If there are absorbing states (for example, failed states) in the system, then the system eventually reaches one of the absorbing states. This shows that system reliability at infinite time is zero. In some cases, the consequences of (damage due to) different failure modes may be different. In order to evaluate the system further (to find overall failure cost for example), the probability of reaching each absorbing state must be found. This can be achieved by solving individual state probabilities at infinite time. However, this can be more easily found using the following procedure.

Find matrix A, which is the product of matrix M=-Qt-1 and matrix St:

It should be noted that the element zij of matrix Zrepresents the probability that the system eventually reaches absorbing state j when the system is initially in state I.

Consider the 2-unit series system shown in Figure 8-8. State 2 (3) presents system failure due to the failure of component 1 (2).

Therefore:

It shows that λ1/(λ1+λ2) is the probability that the system reaches a failed state due to the failure of component 1. Similarly, λ2/(λ1+λ2) is the probability that the system reaches a failed state due to the failure of component 2.