# Average duration of a double outage in a system with exponentially-distributed failure and repair times

Assume that we have a system with two units. For each unit, its failure frequency follows an exponential distribution with mean $\lambda_1$ and its repair time follows an exponential distribution with mean $\lambda_2$. In addition, we assume that these units fail indendently. My question is: how do i calculate the average (and ideally the distribution of) time spent in double outage conditions (i.e. both units have failed)?

OK so it turns out that the answer to my question is straightforward. By modelling the above situation as a four-state Markov process (states 11, 10, 01 and 00 where first digit denotes status of first component etc.), theory states that the mean time spent in each state is the inverse exit rate. The exit rate for the last state 00 where both components have failed is $2/(\lambda_2)$. This is because any of the two components being repaired is enough to change state to 01 or 10. As a result, mean time spent in 00 is $\lambda_2/2$