Reflected Brownian Motion on an Orthant

Harrison, J. Michael; Reiman, Martin I.

doi:10.1214/aop/1176994471

Cited by 385 publications

(431 citation statements)

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“…In order to prove (14) we can assume without loss of generality that its right-hand side is finite. For ε > 0 we can therefore find values t < u < 0, x < κ switch t , and y > λ s −µ s u , such that The last line follows from (9).…”

Section: Appendix a Proof Of Theoremmentioning

confidence: 99%

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Large deviation properties of constant rate data streams sharing a buffer with long-range dependent traffic in critical loading

Majewski¹

2007

Adv. Appl. Probab.

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We consider a constant rate traffic which shares a buffer with a random cross traffic. A first come first served or priority service discipline is applied at the buffer. After service at the first buffer the constant rate traffic moves to a play-out buffer. Both buffers provide service at constant rate and infinite waiting room. We investigate logarithmic large and moderate deviation asymptotics for the tail probabilities of the steady-state queue length distribution at the play-out buffer for long-range dependent cross traffic in critical loading. We characterize the asymptotic behavior of the cross traffic which leads to a large queue length at the play-out buffer and compare it to the one for renewal cross traffic.

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Section: Appendix a Proof Of Theoremmentioning

confidence: 99%

“…We call a process reflected if it is the image of a process under a Skorokhod map which restricts the movements of the process to the positive orthant by shifting it at the border [9], [14]. (A definition of the onedimensional Skorokhod map is contained in Section 2.)…”

Section: Introductionmentioning

confidence: 99%

Large deviation properties of constant rate data streams sharing a buffer with long-range dependent traffic in critical loading

Majewski¹

2007

Adv. Appl. Probab.

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“…The matrix R defined by (27) has a special structure in that it satisfies the HarrisonReiman (HR) condition [10]. We use this structure in proving the convergence of the diffusion-scaled workload process.…”

Section: Covariance and Reflection Matricesmentioning

confidence: 99%

“…Remark It is known from the work of Harrison and Reiman [10] that when R satisfies the HR condition, there is strong existence and uniqueness (and hence weak existence and uniqueness) for an SRBM given the data (R N + , θ, Γ, R) and the initial distribution ν.…”

Section: Definition Of An Srbmmentioning

confidence: 99%

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Diffusion approximation for a heavily loaded multi-user wireless communication system with cooperation

Bhardwaj

Williams

2009

Queueing Syst

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A cellular wireless communication system in which data is transmitted to multiple users over a common channel is considered. When the base stations in this system can cooperate with each other, the link from the base stations to the users can be considered a multi-user multiple-input multiple-output (MIMO) downlink system. For such a system, it is known from information theory that the total rate of transmission can be enhanced by cooperation. The channel is assumed to be fixed for all transmissions over the period of interest and the ratio of anticipated average arrival rates for the users, also known as the relative traffic rate, is fixed. A packet-based model is considered where data for each user is queued at the transmit end. We consider a simple policy which, under Markovian assumptions, is known to be throughput-optimal for this coupled queueing system. Since an exact expression for the performance of this policy is not available, as a measure of performance, we establish a heavy traffic diffusion approximation. To arrive at this diffusion approximation, we use two key properties of the policy; we posit the first property as a reasonable manifestation of cooperation, and the second property follows from coordinate convexity of the capacity region. The diffusion process is a semimartingale reflecting Brownian motion (SRBM) living in the positive orthant of N -dimensional space (where N is the number of users). This SRBM has one direction of reflection associated with each of the 2 N − 1 boundary faces, but show that, in fact, only those directions associated withThe research of S. Bhardwaj was supported by a UCSD ECE departmental dissertation fellowship for 2007-08.

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B rownian Motion and Queueing Applications

Yao

2011

Wiley Encyclopedia of Operations Research and Management Science

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The objective of this article is to present an elementary introduction to the applications of Brownian motion in queueing theory. We start with an overview of some of the basics of Brownian motion, followed by Donsker's theorem and the Skorohod problem. We then discuss the fluid and diffusion scaling of queueing processes, taking the single‐server GI/GI/1 queue as an example. We illustrate why the limiting regime under the diffusion scaling is a reflected Brownian motion, and how it leads to a diffusion approximation of the original queue model.

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Reflected Brownian Motion on an Orthant

Cited by 385 publications

References 5 publications

Large deviation properties of constant rate data streams sharing a buffer with long-range dependent traffic in critical loading

Large deviation properties of constant rate data streams sharing a buffer with long-range dependent traffic in critical loading

Diffusion approximation for a heavily loaded multi-user wireless communication system with cooperation

B rownian Motion and Queueing Applications

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