Growing open Markovian Jackson networks: Fluid limit and infinite-dimensional Skorokhod problem

TL;DR

This paper develops a new infinite-dimensional fluid limit and Skorokhod problem framework for growing open Jackson networks, providing theoretical tools for analyzing large-scale queueing systems.

Contribution

It introduces a novel theory for infinite-dimensional Skorokhod problems with broader reflection operators, establishing existence, uniqueness, and Lipschitz continuity.

Findings

Proved convergence of the queueing process to the infinite-dimensional fluid limit.

Established Lipschitz continuity of the Skorokhod mapping under spectral radius conditions.

Extended convergence results to empirical measures and performance functionals.

Abstract

We study growing open Jackson networks where each station is a single-server queue that follows the first-come first-served discipline with Poisson arrivals and exponentially distributed service times, characterized by node-specific rates. In applying a fluid scaling to the queue-length process, we show that under certain conditions the queueing system can be approximated by an infinite-dimensional fluid limit with a kernel function in place of the transition matrix. This limiting process can be characterized by an infinite-dimensional Skorokhod problem, for which we develop a new theory by considering a broader class of reflection operators and general infinite-dimensional processes. We establish existence and uniqueness of a solution along with Lipschitz continuity provided the reflecting operator has a spectral radius less than 1.By introducing an intermediate process in which the compensated Poisson components are removed, and then lifting this to an infinite-dimensional process, we exploit the new Lipschitz property of the infinite-dimensional Skorokhod mapping to prove convergence of the intermediate process. We then prove the necessary estimates for the difference between the original and intermediate processes by using martingale properties. Finally, we consider the empirical measure of the queueing processes, for which we show convergence to the measure associated with the path of the infinite-dimensional fluid limit, extending to the convergence of specific performance-related functionals.