Parallel discretization of the Markov chain approximation for the autoregressive moving average chart

In the Markov chain model of an autoregressive moving average chart, the post-transition states of nonzero transition probabilities are distributed along one-dimensional lines of a constant gradient over the state space. By considering this characteristic, we propose discretizing the state space parallel to the gradient of these one-dimensional lines. We demonstrate that our method substantially reduces the computational cost of the Markov chain approximation for the average run length in two- and three-dimensional state spaces. Also, we investigate the effect of these one-dimensional lines on the computational cost. Lastly, we generalize our method to state spaces larger than three dimensions.