The function space D([O, ∞)ρ, E)

Ivanoff, B. Gail

doi:10.2307/3315230

Cited by 22 publications

(25 citation statements)

References 7 publications

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“…This lemma is an easy consequence of Corollary 4.2 of Ivanoff (1980) and Theorem 3.1 of Ivanoff (1983).…”

Section: Lemmasmentioning

confidence: 97%

Poisson convergence for point processes on the plane

Ivanoff

1985

J Aust Math Soc A

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A compensator is defined for a point process in two dimensions. It is shown that a Poisson process is characterized by a continuous deterministic compensator. Sufficient conditions are given for convergence in distribution of a sequence of two-dimensional point processes in the Skorokhod topology to a Poisson process when the corresponding sequence of compensators converges pointwise in probability to a continuous deterministic function.1980 Mathematics subject classification (Amer. Math. Soc): primary 60 G 55; secondary 60 F 17, 60 G 48.

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“…This lemma is an easy consequence of Corollary 4.2 of Ivanoff (1980) and Theorem 3.1 of Ivanoff (1983).…”

Section: Lemmasmentioning

confidence: 97%

Poisson convergence for point processes on the plane

Ivanoff

1985

J Aust Math Soc A

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“…is endowed with the Skorohod J 1 topology, that is, both inside and outside D spaces are endowed with the Skorohod J 1 topology. For a complete separable metric space S, the space D([0, ∞) 2 , S) is the space of all S-valued "continuous from above with limits from below" functions on [0, ∞) 2 , and is endowed with the same metric as defined by [19].…”

Section: Introductionmentioning

confidence: 99%

Heavy-Traffic Limits for a Fork-Join Network in the Halfin-Whitt Regime

Pang

2016

Stochastic Systems

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We study a fork-join network with a single class of jobs, which are forked into a fixed number of parallel tasks upon arrival to be processed at the corresponding multi-server stations. After service completion, each task will join a buffer associated with the service station waiting for synchronization, called “unsynchronized queue”. The synchronization rule requires that all tasks from the same job must be completed, referred to as “non-exchangeable synchronization”. Once synchronized, jobs will leave the system immediately. Service times of the parallel tasks of each job can be correlated and form a sequence of i.i.d. random vectors with a general continuous joint distribution function. We study the joint dynamics of the queueing and service processes at all stations and the associated unsynchronized queueing processes. The main mathematical challenge lies in the “resequencing” of arrival orders after service completion at each station. As in Lu and Pang (2015) for the infinite-server fork-join network model, the dynamics of all the aforementioned processes can be represented via a multiparameter sequential empirical process driven by the service vectors for the parallel tasks of each job. We consider the system in the Halfin-Whitt regime, and prove a functional law of large number and a functional central limit theorem for queueing and synchronization processes. In this regime, although the delay for service at each station is asymptotically negligible, the delay for synchronization is of the same order as the service times.

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“…Recall from Theorem 3.1 of [33] that the processesÜ n := {Ü (t, [19] we have that there exists a sequence {α l 0 ∈ R + : l ≥ 1} satisfying α l 0 → ∞ as l → ∞ such that (i') for each α l 0 and every > 0 there exists a compact set…”

Section: (H(x ∧ Y) − H(x)h(y))mentioning

confidence: 99%

“…) endowed with a generalized Skorohod J 1 topology defined in [19] in Proposition 4.1. This proposition generalizes Lemma 3.1 of [29] to the multiparameter setting and Theorem 3.1 in [33], and its proof is provided in §6.1.…”

Section: Assumption 8 the Residual Waiting Times Of The Tasks In Quementioning

confidence: 99%

“…S k is used to represent kfold product space of any complete and separable metric space S for k ∈ N. For a complete separable metric space S, D([0, ∞), S) denotes the space of all S-valued càdlàg functions on [0, ∞), and is endowed with the Skorohod J 1 topology (see, e.g., [7,14,58] is the space of all S-valued "continuous from above with limits from below" functions on [0, ∞) 2 , and is endowed with the same metric as defined by [19]. D 2 ≡ D([0, 1] 2 , R) is denoted as the space of all "continuous from above with limits from below" functions on the unit square [0, 1] 2 in the sense of Neuhaus [39], and is endowed with the same metric d D 2 as in [39].…”

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confidence: 99%

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Heavy-traffic limits for a fork-join networkin the Halfin-Whitt regime

Lu¹,

Pang²

2016

Stoch. Syst.

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We study a fork-join network with a single class of jobs, which are forked into a fixed number of parallel tasks upon arrival to be processed at the corresponding multi-server stations. After service completion, each task will join a buffer associated with the service station waiting for synchronization, called "unsynchronized queue". The synchronization rule requires that all tasks from the same job must be completed, referred to as "non-exchangeable synchronization". Once synchronized, jobs will leave the system immediately. Service times of the parallel tasks of each job can be correlated and form a sequence of i.i.d. random vectors with a general continuous joint distribution function. We study the joint dynamics of the queueing and service processes at all stations and the associated unsynchronized queueing processes.The main mathematical challenge lies in the "resequencing" of arrival orders after service completion at each station. As in Lu and Pang (2015) for the infinite-server fork-join network model, the dynamics of all the aforementioned processes can be represented via a multiparameter sequential empirical process driven by the service vectors for the parallel tasks of each job. We consider the system in the Halfin-Whitt regime, and prove a functional law of large number and a functional central limit theorem for queueing and synchronization processes. In this regime, although the delay for service at each station is asymptotically negligible, the delay for synchronization is of the same order as the service times.

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The function space D([O, ∞)ρ, E)

Cited by 22 publications

References 7 publications

Poisson convergence for point processes on the plane

Poisson convergence for point processes on the plane

Heavy-Traffic Limits for a Fork-Join Network in the Halfin-Whitt Regime

Heavy-traffic limits for a fork-join networkin the Halfin-Whitt regime

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