Memory-efficient kronecker algorithms with applications to the modelling of parallel systems

Parallel and distributed systems can be modelled as a set of interacting components. This has an impact on the mathematical structure of the model, namely it induces a product form represented by a tensor product. We present a new algorithm for computing the solution of large Markov chain models whose generators can be represented in the form of a generalized tensor algebra, such as networks of stochastic automata. The tensor structure inherently involves a product state space but inside this product state space, the actual reachable state space can be much smaller. For such cases, we propose an improvement of the standard numerical algorithm, the so-called "shuffle algorithm", which necessitates only vectors of the size of the actual state space. With this contribution, numerical algorithms based on tensor products can now handle much larger models, even with functional rates and synchronizing events.