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

Masetti G., Robol L.

Markov chain  CTMC  Stochastic modeling  Kronecker algebra  Tensor  Exponential sums  Matrix functions  Stochastic automata networks 

We are concerned with the computation of the mean-time-to-failure(MTTF) for a large system of loosely interconnected components, mod-eled as continuous time Markov chains. In particular, we show that split-ting the local and synchronization transitions of the smaller subsystemsallows to formulate an algorithm for the computation of the MTTF whichis proven to be linearly convergent. Then, we show how to modify themethod to make it quadratically convergent, thus overcoming the difficul-ties for problems with convergent rate close to1.In addition, it is shown that this decoupling of local and synchroniza-tion transitions allows to easily represent all the matrices and vectors in-volved in the method in the tensor-train (TT) format -- and we providenumerical evidence showing that this allows to treat large problems withup to billions of states -- which would otherwise be unfeasible.

Source: ISTI Technical reports, 2019