In this talk we discuss the model of quantum Markov chains, due to S. Gudder, and most particularly the subset of open quantum walks, due to S. Attal et al., acting on finite graphs. As an iterative process, we use a monitoring procedure to determine the mean time for a quantum walker to visit some chosen vertex for the first time. We are interested in ways of calculating such hitting times besides making direct use of its definition and here we notice algebraic similarities and differences with the classical case. The case of unitary quantum walks remains an interesting open problem for which a solution could have potential applications to the associated theory of Schur functions.