Expected Number of Failures
For each possible outcome of the event tree, the expected number of failures (W) estimates the expected number of failures during the period of time 0 to time t. For a given sequence at time t, this calculation is performed by doing the following:
1. Compute for each column the expected number of failures during the period, Wk(t), for the k th column along the sequence.
2. Compute W(t) for the possible outcome by combining these columnwise Wk(t) values.
For the kth column, Wk(t) is computed by the following set of formulae:
• For a column linked to a fault tree gate or event, Wk(t) is equal to the failure rate for that gate or event multiplied by t.
• For a constant probability column, Wk(t) ) is equal to 0.
• For failure rate/MTBF column or failure with repair column with instantaneous repair (mean corrective time = 0), W k (t) = Q k (t), where Q k (t) is the column unavailability at time t.
• For a failure with repair column with non-instantaneous repair (mean corrective time > 0), Wk(t)=λ/(λ+μ)*(μ*t+Q k(t)), where λ is the conditional failure intensity derived from frequency, failure rate, or MTBF, μ = 1/mean corrective time, and Qk(t) is the column unavailability at time t.
The calculation of w(t) for the possible outcome uses the formula:
Where K is the collection of active columns along the sequence
This formula excludes the possibility of simultaneous failures of multiple columns. Also, the set of time indices, tn, is taken from the user-specified time settings for event tree calculations. The actual indices are determined by the values specified for Start time, Through end time, and Number of data points in the Event Tree calculation properties. For more information, see Event Tree Calculate Property Page.
The outer summation of the difference over these indices, (Wk(tn))−(Wk(tn−1)), is actually an approximation for the corresponding integral of Wk(S) over the interval from time 0 to time t. Therefore, to evaluate W(t) accurately, a few points need to be observed. The value for From start time should be 0, and the value for Primary time point should be within the range specified for From start time and Through end time. The accuracy of W(t) may be checked by repeatedly increasing the value for Number of data points and recalculating until W(t) is sufficiently stable.