Size expansions of mean field approximation: Transient and steady-state analysis

Abstract

Mean field approximation is a powerful tool to study the performance of large stochastic systems that is known to be exact as the system's size N goes to infinity. Recently, it has been shown that, when one wants to compute expected performance metric in steady-state, this approximation can be made more accurate by adding a term V /N to the original approximation. This is called a refined mean field approximation in [21]. In this paper, we improve this result in two directions. First, we show how to obtain the same result for the transient regime. Second, we provide a further refinement by expanding the term in 1/N 2 (both for transient and steady-state regime). Our derivations are inspired by moment-closure approximation, a popular technique in theoretical biochemistry. We provide a number of examples that show: (1) that this new approximation is usable in practice for systems with up to a few tens of dimensions, and (2) that it accurately captures the transient and steady state behavior of such systems.

Bounding Mean First Passage Times in Population Continuous-Time Markov Chains

“…T = ∞. This is achieved by dropping the moments y 2 of measure ν 2 in the linear constraints (20). The empirical FPT distribution based on 100,000 SSA simulations is given in Figure 2a and the bounds, given different moment orders, are given in Figure 2b.…”

“…This subclass of Continuous-Time Markov Chains (CTMCs) is used to describe the stochastic dynamics of systems in various domains. Prominent applications are chemical reaction networks in quantitative biology [50], epidemic spreading [42], performance analysis of technical and information systems [9,20] as well as the behavior of collective adaptive systems [7].…”

“…Recent work concentrates on numerical methods for PCTMCs that approximate the statistical moments of the system without the need to consider the probability of each state. For groups of identically behaving agents, it is possible to derive systems of differential equations for the evolution of the statistical population moments [8,47,10,19,46,20]. However, as the system of exact moment equations is not closed, approximation schemes typically rely on certain assumptions about the underlying probability distribution.…”

Mixture distributions in a stochastic gene expression model with delayed feedback: a WKB approximation approach

“…Generally, however, explicit solutions are unavailable or intractable and one resorts to stochastic simulation or seeks a numerical solution to a finite truncation of the master equation (Munsky and Khammash 2006;Borri et al 2016;Gupta et al 2017). An alternative approach, which often provides useful qualitative insights into the model behaviour, is based on reduction techniques such as quasi-steady-state (Srivastava et al 2011;Kim et al 2014;Plesa et al 2019) and adiabatic reductions (Bruna et al 2014;Popovic et al 2016), piecewise-deterministic framework (Lin and Doering 2016;Lin and Buchler 2018), linear-noise approximation (Schnoerr et al 2017;Modi et al 2018), or moment closure (Singh and Hespanha 2007;Andreychenko et al 2017;Gast et al 2019).…”

Mixture distributions in a stochastic gene expression model with delayed feedback

“…Generally, however, explicit solutions are unavailable or intractable and one resorts to stochastic simulation or seeks a numerical solution to a finite truncation of the master equation (Munsky and Khammash, 2006;Borri et al, 2016;Gupta et al, 2017). An alternative approach, which often provides useful qualitative insights into the model behaviour, is based on reduction techniques such as quasi-steadystate (Srivastava et al, 2011;Kim et al, 2014) and adiabatic reductions (Bruna et al, 2014;Popovic et al, 2016), piecewise-deterministic framework (Lin and Doering, 2016;Lin and Buchler, 2018), linear-noise approximation (Schnoerr et al, 2017;Modi et al, 2018), or moment closure (Singh and Hespanha, 2007;Andreychenko et al, 2017;Gast et al, 2019).…”