I have a function $a(t)$ which expresses the total to first failure of machine $1$. At the same time I have on stand by machine $2$ which is identical to machine $1$ in every possible way. As soon as machine $1$ fails, machine $2$ turns on.
I need to find the function describing the total to failure of the sequence of both of the machines (e.g. failure of $M_1$ then failure of $M_2$).
Should I express $T(t)$ as $a(t) + a(t)$ which is the summation of machine $1$ to failure and then machine $2$ to failure? Or can it be expressed using conditional probability as $T(M_2=\text{fail}/M_1=\text{fail})$ which in turns can be described as $T(a(t)/a(t))$
EDIT
Expression of the function is u.e^(-ux) where u is replacing the parameter (1/theta).
It's an auxiliary circuit. The resulting failure rv is the sum of the two.
For example, assuming independence, if the failure density of the two machines $X,Y$ is exponential with mean $1/\theta$, the resulting failure density is
$$Z=(X+Y)\sim \text{Gamma}[2;\theta]$$
where $\theta$ here represents the rate parameter
If you tell us the actual expression of your failure function I can try to give you more hints