A machine consists of two independent components, the $i$th of which functions for an exponential time with rate $\lambda_i$. The machine functions as long as at least one of these components function. When a machine fails, a new machine having both components working is put into use. A cost $K$ is incurred whenever a machine failure occurs; operating costs at rate $c_i$per unit time are incurred whenever the machine in use has $i$ working components, $i=1,2$. Find the long-run average cost per unit time.
Denote $T_1,T_2$ as the working time of component $1$ ans component $2$, respectively. Let $T$ be the time of a cycle. I was confused about $E[T]$.
My solution: $$T_{\max}=\max\{T_1,T_2\}$$ $$E[T]=E[T_{\max}]=1/\lambda_1+1/\lambda_2-1/{(\lambda_1+\lambda_2)}$$ But the answer gives: $$E[T]=E[\min\{T_1,T_2\}]+E[Y|T_1\ge T_2]P\{T_1\ge T_2\}+E[Y|T_1\lt T_2]P\{T_1\lt T_2\}$$ $$E[T] = \frac{1}{\lambda_{1} + \lambda_{2}} + \frac{1}{\lambda_{1}}\cdot\frac{\lambda_{2}}{\lambda_{1} + \lambda_{2}} + \frac{1}{\lambda_{2}}\cdot\frac{\lambda_{1}}{\lambda_{1} + \lambda_{2}}$$ and the $Y$ denotes the working time with only one component.
I can't find any mistakes in these two solutions, any help for this would be much appreciated :)