On Markov Chains with Uncertain Data

In this paper, a general method is described to determine uncertainty intervals for performance measures of Markov chains given an uncertainty region for the parameters of the Markov chains. We investigate the effects of uncertainties in the transition probabilities on the limiting distributions, on the state probabilities after n steps, on mean sojourn times in transient states, and on absorption probabilities for absorbing states. We show that the uncertainty effects can be calculated by solving linear programming problems in the case of interval uncertainty for the transition probabilities, and by second order cone optimization in the case of ellipsoidal uncertainty. Many examples are given, especially Markovian queueing examples, to illustrate the theory.