A Skorokhod Map on Measure-Valued Paths with Applications to Priority Queues

TL;DR

This paper introduces a measure-valued Skorokhod map that aids in analyzing prioritized queueing systems, providing a unified framework for fluid models, solution uniqueness, and convergence, including new results in time-inhomogeneous contexts.

Contribution

The paper develops a novel measure-valued Skorokhod map and demonstrates its application to prioritized queueing models, establishing regularity, solution uniqueness, and convergence results.

Findings

Unified framework for prioritized queueing models

Proved solution uniqueness for fluid models

Established convergence in time-inhomogeneous settings

Abstract

The Skorokhod map on the half-line has proved to be a useful tool for studying processes with non-negativity constraints. In this work we introduce a measure-valued analog of this map that transforms each element $\zeta$ of a certain class of c\`{a}dl\`{a}g paths that take values in the space of signed measures on the half-line to a c\`{a}dl\`{a}g path that takes values in the space of non-negative 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.