An effective design of effective and efficient self-adaptive systems may rely on several existing approaches. Software models and model checking techniques at run time represent one of them since they support automatic reasoning about such changes, detect harmful configurations, and potentially enable appropriate (self-)reactions. However, traditional model checking techniques and tools may not be applied as they are at run time, since they hardly meet the constraints imposed by on-the-fly analysis, in terms of execution time and memory occupation. For this reason, efficient run-time model checking represents a crucial research challenge. This paper precisely addresses this issue and focuses on probabilistic run-time model checking in which reliability models are given in terms of Discrete Time Markov Chains which are verified at run-time against a set of requirements expressed as logical formulae. In particular, the paper discusses the use of probabilistic model checking at run-time for self-adaptive systems by surveying and comparing the existing approaches divided in two categories: state-elimination algorithms and algebra-based algorithms. The discussion is supported by a realistic example and by empirical experiments
