This paper deals with a class of partially observable discounted Markov decision processes defined on Borel state and action spaces, under unbounded one-stage cost. The discount rate is a stochastic process evolving according to a difference equation, which is also assumed to be partially observable. Introducing a suitable control model and filtering processes, we prove the existence of optimal control policies. In addition, we illustrate our results in a class of GI/GI/1 queueing systems where we obtain explicitly the corresponding optimality equation and the filtering process.
We are concerned with a class of GI/GI/1 queueing systems with controlled service rates, in which the waiting times are only observed when they take zero value. Applying a suitable filtering process, we show the existence of optimal control policies under a discounted optimality criterion.