### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

_{1}topology.

*Annals of Applied Probability*,

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

}

_{1}topology',

*Annals of Applied Probability*, vol. 20, no. 1, pp. 214-237. https://doi.org/10.1214/09-AAP611

**Continuity of a queueing integral representation in the M _{1} topology.** / Pang, Guodong; Whitt, Ward.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Continuity of a queueing integral representation in the M1 topology

AU - Pang, Guodong

AU - Whitt, Ward

PY - 2010/2/1

Y1 - 2010/2/1

N2 - We establish continuity of the integral representation y(t) = x(t) + f 0th(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.

AB - We establish continuity of the integral representation y(t) = x(t) + f 0th(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.

UR - http://www.scopus.com/inward/record.url?scp=76449105590&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=76449105590&partnerID=8YFLogxK

U2 - 10.1214/09-AAP611

DO - 10.1214/09-AAP611

M3 - Article

AN - SCOPUS:76449105590

VL - 20

SP - 214

EP - 237

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 1

ER -

_{1}topology. Annals of Applied Probability. 2010 Feb 1;20(1):214-237. https://doi.org/10.1214/09-AAP611