Authors’ summary: For most repairable systems, the number

$N\left(t\right)$ of failed components at time

$t$ appears to be a good quality parameter, so it is critical to study this random function. Here the components are assumed to be independent and both their lifetime and their repair time are exponentially distributed. Moreover, the system is considered new at time 0. Our aim is to compare the random variable

$N\left(t\right)$ with

$N\left(\infty \right)$, especially in terms of total variation distance. This analysis is used to prove a cut-off phenomenon in the same way as

*B. Ycart* [ESAIM, Probab. Stat. 3, 89-106 (1999;

Zbl 0932.60077)] but without the assumption of identical components.