Optimal Time-Abstract Schedulers for CTMDPs and Markov Games

TL;DR

This paper proves the existence of optimal time-abstract schedulers for CTMDPs and Markov games, providing constructive methods to compute simple, finite-memory strategies that converge to memoryless policies.

Contribution

It establishes the existence of optimal schedulers for all CTMDPs and Markov games and offers a constructive approach to compute finite-memory strategies that simplify to memoryless policies.

Findings

Optimal schedulers exist for all CTMDPs.

Constructive methods to compute finite-memory strategies.

Strategies converge to simple memoryless policies.

Abstract

We study time-bounded reachability in continuous-time Markov decision processes for time-abstract scheduler classes. Such reachability problems play a paramount role in dependability analysis and the modelling of manufacturing and queueing systems. Consequently, their analysis has been studied intensively, and techniques for the approximation of optimal control are well understood. From a mathematical point of view, however, the question of approximation is secondary compared to the fundamental question whether or not optimal control exists. We demonstrate the existence of optimal schedulers for the time-abstract scheduler classes for all CTMDPs. Our proof is constructive: We show how to compute optimal time-abstract strategies with finite memory. It turns out that these optimal schedulers have an amazingly simple structure - they converge to an easy-to-compute memoryless scheduling policy after a finite number of steps. Finally, we show that our argument can easily be lifted to Markov games: We show that both players have a likewise simple optimal strategy in these more general structures.