A Refined Mean Field Approximation

International audienceStochastic models have been used to assess the performance of computer (and other) systems for many decades. As a direct analysis of large and complex stochastic models is often prohibitive, approximations methods to study their behavior have been devised. One very popular approximation method relies on mean field theory. Its widespread use can be explained by the relative ease involved to define and solve a mean field model in combination with its high accuracy for large systems. Although mean field theory provides no guarantees with respect to the accuracy of a finite system of size N , there are many examples where the accuracy was demonstrated (using simulation) to be quite high even for systems of moderate size, e.g., for N ≈ 50. Nevertheless this accuracy very much depends on the exact parameter settings, for instance mean field models for load balancing systems are known to be quite inaccurate under high loads even for moderate sized systems. The main contribution of this paper is to establish a result that applies to a broad class of mean field models, including density dependent population processes [3] and the class of discrete-time models of [1], and that provides a significantly more accurate approximation for finite N than the classical mean field approximation. When considering a mean field model X (N) described by the density-dependent population process of Kurtz, where X (N) i (t) is the fraction of objects in state i at time t, the classical mean field approximation shows that if the corresponding ODE model has a unique attractor π , then the stationary measure of X (N) concentrates on π. We establish conditions that show that for a function h, there exists a constant V h such that E (N) h(X (N)) = h(π) + V h N + o 1 N. The constant V h is expressed as a function of the Jacobian and the Hessian matrices of the drift (in π) and the solution of a single Lyapunov equation. As such it can be computed efficiently even when the number of possible states i of an object is large (less than one second for a 500-dimensional model). The refined mean field model that we propose consists in approximating E (N) [h(X (N))] by h(π) + V h /N and maintains many of the attractive features of the classic mean field approximation that resulted in its widespread use, but significantly improves upon its accuracy for finite N. We demonstrate that this refined approximation significantly improves the accuracy of the mean field approximation by using a variety of mean field models that include the coupon replication model [4] and a number of load balancing schemes [5-7]. In some special cases the constant V h can be expressed in closed form, while for remaining cases the computation time for a numerical evaluation of V h is negligible. To give a flavor of the accuracy of the refined approximation, we provide some examples in Table 1 picked from the full version of the paper [2]. This table illustrates that the refined approximation is very accurate, even for N = 10. Moreover, it is much more accurate than the classic mean field approximation

A Refined Mean Field Approximation

International audienceMean field 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 field 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 fixed point and the solution of a single Lyapunov equation. This allows us to propose a refined mean field approximation. By considering a variety of applications, that include coupon collector, load balancing and bin packing problems, we illustrate that the proposed refined mean field approximation is significantly more accurate that the classic mean field approximation for small and moderate values of N: relative errors are often below 1% for systems with N=10

Analysis of the Refined Mean-Field Approximation for the 802.11 Protocol Model

Mean-field approximation is a method to investigate the behavior of stochastic models formed by a large number of interacting objects. A new approximation was recently established, i.e., the refined mean-field approximation, and its high accuracy when the number of objects is small has been shown. In this work, we consider the model of the 802.11 protocol, which is a discrete-time model and show how the refined mean-field approximation can be adapted to this model. Our results confirm the accuracy of the refined mean-field approximation when the model with N objects is in discrete time.This research was founded by the Department of Education of the Basque Government, Spain, through the Consolidated Research Group MATHMODE (IT1456-22) and by the Marie Sklodowska-Curie, grant agreement number 777778

A Refined Mean Field Approximation for Synchronous Population Processes

International audienceMean field approximation is a popular method to study the behaviour of stochastic models composed of a large number of interacting objects. When the objects are asynchronous, the mean field approximation of a population model can be expressed as an ordinary differential equation. When the objects are synchronous the mean field approximation is a discrete time dynamical system. In this paper, we focus on the latter. We show that, similarly to the asynchronous case, the mean field approximation of a synchronous population can be refined by a term in 1/N. Our result holds for finite time-horizon and steady-state. We provide two examples that illustrate the approach and its limit

Bias and Refinement of Multiscale Mean Field Models

Mean field approximation is a powerful technique which has been used in many settings to study large-scale stochastic systems. In the case of two-timescale systems, the approximation is obtained by a combination of scaling arguments and the use of the averaging principle. This paper analyzes the approximation error of this `average' mean field model for a two-timescale model $(\boldsymbol{X}, \boldsymbol{Y})$, where the slow component $\boldsymbol{X}$ describes a population of interacting particles which is fully coupled with a rapidly changing environment $\boldsymbol{Y}$. The model is parametrized by a scaling factor $N$, e.g. the population size, which as $N$ gets large decreases the jump size of the slow component in contrast to the unchanged dynamics of the fast component. We show that under relatively mild conditions the `average' mean field approximation has a bias of order $O(1/N)$ compared to $\mathbb{E}[\boldsymbol{X}]$. This holds true under any continuous performance metric in the transient regime as well as for the steady-state if the model is exponentially stable. To go one step further, we derive a bias correction term for the steady-state from which we define a new approximation called the refined `average' mean field approximation whose bias is of order $O(1/N^2)$. This refined `average' mean field approximation allows computing an accurate approximation even for small scaling factors, i.e., $N\approx 10 -50$. We illustrate the developed framework and accuracy results through an application to a random access CSMA model.Comment: 28 page

rmftool - A library to Compute (Refined) Mean Field Approximation(s)

International audienceMean field approximation is a powerful technique to study the performance of large stochastic systems represented as systems of interacting objects. Applications include load balancing models, epidemic spreading, cache replacement policies, or large-scale data centers, for which mean field approximation gives very accurate estimates of the transient or steady-state behaviors. In a series of recent papers [9, 7], a new and more accurate approximation, called the refined mean field approximation is presented. Yet, computing this new approximation can be cumbersome. The purpose of this paper is to present a tool, called rmf tool, that takes the description of a mean field model, and can numerically compute its mean field approximations and refinement

On quantum mean-field models and their quantum annealing

This paper deals with fully-connected mean-field models of quantum spins with p-body ferromagnetic interactions and a transverse field. For p=2 this corresponds to the quantum Curie-Weiss model (a special case of the Lipkin-Meshkov-Glick model) which exhibits a second-order phase transition, while for p>2 the transition is first order. We provide a refined analytical description both of the static and of the dynamic properties of these models. In particular we obtain analytically the exponential rate of decay of the gap at the first-order transition. We also study the slow annealing from the pure transverse field to the pure ferromagnet (and vice versa) and discuss the effect of the first-order transition and of the spinodal limit of metastability on the residual excitation energy, both for finite and exponentially divergent annealing times. In the quantum computation perspective this quantity would assess the efficiency of the quantum adiabatic procedure as an approximation algorithm.Comment: 44 pages, 23 figure