Probabilistic model checking can provide formal guarantees on the behavior of stochastic models relating to a wide range of quantitative properties, such as runtime, energy consumption or cost. But decision making is typically with respect to the expected value of these quantities, which can mask important aspects of the full probability distribution such as the possibility of high-risk, low-probability events or multimodalities. We propose a distributional extension of probabilistic model checking, applicable to discrete-time Markov chains (DTMCs) and Markov decision processes (MDPs). We formulate distributional queries, which can reason about a variety of distributional measures, such as variance, value-at-risk or conditional value-at-risk, for the accumulation of reward until a co-safe linear temporal logic formula is satisfied. For DTMCs, we propose a method to compute the full distribution to an arbitrary level of precision, based on a graph analysis and forward analysis of the model. For MDPs, we approximate the optimal policy with respect to expected value or conditional value-at-risk using distributional value iteration. We implement our techniques and investigate their performance and scalability across a range of benchmark models. Experimental results demonstrate that our techniques can be successfully applied to check various distributional properties of large probabilistic models.

翻译：概率模型检测能够对与运行时间、能耗或成本等广泛定量属性相关的随机模型行为提供形式化保证。但决策通常基于这些数量的期望值，这可能掩盖完整概率分布的重要特征，例如高风险低概率事件或多模态的可能性。我们提出概率模型检测的分布扩展，适用于离散时间马尔可夫链（DTMC）和马尔可夫决策过程（MDP）。我们形式化分布查询，可推理方差、风险价值或条件风险价值等各类分布度量，用于累积奖励直至满足安全线性时序逻辑公式。对于DTMC，我们提出基于图分析和正向分析的方法，以任意精度计算完整分布。对于MDP，我们使用分布价值迭代来逼近关于期望值或条件风险价值的最优策略。我们实现了所提技术，并在多个基准模型上研究其性能与可扩展性。实验结果表明，我们的技术可成功应用于检测大规模概率模型的各种分布属性。