Continuity of a queueing integral representation in the ${M}_{\mathbf{1}}$ topology

TL;DR

This paper proves the continuity of a queueing integral representation under the Skorohod M1 topology and applies it to derive heavy-traffic limits for many-server queues with jump discontinuities.

Contribution

It establishes the M1 continuity of an integral mapping and uses this to analyze heavy-traffic limits in complex queueing models with jumps.

Findings

Proves M1 continuity of the integral representation.

Applies the result to heavy-traffic limits in many-server queues.

Provides a new characterization of M1 convergence with absolutely continuous parametrizations.

Abstract

We establish continuity of the integral representation $y(t)=x(t)+\int_0^th(y(s)) ds$, $t\ge0$, mapping a function $x$ into a function $y$ when the underlying function space $D$ is endowed with the Skorohod $M_1$ topology. We apply this integral representation with the continuous mapping theorem to establish heavy-traffic stochastic-process limits for many-server queueing models when the limit process has jumps unmatched in the converging processes as can occur with bursty arrival processes or service interruptions. The proof of $M_1$-continuity is based on a new characterization of the $M_1$ convergence, in which the time portions of the parametric representations are absolutely continuous with respect to Lebesgue measure, and the derivatives are uniformly bounded and converge in $L_1$.