Timing analysis of safety properties using fault trees with time dependencies and timed state-charts

Behavior in time domain is often crucial for safety critical systems. Standard fault trees cannot express time-dependent behavior. In the paper, timing analysis of safety properties using fault trees with time dependencies (FTTDs) and timed state-charts is presented. A new version of timed state-charts (TSCs) is also proposed. These state-charts can model the dynamics of technical systems, e.g. controllers, controlled objects, and people. In TSCs, activity and communication times are represented by time intervals. In the proposed approach the structure of FTTD is fixed by a human. Time properties of events and gates of FTTD are expressed by time intervals, and are calculated using TSCs. The minimal and maximal values of these time intervals of FTTD can be calculated by finding paths with minimal and maximal time lengths in TSCs, which is an NP-hard problem. In order to reduce the practical complexity of computing the FTTD time parameters, some reductions of TSCs are defined in the paper, such as sequential, alternative, loop (iteration), and parallel. Some of the reductions are intuitive, in case of others—theorems are required. Computational complexity of each reduction is not greater than linear in the size of reduced TSC. Therefore, the obtained results enable decreasing of the costs of FTTD time parameters calculation when system dynamics is expressed by TSCs. Case study of a railroad crossing with a controller that controls semaphores, gate, light-audio signal close to the gate will be analyzed.