Algorithms for some infinite-state MDPs and stochastic games

I will survey a body of work, developed over the past decade or so, on algorithms for, and the computational complexity of, analyzing and model checking some important families of countably infinite-state Markov chains, Markov decision processes (MDPs), and stochastic games. These models arise by adding natural forms of recursion, branching, or a counter, to finite-state models, and they correspond to probabilistic/control/game extensions of classic automata-theoretic models like pushdown automata, context-free grammars, and one-counter automata. They subsume some classic stochastic processes such as multi-type branching processes and quasi-birth-death processes. They also provide a natural model for probabilistic procedural programs with recursion.