Consider $n$ switches, which activate at rate $a_i$, $1\leq i\leq n$. The time it takes for a specific switch to activate, $T_i$, satisfies $T_i\sim \text{Exp}(a_i)$. In particular, $E[T_i]=1/a_i$.
Now, imagine there are $m$ activating factors (available at any time for the $n$ switches) so that each switch only activates once it is bound to exactly $1$ activating factor (say, at a rate $b$). Once a switch is activated, the activating factor is recycled at rate $r$.
For example, if $b$ is high and $r$ is low, we expect early activation of $m$ switches (if $m<n$), followed by a period of no activation.
What distribution does $T_i$ now follows? Is it possible to work out its expectation?
Edit: Following a comment, I would just add that both binding and recycling times are exponentially distributed with the given rates, and the switch an activating factor binds to is chosen uniformly at random among those not yet activated.