### Abstract

We establish continuity of the integral representation y(t) = x(t) + f _{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 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}.

Original language | English (US) |
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Pages (from-to) | 214-237 |

Number of pages | 24 |

Journal | Annals of Applied Probability |

Volume | 20 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2010 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

_{1}topology.

*Annals of Applied Probability*,

*20*(1), 214-237. https://doi.org/10.1214/09-AAP611