An exponential distribution is often used to model the lifespan of a component with a constant failure rate. What is a suitable distribution to use in the situation where a component has a constant failure rate $\lambda$ but that after a certain time $T$ if the part has not failed it is replaced anyway?
The question is whether there is a continuous distribution that approximates the following: $$f(x;\lambda) = \begin{cases} 0 & x < 0,\\ \lambda e^{-(\lambda x)} & x \ge 0 <T \\ 1 & x\ge T. \end{cases}$$
My best guess would be to use an appropriately scaled $Beta(\alpha,\beta)$ distribution.
I haven't worked it out fully but $\alpha$ and $\beta$ would obviously need to depend on $\lambda$ and $T$. For instance when $T >> \lambda$ the exponential distribution again would be a fair approximation. 
Edited: added explicit definition of the distribution following 5201314's contribution.
For a time to failure $X$, constant failure rate $\lambda$ and given that it is replaced after a certain time $T$ if it hasn't failed, $$F_X(x)=\begin{cases} 1-e^{-\lambda x}, & x<T\\ 1, & x\ge T \end{cases}$$