Stein’s method for steady-state diffusion approximations of $M/\mathit{Ph}/n+M$ systems

Braverman, Anton; Dai, J. G.

doi:10.1214/16-aap1211

Cited by 85 publications

(122 citation statements)

References 59 publications

(92 reference statements)

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“…The use of Stein's method to compute bounds for stationary distributions has been recently popularized in [4,12]. This methodology has then been used in [31] and further extended in [9,32] to establish the rate of convergence of stochastic processes to their mean eld approximation, in light or heavy trac.…”

Section: :3mentioning

confidence: 99%

“…Note that the idea of using h( ) + V h /N as an improved approximation was already proposed in [9] for the two-choice model, where the constant V h was estimated by simulation. The idea of improving classical approximations via Stein's method was also presented in [4] but, according to the authors, their computations seem hard to generalize to high dimensional settings. On the contrary, our method scales as the cube of the number of dimensions of the model which makes it applicable to models with hundreds of dimensions.…”

Section: :3mentioning

confidence: 99%

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A Refined Mean Field Approximation

Gast

Houdt

2018

Abstracts of the 2018 ACM International Conference on Measurement and Modeling of Computer Systems

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Mean eld models are a popular means to approximate large and complex stochastic models that can be represented as N interacting objects. Recently it was shown that under very general conditions the steady-state expectation of any performance functional converges at rate O(1/N ) to its mean eld approximation. In this paper we establish a result that expresses the constant associated with this 1/N term. This constant can be computed easily as it is expressed in terms of the Jacobian and Hessian of the drift in the xed point and the solution of a single Lyapunov equation. This allows us to propose a rened mean eld approximation. By considering a variety of applications, that include coupon collector, load balancing and bin packing problems, we illustrate that the proposed rened mean eld approximation is signicantly more accurate that the classic mean eld approximation for small and moderate values of N : relative errors are often below 1% for systems with N = 10.

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Section: :3mentioning

confidence: 99%

Section: :3mentioning

confidence: 99%

A Refined Mean Field Approximation

Gast

Houdt

2018

Abstracts of the 2018 ACM International Conference on Measurement and Modeling of Computer Systems

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“…In contrast, the density function of

Y_{\infty}

has an explicit form and takes almost no time to evaluate. Following Braverman and Dai (), our article also serves as a demonstration that the Stein's method framework provides not only a powerful tool to characterize the error bounds, but also an engineering tool to identify a good approximation for the steady‐state distribution. The latter is particularly helpful for systems without known or explicit steady‐state distributions, such as our discrete queue.…”

Section: Introductionmentioning

confidence: 99%

Steady‐state diffusion approximations for discrete‐time queue in hospital inpatient flow management

Feng

Shi

2018

Naval Research Logistics

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In this article, we analyze a discrete‐time queue that is motivated from studying hospital inpatient flow management, where the customer count process captures the midnight inpatient census. The stationary distribution of the customer count has no explicit form and is difficult to compute in certain parameter regimes. Using the Stein's method framework, we identify a continuous random variable to approximate the steady‐state customer count. The continuous random variable corresponds to the stationary distribution of a diffusion process with state‐dependent diffusion coefficients. We characterize the error bounds of this approximation under a variety of system load conditions—from lightly loaded to heavily loaded. We also identify the critical role that the service rate plays in the convergence rate of the error bounds. We perform extensive numerical experiments to support the theoretical findings and to demonstrate the approximation quality. In particular, we show that our approximation performs better than those based on constant diffusion coefficients when the number of servers is small, which is relevant to decision making in a single hospital ward.

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“…Our use of Stein's method for the rate of convergence was inspired by the work by Braverman and Dai (2017), in which they developed a modular framework with three components for steady-state diffusion approximations and established the rate of convergence to diffusion models for M/Ph/n + M queuing systems. Please see Braverman et al (2016) for an introduction of Stein's method for steady-state diffusion approximation and its connection to classical applications of Stein's method.…”

Section: Introductionmentioning

confidence: 99%

On the Approximation Error of Mean-Field Models

Lü

2018

Stochastic Systems

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Abstract. Mean-field models have been used to study large-scale and complex stochastic systems, such as large-scale data centers and dense wireless networks, using simple deterministic models (dynamical systems). This paper analyzes the approximation error of mean-field models for continuous-time Markov chains (CTMCs) and focuses on mean-field models that are represented as finite-dimensional dynamical systems with a unique equilibrium point. By applying Stein's method and the perturbation theory, the paper shows that under some mild conditions, if the mean-field model is globally asymptotically stable and locally exponentially stable, the mean-square difference between the stationary distribution of the stochastic system of size M (i.e., with M nodes/particles) and the equilibrium point of the corresponding mean-field system is O(1/M). The result of this paper establishes a general theorem for establishing the convergence and the approximation error (i.e., the rate of convergence) of a large class of CTMCs to their mean-field limits by mainly looking into the stability of the mean-field model, which is a deterministic system and is often easier to analyze than the CTMCs. Two applications of mean-field models in data center networks are presented to demonstrate the novelty of our results.

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Stein’s method for steady-state diffusion approximations of $M/\mathit{Ph}/n+M$ systems

Cited by 85 publications

References 59 publications

A Refined Mean Field Approximation

A Refined Mean Field Approximation

Steady‐state diffusion approximations for discrete‐time queue in hospital inpatient flow management

On the Approximation Error of Mean-Field Models

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