## Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to QueuesThis book is about stochastic-process limits - limits in which a sequence of stochastic processes converges to another stochastic process. These are useful and interesting because they generate simple approximations for complicated stochastic processes and also help explain the statistical regularity associated with a macroscopic view of uncertainty. This book emphasizes the continuous-mapping approach to obtain new stochastic-process limits from previously established stochastic-process limits. The continuous-mapping approach is applied to obtain heavy-traffic-stochastic-process limits for queueing models, including the case in which there are unmatched jumps in the limit process. These heavy-traffic limits generate simple approximations for complicated queueing processes and they reveal the impact of variability upon queueing performance. This book was the recipient of the 2003 INFORMS Frederick W. Lanchester Prize, awarded annually by the Institute for Operations Research and the Management Sciences (INFORMS) for the best contribution to operations research and the management sciences published in English. |

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### Contents

Experiencing Statistical Regularity Iw_Dl | 1 |

StochasticProcess Limits | 23 |

2 | 48 |

Preservation of Pointwise Convergence | 51 |

An Application to Simulation | 73 |

The Framework for StochasticProcess Limits | 75 |

A Panorama of StochasticProcess Limits 95 | 94 |

HeavyTraffic Limits for Queues | 97 |

Fluid Queues with OnOff Sources | 243 |

Errors Discovered in the Book | 281 |

SingleServer Queues | 287 |

Multiserver Queues 341 | 340 |

More on the Mathematical Framework | 367 |

The Space D | 391 |

Useful Functions | 427 |

Queueing Networks | 457 |

The Space D | 113 |

HeavyTraffic Limits for Fluid Queues | 137 |

Useful Functions | 163 |

Unmatched Jumps in the Limit Process | 193 |

Queueing Networks | 195 |

More StochasticPro cess Limits | 225 |

Nonlinear Centering and Derivatives | 235 |

The Spaces E and F 515 | 514 |

References | 541 |

Appendix A Regular Variation | 569 |

577 | |

585 | |

### Other editions - View all

Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and ... Ward Whitt No preview available - 2011 |

Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and ... Ward Whitt No preview available - 2013 |

### Common terms and phrases

apply Theorem arrival process associated assume asymptotic buffer ccdf centered random walk Chapter characterized consider continuous sample paths continuous-mapping approach continuous-time convergence in distribution Corollary counting process deﬁned deﬁnition departure process dependence deterministic Donsker’s equivalent establish example exponential FCLT Figure ﬁnal position ﬁnite ﬁrst ﬁuid ﬁxed G oo Gaussian process heavy-tailed heavy-traﬁic limits heavy-traﬂic stochastic-process limits implies independent inﬁnite input interarrival Internet Supplement interval Lemma Lévy processes limit process limits for queues Lipschitz local uniform convergence M1 topology metric space obtain parametric representations partial sums plots Poisson process probability measures product topology queue-length process queueing models queueing networks random elements random variables random walk reﬁection map renewal process Section self-similarity servers signiﬁcant simulation Skorohod sources Speciﬁcally stable law stable Lévy motion stationary statistical regularity stochastic processes stochastic-process limits subset superposition Taqqu traﬁic intensity unmatched jumps variance Whitt workload process