Large-scale Stochastic System Research Articles

Exponential stability of large-scale stochastic reaction-diffusion equations

In this paper, we consider a class of large-scale stochastic reaction-diffusion systems. To prove the exponential stability of the system, we introduce the corresponding isolated subsystems. We show that the exponential stability of the isolated systems implies the exponential stability of the large-scale stochastic reaction-diffusion system under some conditions. Furthermore, we discuss a special case where the large-scale stochastic reaction-diffusion system is described in a hierarchical form. In this case, we prove that the original system is exponentially stable if and only if the corresponding subsystems are exponentially stable.

Toward carbon-neutral electricity and mobility: Is the grid infrastructure ready?

Toward carbon-neutral electricity and mobility: Is the grid infrastructure ready?

Energy estimates and model order reduction for stochastic bilinear systems

ABSTRACT In this paper, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique called balanced truncation (BT) to reduce the order of spatially-discretised systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. When BT is applied, the reduced system is obtained by removing these states from the original system. The main contribution of this paper is an -error bound for BT for stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far.

Stein's Method for Mean Field Approximations in Light and Heavy Traffic Regimes

Mean-field analysis is an analytical method for understanding large-scale stochastic systems such as large-scale data centers and communication networks. The idea is to approximate the stationary distribution of a large-scale stochastic system using the equilibrium point (called the mean-field limit) of a dynamical system (called the mean-field model). This approximation is often justified by proving the weak convergence of stationary distributions to its mean-field limit. Most existing mean-field models concerned the light-traffic regime where the load of the system, denote by ρ, is strictly less than one and is independent of the size of the system. This is because a traditional mean-field model represents the limit of the corresponding stochastic system. Therefore, the load of the mean-field model is ρ= limN→ ∞ρ(N), where ρ(N)is the load of the stochastic system of sizeN.Now if ρ(N)→ 1asN→ ∞ (i.e., in the heavy-traffic regime), then ρ=1. For most systems, the mean-field limits when ρ=1 are trivial and meaningless. To overcome this difficulty of traditional mean-field models, this paper takes a different point of view on mean-field models. Instead of regarding a mean-field model as the limiting system of large-scale stochastic system, it views the equilibrium point of the mean-field model, called a mean-field solution, simply as an approximation of the stationary distribution of the finite-size system. Therefore both mean-field models and solutions can be functions ofN.This paper first outlines an analytical method to bound the approximation error based on Stein's method and the perturbation theory. We further present two examples: theM/M/Nqueueing system and the supermarket model under the power-of-two-choices algorithm. For both applications, the method enables us to characterize the system performance under a broad range of traffic loads. For the supermarket model, this is the first paper that rigorously quantifies the steady-state performance of the-power-of-two-choices in the heavy-traffic regime. These results in the heavy-traffic regime cannot be obtained using the traditional mean-field analysis and the interchange of the limits.

Stein's Method for Mean-Field Approximations in Light and Heavy Traffic Regimes

Mean-field analysis is an analytical method for understanding large-scale stochastic systems such as large-scale data centers and communication networks. The idea is to approximate the stationary distribution of a large-scale stochastic system using the equilibrium point (called the mean-field limit) of a dynamical system (called the mean-field model). This approximation is often justified by proving the weak convergence of stationary distributions to its mean-field limit. Most existing mean-field models concerned the light-traffic regime where the load of the system, denote by ρ, is strictly less than one and is independent of the size of the system. This is because a traditional mean-field model represents the limit of the corresponding stochastic system. Therefore, the load of the mean-field model is ρ=lim N-> ∞ ρ (N) , where ρ (N) is the load of the stochastic system of size N. Now if ρ (N) -> 1 as N -> ∞ (i.e., in the heavy-traffic regime), then ρ=1. For most systems, the mean-field limits when ρ=1 are trivial and meaningless. To overcome this difficulty of traditional mean-field models, this paper takes a different point of view on mean-field models. Instead of regarding a mean-field model as the limiting system of large-scale stochastic system, it views the equilibrium point of the mean-field model, called a mean-field solution, simply as an approximation of the stationary distribution of the finite-size system. Therefore both mean-field models and solutions can be functions of N. The proposed method focuses on quantifying the approximation error. If the approximation error is small (as we will show in two applications), then we can conclude that the mean-field solution is a good approximation of the stationary distribution.

Analysis of some large-scale nonlinear stochastic dynamic systems with subspace-EPC method

The probabilistic solutions to some nonlinear stochastic dynamic (NSD) systems with various polynomial types of nonlinearities in displacements are analyzed with the subspace-exponential polynomial closure (subspace-EPC) method. The space of the state variables of the large-scale nonlinear stochastic dynamic system excited by Gaussian white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system are then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in the other subspace is formulated. Therefore, the FPK equations in low dimensions are obtained from the original FPK equation in high dimensions and the FPK equations in low dimensions are solvable with the exponential polynomial closure method. Examples about multi-degree-offreedom NSD systems with various polynomial types of nonlinearities in displacements are given to show the effectiveness of the subspace-EPC method in these cases.

On the behavior of ECN/RED gateways under a large number of TCP flows: Limit theorems

We consider a discrete-time stochastic model of an ECN/RED gateway where competing TCP sources share the link capacity. As the number of competing flows becomes large, the asymptotic queue behavior (normalized by the number of flows) at the gateway can be described by a simple recursion and the throughput behavior of individual TCP flows becomes asymptotically independent. A Central Limit Theorem complement is also presented, yielding a more accurate characterization of the asymptotic queue size. These results suggest a scalable yet accurate model of this complex large-scale stochastic feedback system, and crisply reveal the sources of queue fluctuations.

DECENTRALIZED HIERARCHICAL STOCHASTIC CONTROL IN A LARGE SCALE STATIC SYSTEM

In the paper a synthesis of control law for a large scale stochastic system is presented. The large scale system composed of M linear static subsystems with an-iteraction and quadratic performance index are considered. A two-level hierarchical control structure is assumed, in which a coordinator and local controllers have access to different information. A suboptimal algorithm, in which it is possible to partially decompose calculations and to realize decentralized control, is proposed. A simple example is presented.

Two-level hierarchical control in a large-scale stochastic dynamic system

Two-level hierarchical control in a large-scale stochastic dynamic system

Two-level hierarchical control in a large-scale stochastic dynamic system

In this paper a two-level hierarchical control for a resource allocation problem is considered. It is assumed that a coordinator on the upper level and local controllers on the lower level have different information. Soft constraints on control and state variables are taken into account. For a dynamic scalar system and a quadratic performance index, two strategies for the local controllers are proposed.

Exponential stability of large-scale stochastic differential equations

Given a large-scale stochastic system described by a number of subsystems, we introduce the corresponding isolated subsystems. It is shown that the exponential stability of the isolated systems implies the exponential stability of the large-scale system under some hypotheses added on the interconnected terms. We also study a slightly special case where the large-scale system is described in a hierarchical form, i.e., the system consists of several subsystems and each subsystem interacts only with ‘lower’ subsystems but not with ‘higher’ subsystems.

Stability of interconnected stochastic delay systems

A decomposition of a large-scale stochastic functional differential system into a family of isolated subsystems is made. A set of criteria is obtained under which the uniform asymptotic stability with probability one of the trivial solution process of the large-scale system is preserved. These criteria are expressed in terms of the stability of isolated subsystems and bounds on interconnection components. As a tool of our analysis, Razumikhin-type stability conditions for stochastic functional differential systems are also obtained. Several examples are given to show the applicability of our results.