Jump longer to jump less: improving dynamic boundary projection with h-scaling

The master equation describes exactly the dynamics of a Markov Population Process (MPP) by associating one differential equation for each discrete state of the process. It is well known that MPPs are prone to suffer from the so-called curse of dimensionality, making the master equation intractable in most cases. We propose a novel approach, called h-scaling, that covers the state space of an MPP with a smaller number of states by an appropriate re-scaling of the MPP transition rate functions. When the original state space is bounded, this procedure may significantly reduce the number of the states while returning an approximate master equation that still retains good accuracy. We present h-scaling together with some theoretical results on asymptotic correctness and numerical examples taken from the performance evaluation literature. Moreover, we show that h-scaling can be combined with a recently proposed framework called dynamic boundary projection, which couples subsets of the master equation with mean-field approximations, to further reduce the number of equations without penalizing accuracy.