Markov decision processes with delays and asynchronous cost collection

Markov decision processes (MDPs) may involve three types of delays. First, state information, rather than being available instantaneously, may arrive with a delay (observation delay). Second, an action may take effect at a later decision stage rather than immediately (action delay). Third, the cost induced by an action may be collected after a number of stages (cost delay). We derive two results, one for constant and one for random delays, for reducing an MDP with delays to an MDP without delays, which, differs only in the size of the state space. The results are based on the intuition that costs may be collected asynchronously, i.e., at a stage other than the one in which they are induced, as long as they are discounted properly.