Lyapunov Functions for Semimartingale Reflecting Brownian Motions

Dupuis, Paul; Williams, Ruth

doi:10.1214/aop/1176988725

Cited by 145 publications

(191 citation statements)

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“…[8]) of the above assumption is that for each λ ∈ Λ,λ = ∅ there exists a vector n λ such that n λ ∈ n(x) for all x ∈ G satisfying In(x) = λ and…”

Section: Setting and Resultsmentioning

confidence: 99%

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Stability Properties of Constrained Jump-Diffusion Processes

Atar

Budhiraja²

2002

Electron. J. Probab.

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We consider a class of jump-diffusion processes, constrained to a polyhedral cone G ⊂ IR n , where the constraint vector field is constant on each face of the boundary. The constraining mechanism corrects for "attempts" of the process to jump outside the domain. Under Lipschitz continuity of the Skorohod map Γ, it is known that there is a cone C such that the image Γφ of a deterministic linear trajectory φ remains bounded if and only ifφ ∈ C. Denoting the generator of a corresponding unconstrained jump-diffusion by L, we show that a key condition for the process to admit an invariant probability measure is that for x ∈ G, L id(x) belongs to a compact subset of C o .

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“…[8]) of the above assumption is that for each λ ∈ Λ,λ = ∅ there exists a vector n λ such that n λ ∈ n(x) for all x ∈ G satisfying In(x) = λ and…”

Section: Setting and Resultsmentioning

confidence: 99%

“…Now the proof of (4.26) is identical to the proof of Proposition 3.7 of [8] on observing that if φ ∈ H a (x) then for c ∈ (0, ∞), the trajectory θ c (·), defined as θ c (t) . ((1 + c)x).…”

Section: Whenever G(·)mentioning

confidence: 93%

Stability Properties of Constrained Jump-Diffusion Processes

Atar

Budhiraja²

2002

Electron. J. Probab.

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“…A Lyapunov function similar in spirit to (1) was used in [5] to establish a sufficient condition for positive recurrence of a semimartingale reflecting Brownian motion in the positive orthant. (In [5], the solutions to the Skorohod problem, for the trajectories determined by the process drift alone, are the DFLs in our terminology.)…”

Section: (Function G(·)mentioning

confidence: 99%

Section: (Function G(·)mentioning

confidence: 99%

“…(In [5], the solutions to the Skorohod problem, for the trajectories determined by the process drift alone, are the DFLs in our terminology.) Obtaining the Lyapunov function second derivative bounds is also a key part of the analysis in [5]. We note, however, that our basic model, the problem, the structure of the (family of) process(es) and corresponding DFLs, the form of function g (·), and the analysis of the Lyapunov function derivatives are completely different.…”

Section: (Function G(·)mentioning

confidence: 99%

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Tightness of Stationary Distributions of a Flexible-Server System in the Halfin-Whitt Asymptotic Regime

Stolyar

2015

Stochastic Systems

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A large-scale flexible service system with two large server pools and two types of customers is considered. Servers in pool 1 can only serve type 1 customers, while server in pool 2 are flexible -they can serve both types 1 and 2. (This is a so-called "N-system.") The customer arrival processes are Poisson and customer service requirements are independent exponentially distributed. The service rate of a customer depends both on its type and the pool where it is served. A priority service discipline, where type 2 has priority in pool 2, and type 1 prefers pool 1, is considered. We consider the Halfin-Whitt asymptotic regime, where the arrival rate of customers and the number of servers in each pool increase to infinity in proportion to a scaling parameter n, while the overall system capacity exceeds its load by O( √ n). For this system we prove tightness of diffusion-scaled stationary distributions. Our approach relies on a single common Lyapunov function G (n) (x), depending on parameter n and defined on the entire state space as a functional of the drift-based fluid limits (DFL). Specifically, G (n) (x) = ∞ 0 g(y (n) (t))dt, where y (n) (·) is the DFL starting at x, and g(·) is a "distance" to the origin. (g(·) is same for all n). The key part of the analysis is the study of the (first and second) derivatives of the DFLs and function G (n) (x). The approach, as well as many parts of the analysis, are quite generic and may be of independent interest.

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Reflected B rownian Motion

Dieker

2011

Wiley Encyclopedia of Operations Research and Management Science

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Reflected Brownian motion is a continuous‐time continuous‐state Markov process, which is constrained to its finite‐dimensional state space S by a pushing mechanism on the boundary ∂ S of S . In the context of Operations Research and Management Science, the state space S is typically the nonnegative orthant. This article reviews several aspects of reflected Brownian motion on this orthant: its construction, its positive recurrence, and its stationary distribution.

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Lyapunov Functions for Semimartingale Reflecting Brownian Motions

Cited by 145 publications

References 0 publications

Stability Properties of Constrained Jump-Diffusion Processes

Stability Properties of Constrained Jump-Diffusion Processes

Tightness of Stationary Distributions of a Flexible-Server System in the Halfin-Whitt Asymptotic Regime

Reflected B rownian Motion

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