The Annals of Applied Probability

A Skorokhod map on measure-valued paths with applications to priority queues

Rami Atar, Anup Biswas, Haya Kaspi, and Kavita Ramanan

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Abstract

The Skorokhod map on the half-line has proved to be a useful tool for studying processes with nonnegativity constraints. In this work, we introduce a measure-valued analog of this map that transforms each element $\zeta$ of a certain class of càdlàg paths that take values in the space of signed measures on $[0,\infty)$ to a càdlàg path that takes values in the space of nonnegative measures on $[0,\infty)$ in such a way that for each $x>0$, the path $t\mapsto\zeta_{t}[0,x]$ is transformed via a Skorokhod map on the half-line, and the regulating functions for different $x>0$ are coupled. We establish regularity properties of this map and show that the map provides a convenient tool for studying queueing systems in which tasks are prioritized according to a continuous parameter. Three such well-known models are the earliest-deadline-first, the shortest-job-first and the shortest-remaining-processing-time scheduling policies. For these applications, we show how the map provides a unified framework within which to form fluid model equations, prove uniqueness of solutions to these equations and establish convergence of scaled state processes to the fluid model. In particular, for these models, we obtain new convergence results in time-inhomogeneous settings, which appear to fall outside the purview of existing approaches.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 1 (2018), 418-481.

Dates
Received: April 2016
Revised: February 2017
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1520046092

Digital Object Identifier
doi:10.1214/17-AAP1309

Mathematical Reviews number (MathSciNet)
MR3770881

Zentralblatt MATH identifier
06873688

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60G57: Random measures 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

Keywords
Skorokhod map measure-valued Skorokhod map measure-valued processes fluid models fluid limits law of large numbers priority queueing Earliest-Deadline-First Shortest-Remaining-Processing Time Shortest-Job-First

Citation

Atar, Rami; Biswas, Anup; Kaspi, Haya; Ramanan, Kavita. A Skorokhod map on measure-valued paths with applications to priority queues. Ann. Appl. Probab. 28 (2018), no. 1, 418--481. doi:10.1214/17-AAP1309. https://projecteuclid.org/euclid.aoap/1520046092


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