State-dependent response times via fluid limits in shortest remaining processing time queues

Down, Douglas G.; Gromoll, H. Christian; Puha, Amber L.

doi:10.1145/1639562.1639593

Cited by 8 publications

(12 citation statements)

References 6 publications

(9 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One challenge associated with a detailed analysis of SRPT is that, due to the need to keep track of the remaining processing times of all jobs in the system, the state descriptor for an SRPT queue is infinite dimensional, even for exponentially distributed processing times. In order to describe the state of the system, Down, Gromoll, and Puha [8,9] introduce a measure valued process in which the state of the system at a given time is the finite nonnegative Borel measure on the nonnegative real line that puts a unit atom at the remaining processing time of each job in system. Under natural modeling assumptions and asymptotic conditions, they prove a fluid limit theorem (a functional law of large numbers) for this measure valued state descriptor.…”

Section: Introductionmentioning

confidence: 99%

Heavy Traffic Scaling Limits for shortest remaining processing time queues with heavy tailed processing time distributions

Banerjee¹,

Budhiraja²,

Puha³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We study a single server queue operating under the shortest remaining processing time (SRPT) scheduling policy; that is, the server preemptively serves the job with the shortest remaining processing time first. Since one needs to keep track of the remaining processing times of all jobs in the system in order to describe the evolution, a natural state descriptor for an SRPT queue is a measure valued process in which the state of the system at a given time is the finite nonnegative Borel measure on the nonnegative real line that puts a unit atom at the remaining processing time of each job in system. In this work we are interested in studying the asymptotic behavior of the suitably scaled measure valued state descriptors for a sequence of SRPT queuing systems. Gromoll, Kruk, and Puha (2011) have studied this problem under diffusive scaling (time is scaled by r 2 and the mass of the measure normalized by r, where r is a scaling parameter approaching infinity). In the setting where the processing time distributions have bounded support, under suitable conditions, they show that the measures at any time instant converge in distribution to a single atom located at the right edge of the support of the processing time distribution with the size of the atom fluctuating randomly in time. In the setting where the processing time distributions have unbounded support, under suitable conditions, they show that the diffusion scaled measures converge in distribution to the process that is identically zero. In Puha (2015) for the setting where the processing time distributions have unbounded support and light tails, a nonstandard scaling of the queue length process is shown to give rise to a form of state space collapse that results in a nonzero limit.In the current work we consider the case where processing time distributions have finite second moments and regularly varying tails. Results of Puha ( 2015) suggest that the right scaling for the measure valued process is governed by a parameter c r that is given as a certain inverse function related to the tails of the first moment of the processing time distribution. Using this parameter we consider a novel scaling for the measure valued process in which the time is scaled by a factor of r 2 , the mass is scaled by the factor c r /r and the space (representing the remaining processing times) is scaled by the factor 1/c r . We show that the scaled measure valued process converges in distribution (in the space of paths of measures). In a sharp contrast to results for bounded support and light tailed service time distributions, this time there is no state space collapse and the limiting measures are not concentrated on a single atom. Nevertheless, the description of the limit is simple and given explicitly in terms of a certain R+ valued random field which is determined from a single Brownian motion. Along the way we establish convergence of suitably scaled workload and queue length processes. We also show that as the tail of the distribution of job processing times becomes lighter i...

show abstract

Section: Introductionmentioning

confidence: 99%

Heavy Traffic Scaling Limits for shortest remaining processing time queues with heavy tailed processing time distributions

Banerjee¹,

Budhiraja²,

Puha³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The fluid-scaled counterparts of h r x and s r x arē i.e., the left-continuous and the right-continuous inverses ofF r , respectively. The following theorem formalizes and extends the interpretation of s · as the fluid analog of the state-dependent response time (see Down et al [8]). …”

Section: Fluid Limit Theoremsmentioning

confidence: 84%

“…Thus, for the sake of the proof of the fluid limit theorem justifying our fluid model as the first-order approximation of the queueing system under consideration (Theorem 4.6), it becomes necessary to provide exact asymptotics for the state-dependent response times in the pre-limit stochastic models. To this end, in Theorem 4.7 we establish fluid limits for these response times, which may be of independent interest, even in the G/G/1 case, as a theoretical justification of the fluid approximations for these quantities introduced by Down et al [8,9].…”

mentioning

confidence: 94%

Fluid Limits for Multiple-Input Shortest Remaining Processing Time Queues

Kruk¹,

Sokołowska²

2016

Mathematics of OR

View full text Add to dashboard Cite

A single queueing station serving K input streams with renewal arrivals and generally distributed independent and identically distributed service times is considered. Customers are served by the Shortest Remaining Processing Time policy. In the case of a tie, the first-in, first-out policy is utilized. We analyze a fluid model for the evolution of a measure-valued state descriptor of this system, with particular emphasis on its limiting behavior in the critical case as time gets large. We also prove a fluid limit theorem justifying our fluid model as the first-order approximation of the queueing system under consideration. Along the way, we establish fluid limits for the corresponding state-dependent response times.1. Introduction. In a single server queue working under the shortest remaining processing time (SRPT) policy, preemptive priority is given to the job with the shortest remaining processing time. If there are several such jobs, priority is given to the one that was the first one to arrive at the system.A classical result of Schrage [26] asserts that SRPT minimizes the queue length at any given point in time, regardless of any distributional assumptions on the underlying arrival and service processes (see also Smith [28]). Expressions for the mean response time for an M/G/1 SRPT queue were found earlier by Schrage and Miller [27] and extended later in Schassberger [25] and Perera [22]. Pavlov [20] and Pechinkin [21] characterized the heavy traffic limit of the steady state distributions for the queue length of an M/G/1 SRPT queue.In recent years, one may observe renewed interest in SRPT, mainly in computer science. For example, Bansal and Harchol-Balter [1] study the fairness issue for SRPT. Subsequent work seeks to provide a framework for comparing policies in the M/G/1 setting; see, for example, Wierman and Harchol-Balter [32]. Verloop et al. [30] investigate stability properties of size-based scheduling strategies (including SRPT) in multiresource systems, such as bandwidth-sharing networks. There has also been a recent body of work on the tail behavior of single server queues under SRPT; see, for example, Núñez Queija [18] and Nuyens and Zwart [19].Down and Wu [7] use diffusion limits to show certain optimality properties of a multilayered round robin routing policy for a system of parallel servers operating under SRPT, assuming a finitely supported service time distribution. In the case of a general service time distribution, Down et al. [9] developed fluid limits for a singleserver, single-customer class SRPT queue and used these to propose an approximate formula for state-dependent response times (on fluid scale) of jobs entering the system (see also Down et al. [8]). Gromoll and Keutel [11] obtained the same fluid limits in the case of a nonpreemptive variant of SRPT called shortest job first. Gromoll et al.[13] proved a diffusion limit theorem for a G/G/1 SRPT queue under usual heavy traffic assumptions. Recently Puha [24] provided a diffusion limit for this system under nonstandard spacial ...

show abstract

“…A job's state-dependent response time is the time until it exits the system, conditional on the state of the system (the configuration of all current residual service times) when it arrives. The fluid limit Z * (•) can be used to calculate a fluid approximation s(x) to the state-dependent response time of a job of size x, as a function of an initial state ξ; see [5,4]. Thus Theorem 1 implies that, asymptotically on fluid scale, a job of size x arriving to a queue in state ξ will have the same state-dependent response time under SJF or SRPT.…”

Section: Implications For Comparing Performancementioning

confidence: 99%

“…Here, V * (t) = αtν for all t ≥ 0, and Z * (•) is almost surely an SRPT fluid model solution for data (α, ν) and initial condition Z * (0), equal in distribution to Z 0 ; see Theorem 5.16 in [4] as well as Section 2.2 in [4] for a definition of the fluid model solutions. Such fluid model solutions are analyzed in detail in [4,5]. Within the proof of (33), it is assumed by invoking the Skorohod representation theorem that all random elements are defined on a common probability space (Ω * , F * , P * ) such that, almost surely as r → ∞,…”

Section: Invariance Of Fluid Limitmentioning

confidence: 99%

Invariance of fluid limits for the shortest remaining processing time and shortest job first policies

Gromoll

Keutel

2011

Queueing Syst

Self Cite

View full text Add to dashboard Cite

We consider a single-server queue with renewal arrivals and i.i.d. service times, in which the server employs either the preemptive Shortest Remaining Processing Time (SRPT) policy, or its non-preemptive variant, Shortest Job First (SJF). We show that for given stochastic primitives (initial condition, arrival and service processes), the model has the same fluid limit under either policy. In particular, we conclude that the well-known queue length optimality of preemptive SRPT is also achieved, asymptotically on fluid scale, by the simpler-toimplement SJF policy. We also conclude that on fluid scale, SJF and SRPT achieve the same performance with respect to response times of the longest-waiting jobs in the system. AMS 2010 subject classifications. Primary 60K25, 60F17; secondary 60G57, 68M20, 90B22.

show abstract

State-dependent response times via fluid limits in shortest remaining processing time queues

Cited by 8 publications

References 6 publications

Heavy Traffic Scaling Limits for shortest remaining processing time queues with heavy tailed processing time distributions

Heavy Traffic Scaling Limits for shortest remaining processing time queues with heavy tailed processing time distributions

Fluid Limits for Multiple-Input Shortest Remaining Processing Time Queues

Invariance of fluid limits for the shortest remaining processing time and shortest job first policies

Contact Info

Product

Resources

About