## Joint distribution of waiting time and queue size for single server queues |

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### Contents

Introduction | 5 |

Departure process | 19 |

Joint distribution of waiting time and queue size | 32 |

Copyright | |

1 other sections not shown

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3.1 are satisfied a.e. in probability analogue assertion assumptions Borel set bounded variation central limit theorem Cn,k common probability space completes the proof conditions of Lemma continuity of addition convergence in distribution convergence in probability Corollary 3.8 counting renewal process definitions departure process ergodic finite positive numbers finite-dimensional distributions following relation functional central limit Furthermore GI/G/1 queue gives the convergence heavy traffic Hence in view Hence we obtain immediate consequence implies the convergence inequality large numbers law of large Let Qn manner mapping negative exponential nonnegative integer Notice obtain Corollary obtain the convergence oo a.e. P{YkeB Polish space Proposition 2.11 queue Q queueing theory random variables real number right-hand side sample paths sequence kn sequence vk stationary renewal process stationary sequence sup Sk tending to infinity Theorem 5.4 two-dimensional Wiener process uniformly integrable vector virtual waiting Vn,k w/iere