### Citations

5201 |
Convergence of Probability Measures
- Billingsley
- 1999
(Show Context)
Citation Context ...rgence is almost “too strong”. Since it is nonseparable, the associated Borel σ-algebra is strictly larger than the Kolmogorov σ-algebra, which leads to measurability problems, see, e.g., Billingsley =-=[1]-=-. As a consequence, one needs to extend the notion of convergence in distribution. The relevant theory has been developed by Hoffman-Jørgensen and his followers and is expounded upon in van der Vaart ... |

1086 |
Limit Theorems for Stochastic Processes.
- Jacod, Shiryaev
- 1987
(Show Context)
Citation Context ...ce U . We denote this set by D(I, U). If U is, in addition, separable, then it is customary to equip D(I, U) with Skorohod’s J1-topology and a complete separable metric, see, e.g., Jacod and Shiryaev =-=[12]-=-, Ethier and Kurtz [9]. The resulting Polish space will also be denoted by D(I, U). We hope that no confusion will arise out of this ambiguity. For the most part, though, we will be concerned with the... |

514 |
Weak convergence and empirical processes: with applications to statistics,
- Vaart, Wellner
- 1996
(Show Context)
Citation Context ...quence, one needs to extend the notion of convergence in distribution. The relevant theory has been developed by Hoffman-Jørgensen and his followers and is expounded upon in van der Vaart and Wellner =-=[24]-=-. Its primary motivation was to provide tools to tackle convergence in distribution of empirical processes in strong topologies. Empirical processes also play an important part in the study of heavy t... |

309 | Point Processes and Queues: Martingale Dynamics, - Bremaud - 1981 |

274 |
Classical Potential Theory and Its Probabilistic Counterpart, Grundlehren der mathematischen Wissenschaften 262
- Doob
- 1984
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Citation Context ... uncorrelated, hence, independent increments. It is, thus, a zero-mean Gaussian martingale. By the independence-of-increments property, the natural filtration of N is rightcontinuous, see, e.g., Doob =-=[6]-=-. Consequently, N admits a right-continuous with left-hand limits modification, see, e.g., Meyer [19]. Let M(t) = ∫ R2+ 1{s+x≤t} dL(s, x) . (4.16) By Lemma 4.2, the process M = (M(t) , t ∈ R+) admits ... |

273 |
Heavy-traffic limits for queues with many exponential servers.
- Halfin, Whitt
- 1981
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Citation Context ...e system converge in distribution for Skorohod’s J1-topology to a process with continuous trajectories, which may be explicitly identified. This was first accomplished in the work of Halfin and Whitt =-=[10]-=- for the case of exponential service time distributions and has been continued by many additional authors as well for different classes of service time distributions, see, for instance, Jelenkovic, Ma... |

135 |
Theory of Martingales.
- Liptser, Shiryayev
- 1989
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Citation Context ...pology. One can talk of convergence in the M1-topology. Since the process Z is a Gaussian semimartingale, its jumps occur at deterministic times, see Jacod and Shiryaev [12] and Liptser and Shiryayev =-=[16]-=-, so this process is a separable random element of DC(R+,R). Similarly to Krichagina and Puhalskii [15], Z admits a representation as a stochastic integral, which follows from the proof of Lemma 4.4 b... |

77 |
Asymptotic Methods in Queueing Theory
- Borovkov
- 1984
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Citation Context ...the Xn converge in distribution in D(R+,R) to X . For the case of a continuous Y , the latter limit theorem generalizes the earlier results by Borovkov [2] (which result can also be found in Borovkov =-=[3]-=-) and Krichagina and Puhalskii [15]. We recall that Borovkov [2] imposed a Hölder continuity condition on the function(∫ t 0 ( 1− F (t− s)) de(s) , t ∈ R+) and Krichagina and Puhalskii [15] required ... |

70 | Martingale proofs of many-server heavy-traffic limits for Markovian queues.
- Pang, Talreja, et al.
- 2007
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Citation Context ...nce, Jelenkovic, Mandelbaum, and Momcilovic [13], Mandelbaum and Momcilovic [18], Kaspi and Ramanan[14], Puhalskii and Reiman [21], Whitt [25], and the recent survey paper by Pang, Talreja, and Whitt =-=[20]-=-. One exception to the above list is the work of Reed [22] in which no assumptions beyond a first moment are placed on the service time distribution. A related avenue of research concerns infinite ser... |

57 | Capacités et Processus Stochastiques - Dellacherie - 1972 |

53 | Heavy traffic limits for queues with many deterministic servers. Queueing Syst., - Jelenkovic, Mandelbaum, et al. - 2004 |

51 | Heavy-traffic limits for the G/H∗2/n/m queue. - Whitt - 2004 |

29 | On limit laws for service processes in multi-channel systems (in Russian). - Borovkov - 1967 |

29 |
A heavy-traffic analysis of a closed queueing system with a GI/∞ service center.
- Krichagina, Puhalskii
- 1997
(Show Context)
Citation Context ...in which no assumptions beyond a first moment are placed on the service time distribution. A related avenue of research concerns infinite servers in heavy traffic, see, e.g., Krichagina and Puhalskii =-=[15]-=-, Pang, Talreja, and Whitt [20], and references therein. The purpose of this paper is to extend the aforementioned results to allow noncritical loading, generally distributed service times, and genera... |

24 | Queues with many servers: The virtual waiting-time process in the QED regime.
- Mandelbaum, Momcilovic
- 2008
(Show Context)
Citation Context ...nd has been continued by many additional authors as well for different classes of service time distributions, see, for instance, Jelenkovic, Mandelbaum, and Momcilovic [13], Mandelbaum and Momcilovic =-=[18]-=-, Kaspi and Ramanan[14], Puhalskii and Reiman [21], Whitt [25], and the recent survey paper by Pang, Talreja, and Whitt [20]. One exception to the above list is the work of Reed [22] in which no assum... |

11 |
Uniform Central Limit Theorems, Volume 23.
- Dudley
- 1999
(Show Context)
Citation Context ...remark, there are no known examples of nonseparable Borel probability measures and the axiom that such measures do not exist can be consistently added to the Zermelo–Frenkel set of axioms, see Dudley =-=[8]-=- and van der Vaart and Wellner [24] for these and other observations. Our results, however, make no use of this fact. Another ingredient in the proof of the main theorem is a martingale argument which... |

11 | SPDE limits for many-server queues,
- Kaspi, Ramanan
- 2009
(Show Context)
Citation Context ...y many additional authors as well for different classes of service time distributions, see, for instance, Jelenkovic, Mandelbaum, and Momcilovic [13], Mandelbaum and Momcilovic [18], Kaspi and Ramanan=-=[14]-=-, Puhalskii and Reiman [21], Whitt [25], and the recent survey paper by Pang, Talreja, and Whitt [20]. One exception to the above list is the work of Reed [22] in which no assumptions beyond a first m... |

10 | The G/GI/N queue in the Halfin-Whitt regime I: Infinite server queue system equations
- Reed
- 2007
(Show Context)
Citation Context ...um and Momcilovic [18], Kaspi and Ramanan[14], Puhalskii and Reiman [21], Whitt [25], and the recent survey paper by Pang, Talreja, and Whitt [20]. One exception to the above list is the work of Reed =-=[22]-=- in which no assumptions beyond a first moment are placed on the service time distribution. A related avenue of research concerns infinite servers in heavy traffic, see, e.g., Krichagina and Puhalskii... |

9 | Nonlinear functionals of empirical measures and the bootstrap - Dudley - 1990 |

9 | The G/GI/N queue in the Halfin-Whitt regime II: Idle time system equations - Reed - 2007 |

5 |
Gaussian random processes, volume 9 of Applications of Mathematics
- Ibragimov, Rozanov
- 1978
(Show Context)
Citation Context ...robability. By (4.18), (4.21), (4.23), (4.24), and (4.25), Z(t) = P– liml→∞ Zl(t) . Since the Zl are Gaussian processes, the latter limit holds in the mean square sense too, see Ibragimov and Rozanov =-=[11]-=-. By (4.22), we can write EZl(t)Zl(s) = ∞∫ 0 F (t ∧ s− rl(u))(1− F (t ∨ s− rl(u))) 1{rl(u)≤t∧s} da(u) , 22 where rl(u) = ∞∑ i=1 sli−1 1{u∈(sli−1,sli]} . Since maxi(sli − sli−1) → 0 as l → ∞, rl(u) → u... |

4 |
The multiclass GI/PH/N queue in the Haflin–Whitt regime
- Puhalskii, Reiman
(Show Context)
Citation Context ...s well for different classes of service time distributions, see, for instance, Jelenkovic, Mandelbaum, and Momcilovic [13], Mandelbaum and Momcilovic [18], Kaspi and Ramanan[14], Puhalskii and Reiman =-=[21]-=-, Whitt [25], and the recent survey paper by Pang, Talreja, and Whitt [20]. One exception to the above list is the work of Reed [22] in which no assumptions beyond a first moment are placed on the ser... |

3 | Large finite population queueing systems. I. The infinite server model - Louchard - 1988 |