Stein’s Method for Steady-state Diffusion Approximations: An Introduction through the Erlang-A and Erlang-C Models

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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“…Their results and approaches are then refined for some multiclass networks in [26,29,30,34,43]. Other related works involve the manyserver regime, including [7,8,16,23,24,[36][37][38]. This is a different limiting regime and often calls for a very different approach (from those used in MQN), such as Stein's method in [7,8].…”

Section: Related Literaturementioning

confidence: 99%

Justifying diffusion approximations for multiclass queueing networks under a moment condition

Ye

¹

,

Yao

²

2018

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Multiclass queueing networks (MQN) are, in general, difficult objects to study analytically. The diffusion approximation refers to using the stationary distribution of the diffusion limit as an approximation of the diffusion-scaled process (say, the workload) in the original MQN. To validate such an approximation amounts to justifying the interchange of two limits, t → ∞ and k → ∞, with t being the time index and k, the scaling parameter. Here, we show this interchange of limits is justified under a p * th moment condition on the primitive data, the interarrival and service times; and we provide an explicit characterization of the required order (p * ), which depends naturally on the desired order of moment of the workload process.

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“…Our results are obtained by using Stein's method (see [5] for a good introduction to Stein's method). The use of Stein's method to compute bounds for stationary distributions has been recently popularized in [4,12].…”

Section: :3mentioning

confidence: 99%

A Refined Mean Field Approximation

Gast

¹

,

Houdt

²

2017

Proc. ACM Meas. Anal. Comput. Syst.

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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.

show abstract

Stein’s Method for Steady-state Diffusion Approximations: An Introduction through the Erlang-A and Erlang-C Models

Cited by 77 publications

References 68 publications

A Refined Mean Field Approximation

A Refined Mean Field Approximation

Justifying diffusion approximations for multiclass queueing networks under a moment condition

A Refined Mean Field Approximation

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