10.2 CTMC‐Based Method
As discussed in Section 2.5, system states and state transitions are two essential concepts of the Markov‐based method [29]. A system state is defined by a specific combination of component state variables at a given instant of time. A state transition occurs due to the failure or repair of a system component [30]. The CTMC‐based methods assume exponential time‐to‐failure and time‐to‐repair distributions for the system components.
The solution to a CTMC model with n different states includes the probability of the system being in each state, particularly, Pj(t) that denotes the probability that the system is in state j at time t (j = 1,…,n). They can be obtained by solving a set of differential equations in the form of (2.26), which is
(10.1)
− α 11 α 21 α 31 … α n 1 α 12 − α 22 α 32 … α n 2 ...