We consider in this paper constrained Markov decision processes. This type of control model has many applications in telecommunications and other fields [E. Altman and A. Shwartz, IEEE Trans. Automat. Control, 34 (1989), pp. 1089-1102, E. A. Feinberg and M. I. Reiman, Probab. Engrg. Inform. Sci., 8 (1994), pp. 463-489, A. Hordijk and F. Spieksma, Adv. in Appl. Probab., 21 (1989), pp. 409-431, A. Lazar, IEEE Trans. Automat. Control, 28 (1983), pp. 1001-1007,. Nain and K. W. Ross, IEEE Trans. Automat. Control, 31 (1986), pp. 883-888, K. W. Ross and B. Chen, IEEE Trans. Automat. Control, 33 (1988), pp. 261-267]. We address the issue of the convergence of the value and optimal policies of the problem with discounted costs, to the ones for the problem with expected average cost. We consider the general multichain ergodic structure. We present two stability results in this paper. We establish the continuity of optimal values and solutions of as well as some type of robustness of some suboptimal solutions in the discount factor. Our proof relies on same general theory on continuity of values and solutions in convex optimization that relies on well-known notions of Gamma-convergence.