Ratio and Weight Objectives in Annotated Markov Chains

In this thesis we investigate the analysis of energy-aware systems modeled by finitestate Markov chains. We deal with Markov chains augmented with transition weights and consider objectives referring to the accumulated weight along finite paths or to the quotient of the accumulated weight of two non-negative weight functions. Beside objectives asking for linear-time properties to be fulfilled while always exceeding a threshold on the accumulated weight or ratio, we allow for properties where the threshold has not to be exceeded globally, but at least once, infinitely often or globally after a finite initialization phase. For these objectives, we aim to exactly compute optimal thresholds such that the corresponding objective is either fulfilled almost-surely or with positive probability (quantiles). To this end, we first state a polynomial-time procedure to decide, whether a weight objective is fulfilled almostsurely or with positive probability. Employing a simple transformation, decision problems referring to ratio objectives are known to be reducible in polynomial time to decision problems related to weight objectives. It turns out that this transformation cannot be applied for quantiles. However, using two different approaches we state polynomial-time computation schemes for both, ratio and weight quantiles. Weight quantiles can be computed by solving the corresponding decision problem within a simple binary search. The algorithm for ratio quantiles relies on establishing a finite set of rational-valued candidates and solving a best-approximation problem using the so-called continued-fraction method.