Fluid Limits for Shortest Remaining Processing Time Queues

Abstract

We consider a single-server queue with renewal arrivals and i.i.d. service times in which the server uses the shortest remaining processing time policy. To describe the evolution of this queue, we use a measure-valued process that keeps track of the residual service times of all buffered jobs. We propose a fluid model (or formal law of large numbers approximation) for this system and, under mild assumptions, prove the existence and uniqueness of fluid model solutions. Furthermore, we prove a scaling limit theorem that justifies the fluid model as a first-order approximation of the stochastic model. The state descriptor of the fluid model is a measure-valued function whose dynamics are governed by certain inequalities in conjunction with the standard workload equation. In particular, these dynamics determine the evolution of the left edge (infimum) of the state descriptor's support, which yields conclusions about response times. We characterize the evolution of this left edge as an inverse functional of the initial condition, arrival rate, and service time distribution. This characterization reveals the manner in which the growth rate of the left edge depends on the service time distribution. By considering varying examples, the authors show that the rate can vary from logarithmic to polynomial.

Stability of linear EDF networks with resource sharing

“…Indeed, there is a deep relation between the EDF and SRPT scheduling strategies. Their similarity was first noticed (at least to our knowledge) by Bender, Chakrabarti and Muthukrishnan [5] and then, more explicitly, by Down, Gromoll and Puha [14], who investigated fluid limits for SRPT queues using an auxiliary process similar to the one introduced by Doytchinov, Lehoczky and Shreve [15] in the context of heavy traffic analysis for EDF queues. In fact, in Sect.…”

“…5. Recall that, by Lemmas 13, 14 and Remark 2, for each such network with initial state x n , there is a random timeσ , bounded by a constant γ , such that (14) holds. If (14) implies (15), then the argument given in Sect.…”

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

“…That is, we obtain from our fluid model a fluid level approximation of the amount of time a job of a given size spends in the system, given an arbitrary system configuration at the time of its arrival. This paper is an extended abstract of results in [2]. In [2], a more thorough introduction is provided, as well as all proofs and a detailed analysis of the fluid model behavior.…”

“…This paper is an extended abstract of results in [2]. In [2], a more thorough introduction is provided, as well as all proofs and a detailed analysis of the fluid model behavior. There is also a rigorous justification of the fluid model as an approximation to the underlying stochastic model.…”

“…There is also a rigorous justification of the fluid model as an approximation to the underlying stochastic model. Here we briefly summarize the stochastic and fluid models and develop some examples that highlight the implications of the results in [2].…”