sufficient conditions under which asymptotic behavior of critical fluid model solutions can be used to prove state space collapse and thereby a heavy traffic limit theorem justifying a diffusion approximation. Although the HL assumption covers a wide variety of service disciplines, including firstin-first-out (FIFO) and static priorities, it requires that service for a given job class is concentrated on the job at the head-of-the-line. Consequently, it does not cover some disciplines that arise naturally in applications, such as the processor sharing discipline. While it is desirable to have a modular approach to proving diffusion approximations for stochastic networks with

Stability of a Subcritical Fluid Model for Fair Bandwidth Sharing with General File Size Distributions

“…The following lemma can be proved in the same manner as lemma 7.12 of Puha and Williams (2016), so we omit the proof.…”

“…This provides a rigorous formulation of a partial differential equation assumed to hold by Paganini et al (2012). A similar smoothing technique was also used by Puha and Williams (2016) in the study of the asymptotic behavior of critical fluid model solutions for a single class processor sharing queue. Our method is a little different from that of Puha and Williams (2016) in that we smooth the entire fluid model solution and not just the initial condition.…”

Asymptotic behavior of a critical fluid model for a multiclass processor sharing queue via relative entropy

“…Traditional processor sharing with general distributional assumptions, as studied here, is not HL and a different notion of relative entropy is needed to carry out our analysis. In [14], Puha and Williams formulated such a notion of relative entropy in order to study the asymptotic behavior of critical fluid model solutions for single class processor sharing queues operating under general distributional assumptions. In the earlier work [13], also on single class processor sharing queues, Puha and Williams used renewal theoretic arguments to obtain rates of convergence results under higher moment assumptions than those required in [14].…”

“…In [14], Puha and Williams formulated such a notion of relative entropy in order to study the asymptotic behavior of critical fluid model solutions for single class processor sharing queues operating under general distributional assumptions. In the earlier work [13], also on single class processor sharing queues, Puha and Williams used renewal theoretic arguments to obtain rates of convergence results under higher moment assumptions than those required in [14]. The renewal theoretic techniques in [13] seem rather specific to a single class and rates of convergence are stronger than what is typically required in state space collapse arguments.…”

“…The renewal theoretic techniques in [13] seem rather specific to a single class and rates of convergence are stronger than what is typically required in state space collapse arguments. Therefore, an important motivation for the work in [14] was to devise a methodology that might not require higher moment assumptions and might more naturally extend to the network setting. The results in this current paper further develop the methodology of [14] in this direction.…”