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Use Standby component prior to their need at a deterministic time?

Posted 5 months ago
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Hi this might be a simple question but i still can't figure it out. Suppose i have component which is critical to the survival of a machine. The component has a lifetime distribution distA if it is active and distI if is not active. There is a identical backup of this component in the system.

Now in a parallel configuration, given the survival of the system till time t, its failure before time s>t is given by:

NProbability[Conditioned[z<=s,z>t],Distributed[z,ReliabilityDistribution[x || y, {{x,distA}, {y,distA}}]]]

Similarly in a standby arrangement the same probability is given by

NProbability[Conditioned[z<=s,z>t],Distributed[z,StandbyDistribution[distA, {{distI, distA}},1]]]

But my question is how do i model the situation where system switching from primary to the backup happens at a fixed time a<t. So i want to calculate the probability given that the primary component has survived for a fixed time a in active mode and secondary has survived till time a in inactive mode but the operator still switched to backup, now what is the probability that the backup failed at some time s>t given that the backup additionally survived till t in active mode after already surviving till a in inactive mode. I know mathematica provides NProbabiltiy and Conditioned. But how do i use these to calculate the require probability?

Conditioned[y1<=s , x>a && y0>a && (y1>t | y1 >a)] where Distributed[x,distA], Distributed[y0,distI], Distributed[y1,distA]

I am not sure where the above expression is even modelling the event correctly. Any help please.

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