Revisiting the matrix‐free solution of Markov regenerative processes

In this paper, we revisit the steady-state solution method for Markov Regenerative Processes (MRP) proposed in the work by German. This method solves the embedded Markov chain P of the MRP without storing the matrix P explicitly. We address three issues left open in German's Work: 1) the solution method is restricted to Power method; 2) it has been defined only for ergodic MRPs; and 3) no preconditioning is available to speed-up the computation. This paper discusses how to lift these limitations by extending the algorithm to preconditioned Krylov-subspace methods and by generalizing it to the non-ergodic case. An MRP-specific preconditioner is also proposed, which is built from a sparse approximation of the MRP matrix, computed via simulation. An experimental assessment of the proposed preconditioner is then provided.