Time-redundant system reliability under randomly constrained time resources

The paper considers the time-redundant system where the system total task is a sequence of n phases and the total task must be executed during constrained time. For every phase, there is its own server, which executes the phase task during randomly distributed time. The server is not perfectly reliable and two types of failure ('open' and 'closed') are possible. Redundant servers may be used in any phase. The time-probability characteristics are introduced for any task, based on which the system reliability is treated as a probability that the system total task will be correctly completed during a corresponding time resource, which also may be randomly distributed. The adequate model is presented and a semi-Markov process is used as a mathematical technique. The closed-form solution was derived based on an acyclic Semi-Markov process. The numerical example of the elaborated approach is presented