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

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