Whit this constraint imposed, the principles used to deduce the equation for parallel system can be extended and the probability of a component in the system being already out for maintenance when a forced outage or failure of another component occurs can be included. The equation for the failure rate of a two component parallel system or second order minimal cut set (Equation 11.19) was
Λp = λ1(λ2λ1)+ λ2(λ1λ2)
In terms of maintenance, this equation can be expressed in words as : failure of the system occurs if (component 1 is out for maintenance failure of component 2 during the maintenance time of component 1 ) or (component 2 is out for maintenance followed by failure of component 1 during the maintenance time of component 2).
Because of the above constraint, the condition: component 1 fails followed by a maintenance outage of component 2 during the repair time of component 1 and vice versa is not included because this would fail the system by the simple action of commencing maintenance.
Defining
Λ”1, λ”2 as maintenance outage rate/year
R”1,r”2 as average maintenance time/outage
Equation 11.19, for the case of a failure event overlapping a maintenance outage, becomes
Λp”= λ1”( λ2r1”)+ λ2”( λ1r2”) (11.23)
The evaluation of overlapping repair time is more complicated in this case unless components 1 and 2 are identical. In the general case when components 1 and 2 are different, the overlapping repair time ra” of the first term of equation 11.23 is different from that, rb”, of the second term. Using the same concept as equation 11.15c
There, since either term 1 or 2 of Equation 11.23 will cause system failure, the principle of series system (Equation 11.18) can be used to give