Decay rates and cutoff for convergence and hitting times of Markov chains with countably infinite state space

For a positive recurrent continuous-time Markov chain on a countable state space, we compare the access time to equilibrium to the hitting time of a particular state. For monotone processes, the exponential rates are ranked. When the process starts far from equilibrium, a cutoff phenomenon occurs at the same instant, in the sense that both the access time to equilibrium and the hitting time of a fixed state are equivalent to the expectation of the latter. In the case of Markov chains on trees, that expectation can be computed explicitly. The results are illustrated on the M/M/infinity queue.