Continuity of a queueing integral representation in the M1 topology

Abstract

We establish continuity of the integral representation y(t)=x(t)+∫₀^th(y(s)) ds, t≥0, mapping a function x into a function y when the underlying function space D is endowed with the Skorohod M₁ 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 M₁-continuity is based on a new characterization of the M₁ 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 L₁.