Tensor methods for the computation of MTTA in large systems of loosely interconnected components

TL;DR

This paper introduces tensor-based algorithms for efficiently computing the mean-time-to-absorption in large Markov systems by decoupling local and synchronization transitions, enabling scalable analysis of systems with billions of states.

Contribution

It presents a novel tensor-train based method with linear and quadratic convergence for calculating MTTA in large interconnected Markov systems, overcoming computational challenges.

Findings

Algorithm achieves linear convergence for MTTA computation.

Quadratic convergence method improves efficiency for near-critical problems.

Tensor-train representation enables handling systems with billions of states.

Abstract

We are concerned with the computation of the mean-time-to-absorption (MTTA) for a large system of loosely interconnected components, modeled as continuous time Markov chains. In particular, we show that splitting the local and synchronization transitions of the smaller subsystems allows to formulate an algorithm for the computation of the MTTA which is proven to be linearly convergent. Then, we show how to modify the method to make it quadratically convergent, thus overcoming the difficulties for problems with convergent rate close to $1$. In addition, it is shown that this decoupling of local and synchronization transitions allows to easily represent all the matrices and vectors involved in the method in the tensor-train (TT) format - and we provide numerical evidence showing that this allows to treat large problems with up to billions of states - which would otherwise be unfeasible.