Fluid Limits for Multiple-Input Shortest Remaining Processing Time Queues

Abstract

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 ...

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

“…These results are consistent with the growth rates obtained in [19] for steady state mean response times. In follow on work, Kruk [17] proves a fluid limit theorem for multiclass SRPT queues that includes convergence of the response times to the expression studied in [8], which justifies it as an approximation. Atar, Biswas, Kaspi and Ramanan [1] develop more general fluid limits for SRPT and other priority queues with time varying arrivals and service rates.…”

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

“…Gromoll and Keutel (2012) obtained the same fluid limits in the case of the Shortest Job First (SJF) protocol, a non-preemptive version of SRPT. Kruk and Sokołowska (2016) generalized the results of Down et al (2009) to SRPT queues with multiple inputs. 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.…”

Diffusion Limits for Shortest Remaining Processing Time Queues with Multiple Customer Types

“…. By (17), there exists a finite constant C such that Ẑr Σ (0)(x * ) ≤ C for all r ∈ R. This means that all the functions Sr k as well as the functions D k are constant on [Cx * , ∞). In this case, convergence in the Skorohod topology to a continuous limit implies convergence in the uniform topology as well ( [5], Section 12).…”

“…Previous research on this topic includes the work of Peterson ([21]) concerning heavy-traffic limit theorems for queueing networks with multiple customers classes divided into two types: high-priority ones having a preemptive priority over low-priority ones, with customers within each of these types served according to the FIFO policy. In [17], Kruk and Sokołowska establish a fluid limit theorem for a single-server queueing model with K classes of customers, served according to the SRPT protocol, with FIFO used as the tie-breaker.…”