Computing Entry-Wise Bounds of the Steady-State Distribution of a Set of Markov Chains

We present two algorithms to find the component-wise upper and lower bounds of the steady-state distribution of an ergodic Markov chain. whose transition matrix\(\mathbf{M}\) is entry-wise larger than matrix \(\mathbf{L}\). The algorithms are faster than Muntz’s approach. They are based on the polyhedral theory developed by Courtois and Semal and on a new iterative algorithm which gives bounds of the steady-state distribution at each iteration.