The vanishing discount approach to constrained continuous-time controlled Markov chains

We consider a denumerable state continuous-time controlled Markov chain (CMC) with possibly unbounded transition and reward rates. We deal with constrained optimality; that is, we want to maximize a discounted reward (an average reward) criterion subject to a constraint on a discounted cost (an average cost). We give conditions ensuring that the average constrained optimal reward and policies can be obtained as the limit, as the discount rate vanishes, of the corresponding discounted constrained optimal reward and policies. This extends to average constrained CMCs the standard results on the vanishing discount approach for the average unconstrained case. We also present an example showing that the vanishing discount results for constrained problems might not hold.