Functional limit theorems for the number of busy servers in a $G/G/\infty$ queue

TL;DR

This paper establishes functional limit theorems for the number of busy servers in a $G/G/ ext{infinity}$ queue, addressing open problems and providing new integral representations for the limiting Gaussian processes.

Contribution

It proves two new functional limit theorems with different centerings, solving previously open problems and advancing the understanding of queue behavior in stochastic processes.

Findings

Proves weak convergence in the Skorokhod $J_1$-topology.

Provides a new integral representation for the limit Gaussian process.

Addresses open problems from Mikosch and Resnick (2006).

Abstract

We discuss weak convergence of the number of busy servers in a $G/G/\infty$ queue in the $J_1$-topology on the Skorokhod space. We prove two functional limit theorems, with random and nonrandom centering, respectively, thereby solving two open problems stated in Mikosch and Resnick (2006}. A new integral representation for the limit Gaussian process is given.