Multicomponent systems with time-dependent transition rates. A note on the representation and number of system states

Abstract~Many applications in reliability and risk assessment ask for a timedependent analysis. In most cases, the Markovian treatment, based on the assumption of constant transition rates between states, is used. However, a realistic modeling should take into account the time at which a component entered its current state, because transition rates actually depend on the time elapsed in this state. When multicomponent systems are considered, the chronological order of the last transition times of the components affects the calculation of the system reliability characteristics. This fact entails a multiplication of the number of system states, compared with the classical Markovian analysis. This work presents a systematic counting of all possible system configurations