Percentile Queries in Multi-dimensional Markov Decision Processes

Markov decision processes (MDPs) with multi-dimensional weights are useful to analyze systems with multiple objectives that may be conflicting and require the analysis of trade-offs. In this paper, we study the complexity of percentile queries in such MDPs and give algorithms to synthesize strategies that enforce such constraints. Given a multi-dimensional weighted MDP and a quantitative payoff function f, thresholds v(i) (one per dimension), and probability thresholds alpha(i), we show how to compute a single strategy to enforce that for all dimensions i, the probability of outcomes rho satisfying f(i) (rho) >= v(i) is at least alpha(i). We consider classical quantitative payoffs from the literature (sup, inf, lim sup, lim inf, mean-payoff, truncated sum, discounted sum). Our work extends to the quantitative case the multi-objective model checking problem studied by Etessami et al. [16] in unweighted MDPs.