Computing lower and upper expected first-passage and return times in imprecise birth–death chains

Abstract

We provide simple methods for computing exact bounds on expected first-passage and return times in finite-state birth–death chains, when the transition probabilities are imprecise, in the sense that they are only known to belong to convex closed sets of probability mass functions. In order to do that, we model these so-called imprecise birth–death chains as a special type of time-homogeneous imprecise Markov chain, and use the theory of sub- and supermartingales to define global lower and upper expectation operators for them. By exploiting the properties of these operators, we construct a simple system of non-linear equations that can be used to efficiently compute exact lower and upper bounds for any expected first-passage or return time. We also discuss two special cases: a precise birth–death chain, and an imprecise birth–death chain for which the transition probabilities belong to linear-vacuous mixtures. In both cases, our methods simplify even more. We end the paper with some numerical examples.