Diffusion Limits for Shortest Remaining Processing Time Queues

Gromoll, H. Christian; Kruk, Łukasz; Puha, Amber L.

doi:10.1287/10-ssy016

Cited by 15 publications

(18 citation statements)

References 17 publications

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“…Down and Wu (2006) used diffusion limits to obtain some optimality properties of a multi-layered round robin routing policy in a system of parallel servers operating under SRPT, with a finitely supported service time distribution. Gromoll et al (2011) established diffusion limits for G/G/1 SRPT queues with general service time distributions. Recently, Puha (2015) and Banerjee et al (2020) obtained diffusion limits for a G/G/1 SRPT system under nonstandard spatial scaling.…”

Section: Summary Of Prior Resultsmentioning

confidence: 99%

“…Recently, Puha (2015) and Banerjee et al (2020) obtained diffusion limits for a G/G/1 SRPT system under nonstandard spatial scaling. Kruk (2019) observed that, under suitable assumptions, the results of Gromoll et al (2011) could be obtained from heavy traffic limits of Kruk (2007) for preemptive G/G/1 Earliest Deadline First (EDF) queues with job service times correlated with their initial lead times. Recall that under the EDF service discipline, priority is given to the task with the shortest lead time.…”

Section: Summary Of Prior Resultsmentioning

confidence: 99%

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Heavy traffic analysis for single-server SRPT and LRPT queues via EDF diffusion limits

Kruk

2021

Ann Oper Res

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Extending the results of Kruk (Queueing theory and network applications. QTNA 2019. Lecture notes in computer science, vol 11688. Springer, Cham, pp 263–275, 2019), we derive heavy traffic limit theorems for a single server, single customer class queue in which the server uses the Shortest Remaining Processing Time (SRPT) policy from heavy traffic limits for the corresponding Earliest Deadline First queueing systems. Our analysis allows for correlated customer inter-arrival and service times and heavy-tailed inter-arrival and service time distributions, as long as the corresponding stochastic primitive processes converge weakly to continuous limits under heavy traffic scaling. Our approach yields simple, concise justifications and new insights for SRPT heavy traffic limit theorems of Gromoll et al. (Stoch Syst 1(1):1–16, 2011). Corresponding results for the longest remaining processing time policy are also provided.

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Section: Summary Of Prior Resultsmentioning

confidence: 99%

Section: Summary Of Prior Resultsmentioning

confidence: 99%

Heavy traffic analysis for single-server SRPT and LRPT queues via EDF diffusion limits

Kruk

2021

Ann Oper Res

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“…compare ( 25), (26). From this point, we follow the analysis of the first case, with η in place of τ 1 .…”

Section: Proof Of Propositionmentioning

confidence: 95%

“…The definition of τ (26), by the time τ 1 a customer leaves buffer 1 arriving into buffer 2, resulting in Q 2 (τ 1 ) + Q 3 (τ 1 ) ≥ 1, which contradicts (52). Thus, by (10), we have…”

Section: Proof Of Propositionmentioning

confidence: 96%

Instability of SRPT, SERPT and SJF multiclass queueing networks

Kruk

Chojecki

2022

Queueing Syst

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We provide two examples of strictly subcritical multiclass queueing networks which are unstable under the shortest remaining processing time (SRPT) service protocol. Both of them are reentrant lines with two servers and eight customer classes. The customer service times in our first system are deterministic, yielding an example of an unstable shortest remaining expected processing time (SERPT) network. In the second one, the service times in one customer class are randomized. Both our examples show also system instability under the shortest job first (SJF) discipline. A simulation study of robustness of our results with respect to changes in the customer interarrival and service times is also provided. Our results indicate that size-based service policies may not use the available resources efficiently in a multiserver network setting and in fact cause instability effects. This is in sharp contrast with their satisfactory performance for single-server queues.

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“…This is in contrast to some other queueing systems, for example, FIFO networks of Kelly type and head-of-the-line proportional processor sharing networks, for which such invariance principles have been established (Bramson [5], Williams [33]). Indeed, in the diffusion approximations for a G/G/1 SRPT queue established in Gromoll et al [13] and Puha [24], the limiting queue length process depends on the tail behavior of the corresponding service time distribution, and hence the invariance principle fails. In the case of a multiclass SRPT system considered in this paper, additional complications arise on the level of distributing the total queue length and workload between different customer classes.…”

Section: Kruk and Sokołowska: Fluid Limits For Multiclass Srpt Queuesmentioning

confidence: 99%

Fluid Limits for Multiple-Input Shortest Remaining Processing Time Queues

Kruk¹,

Sokołowska²

2016

Mathematics of OR

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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 ...

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Diffusion Limits for Shortest Remaining Processing Time Queues

Cited by 15 publications

References 17 publications

Heavy traffic analysis for single-server SRPT and LRPT queues via EDF diffusion limits

Heavy traffic analysis for single-server SRPT and LRPT queues via EDF diffusion limits

Instability of SRPT, SERPT and SJF multiclass queueing networks

Fluid Limits for Multiple-Input Shortest Remaining Processing Time Queues

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