The fluid limit of a heavily loaded processor sharing queue

Gromoll, H. Christian; Puha, Amber L.; Williams, Ruth

doi:10.1214/aoap/1031863171

Cited by 102 publications

(231 citation statements)

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“…Following the general idea of such an approach, Gromoll et al [5,6] established a diffusion approximation for a measure-valued descriptor of a single server processor sharing queue. A key ingredient in that work was an analysis of the long time behavior of the measure-valued solutions of a critical fluid model.…”

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confidence: 99%

“…We consider a critical fluid model for a single server processor sharing queue. This model was first introduced in [6] where it was shown that critical fluid model solutions arise as functional law of large numbers limits of measure-valued processes used to track the residual service times of jobs in heavily loaded processor sharing queues. The critical fluid model has one parameter, a Borel probability measure ν on R + = [0, ∞) that does not charge the origin and that has a finite positive mean 1/α, where α ∈ (0, ∞).…”

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confidence: 99%

“…It was shown in [6] that if ξ ∈ M is continuous, i.e., it does not charge singletons, then there exists a unique fluid model solution μ such that μ 0 = ξ. Here we shall focus on fluid model solutions with continuous initial conditions.…”

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confidence: 99%

“…Section 2 contains the definition of a critical fluid model solution and a summary of its properties proved in [6]. Following that, in Section 3, the relative entropy functional is introduced and the main results of the paper, Theorems 3.1 and 3.2, and Corollary 3.1, are stated.…”

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

Puha¹,

Williams²

2016

Stoch. Syst.

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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

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confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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

Puha¹,

Williams²

2016

Stoch. Syst.

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“…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.…”

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A note on the event horizon for a processor sharing queue

Hampshire

Massey

2008

Queueing Syst

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In this note we identify a phenomenon for processor sharing queues that is unique to ones with time-varying rates. This property was discovered while correcting a proof in Hampshire, Harchol-Balter and Massey [5]. If the arrival rate for processor sharing queue has unbounded growth over time, then it is possible for the number of customers in a processor sharing queue to grow so quickly that a newly entering job never finishes. We define the minimum size for such a job to be the event horizon for a processor sharing queue. We discuss the use of such a concept and develop some of its properties. This short article serves both as errata for [5] and as documentation of a characteristic feature for some processor sharing queues with time varying rates.

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Multiclass Queueing Network Models

Hasenbein

2011

Wiley Encyclopedia of Operations Research and Management Science

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The fluid limit of a heavily loaded processor sharing queue

Cited by 102 publications

References 22 publications

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

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

A note on the event horizon for a processor sharing queue

Multiclass Queueing Network Models

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