Continuity of a queueing integral representation in the $M_1$ topology

Abstract

We establish continuity of the integral representation $y(t)=x(t)+{\int }_0^{t}h(y(s))ds$, $t\ge 0$, 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 $L_1$.

Published in at http://dx.doi.org/10.1214/09-AAP611 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Martingale proofs of many-server heavy-traffic limits for Markovian queues

Cited by in corpus (4)

• Ergodicity of Lévy-driven SDEs arising from multiclass many-server queues
• Stability for gains from large investors' strategies in M1/J1 topologies
• Skorokhod's M1 topology for distribution-valued processes
• Uniform polynomial rates of convergence for a class of Lévy-driven controlled SDEs arising in multiclass many-server queues