Markov decision processes with constrained stopping times

The optimization problem for a stopped Markov decision process is considered to be taken over stopping times tau constrained so that E-tau less than or equal to alpha for some fixed alpha > 0. We introduce the concept of a randomized stationary stopping time which is a mixed extension of the entry time of a stopping region and prove the existence of an optimal constrained pair of stationary policy and stopping time by utilizing a Lagrange multiplier approach. Also, applying the idea of the onestep look ahead (OLA) policy the optimal constrained pair is sought concretely. As an example, constrained Markov deteriorating system is explained.