Order Statistics Distribution for General Multivariate Distribution

A mission‐critical service has three batteries with mean times to failure τ1, τ2, and τ3. Dependencies of the failure times are assumed to be given by Farlie–Gumbel–Morgenstern copula with parameter α. Consider the event of the second battery going out of order.

    
Compute the survival function, i.e. the probability for the system to be in order after epoch t:
Find the expected time to failure:
Plot the survival function for τ18.0, τ25, and τ34.5 for different values of copula parameter α: