Semimartingale Reflecting Brownian Motions in the Orthant

Williams, Ruth

doi:10.1007/978-1-4757-2418-9_7

Cited by 126 publications

(209 citation statements)

References 27 publications

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“…The gap process for finite systems. The results of this subsection are taken from [2,3,22,39]. However, we present an outline of proofs in Section 6 for completeness.…”

Section: Asymmetric Collisionsmentioning

confidence: 99%

“…It turns out that Z is an SRBM in the orthant R N −1 + with reflection matrix R given by (14), drift vector µ as in (15), and covariance matrix As mentioned in [39,Theorem 3.5], this is a necessary and sufficient condition for the stationary distribution to have product-of-exponentials form. This condition can be rewritten for R and A from (14) and (28) as (16). 6.2.…”

Section: Triple Collisions For Infinite Systemsmentioning

confidence: 99%

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Infinite systems of competing Brownian particles

Sarantsev¹

2017

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. Consider a system of infinitely many Brownian particles on the real line. At any moment, these particles can be ranked from the bottom upward. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its current rank. The gaps between consecutive particles form the (infinite-dimensional) gap process. We find a stationary distribution for the gap process. We also show that if the initial value of the gap process is stochastically larger than this stationary distribution, this process converges back to this distribution as time goes to infinity. This continues the work by Pal and Pitman (2008). Also, this includes infinite systems with asymmetric collisions, similar to the finite ones from Karatzas, Pal and Shkolnikov (2016).

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“…The gap process for finite systems. The results of this subsection are taken from [2,3,22,39]. However, we present an outline of proofs in Section 6 for completeness.…”

Section: Asymmetric Collisionsmentioning

confidence: 99%

Section: Triple Collisions For Infinite Systemsmentioning

confidence: 99%

Infinite systems of competing Brownian particles

Sarantsev¹

2017

Ann. Inst. H. Poincaré Probab. Statist.

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“…For more details, see [9,14,40]. We note that K ℓ ≤ 2, but we keep the notation K ℓ for convenience.…”

Section: Asymptotic Coupling Of a Pair Of Processesmentioning

confidence: 99%

On Existence and Uniqueness of Stationary Distributions for Stochastic Delay Differential Equations with Positivity Constraints

Kinnally

Williams

2010

Electron. J. Probab.

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Deterministic dynamic models with delayed feedback and state constraints arise in a variety of applications in science and engineering. There is interest in understanding what effect noise has on the behavior of such models. Here we consider a multidimensional stochastic delay differential equation with normal reflection as a noisy analogue of a deterministic system with delayed feedback and non-negativity constraints. We obtain sufficient conditions for existence and uniqueness of stationary distributions for such equations. The results are applied to an example from Internet rate control and a simple biochemical reaction system. AMS 2000 subject classifications: 34K50, 37H10, 60H10, 60J25, 93E15.

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“…In these cases, traditional methods cannot be readily extended. To overcome the difficulty, the weak convergence method was initially established by authors, such as W. Dai [1,2] , J. G. Dai and W. Dai [3] , for finite buffer networks, then along the similar line by authors like Williams [7,8,16] and Bramson [5,6] for certain family of multiclass networks. By the method, the keys to prove a heavy traffic limit theorem for multiclass networks with feedback are to show a state space collapse property and a completely-S property for some reflection matrix.…”

Section: Introductionmentioning

confidence: 99%

“…Bramson and J. G. Dai [9] further summarized the results in Refs. [5,7,16] and claimed that as long as one can show the uniform convergence for associated fluid model then one can prove the state space collapse property. By this way, they established a heavy traffic limit theorem in Ref.…”

Section: Introductionmentioning

confidence: 99%

Diffusion approximations for multiclass queueing networks under preemptive priority service discipline

戴万阳

2007

Appl Math Mech

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We prove a heavy traffic limit theorem to justify diffusion approximations for multiclass queueing networks under preemptive priority service discipline and provide effective stochastic dynamical models for the systems. Such queueing networks appear typically in high-speed integrated services packet networks about telecommunication system. In the network, there is a number of packet traffic types. Each type needs a number of job classes (stages) of processing and each type of jobs is assigned the same priority rank at every station where it possibly receives service. Moreover, there is no inter-routing among different traffic types throughout the entire network.

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Semimartingale Reflecting Brownian Motions in the Orthant

Cited by 126 publications

References 27 publications

Infinite systems of competing Brownian particles

Infinite systems of competing Brownian particles

On Existence and Uniqueness of Stationary Distributions for Stochastic Delay Differential Equations with Positivity Constraints

Diffusion approximations for multiclass queueing networks under preemptive priority service discipline

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