Continuous approximation of collective system behaviour: A tutorial

Abstract

In this paper we present an overview of the field of deterministic approximation of Markov processes, both in discrete and continuous times. We will discuss mean field approximation of discrete time Markov chains and fluid approximation of continuous time Markov chains, considering the cases in which the deterministic limit process lives in continuous time or discrete time. We also consider some more advanced results, especially those relating to the limit stationary behaviour. We assume a knowledge of modelling with Markov chains, but not of more advanced topics in stochastic processes

Mean Field Approximation of Uncertain Stochastic Models

“…A simpler definition can be obtained by specifying transition classes, similarly to [15], and describing the state of the system by a state vector, counting the size of each population. The idea is to specify the dynamics by a set of possible transitions or events, providing their rate, as a function of the state vector and of the imprecise parameters, and how they change the state vector of the system.…”

“…Mean field based analysis, for the transient and the steady state, have been applied in performance modelling and model-checking tools [13], [14], systems biology, epidemiology [5]. For a gentle introduction, see [15].…”

Spatial Representations and Analysis Techniques

“…Basic definitions can be found in the appendix. An important aspect of our modelling approach is the application of the mean-field technique where the analysis of a population CTMC or DTMC can be approximated by an analysis using ordinary differential equations (ODEs) [58,10]. As the number of states of a Markov chain increases (the "state-space explosion" problem), the analysis of the Markov chain becomes intractable.…”

“…When applied to spatial models, it is a spatial abstraction technique because information about what happens in individual locations is lost. The basic approach is to obtain an ODE for each subpopulation for the ensemble 10 of the average over all locations for that subpopulation. This will then (in most cases) be expressed in terms of the expectation of the product of two variables (a higher order moment).…”

“…This section briefly introduces these concepts, as they would be used in stochastic modelling both without aggregation of state and with aggregation of state (population-based Markov chains) [4,10].…”