Probabilistic approximation of runtime quantitative verification in self-adaptive systems

Abstract

Cyber-physical systems (CPS) are expected to continuously monitor the physical components to autonomously calculate appropriate runtime reactions to deal with the uncertain environmental conditions. Self-adaptation, as a promising concept to fulfill a set of provable rules, majorly needs runtime quantitative verification (RQV). Taking a few probabilistic variables into account to represent the uncertainties, the system configuration will be extremely large. Thus, efficient approaches are needed to reduce the model state-space, preferably with certain bounds on the approximation error. In this paper, we propose an approximation framework to efficiently approximate the entire model of a self-adaptive system. We split up the large model into strongly-connected components (SCCs), apply the approximation algorithm separately on each SCC, and integrate the result of each part using a centralized algorithm. Due to a number of changes in probabilistic variables, it is not possible to use static models. Addressing this issue, we have deployed parametric Markov decision process. In order to apply approximation on the model, the notion of ε-approximate probabilistic bisimulation is utilized that introduces the approximation level ε. We show that our approximation framework offers a certain error bound on each level of approximation. Then, we denote that the approximation framework is appropriate to be applied in decision-making process of self-adaptive systems where the models are relatively large. The results reveal that we can achieve up to 50% size reduction in the approximate model while maintaining the accuracy about 95%. In addition, we discuss about the trade-off between efficiency and accuracy of our approximation framework.