A Poisson process is defined by the probability of encountering $k$ events within an arbitrary time interval $\Delta t$, namely
$P\{N(\Delta t) = k\} = \frac{e^{−λ\Delta t}(λ\Delta t)^k}{k!}$
Further, it can be shown (for example, here), that the time interval between two consecutive events $t_i$ and $t_{i+1}$ follows an exponential distribution. Its probability density is given by
$\rho(t_{i+1} - t_i = T) = \lambda e^{-\lambda T}, \;\forall i$
I am interested in a slightly more complicated distribution for event interval. In my application, the interval follows the exponential distribution, except it can't be less than come constant waiting time $T_0$. Using the Heaviside step function $H(x) = \delta(x > 0)$, I define the interval time probability density as
$\rho(t_{i+1} - t_i = T) = \lambda e^{-\lambda (T-T_0)} H(T-T_0), \;\forall i$
which means it is zero for any interval $T < T_0$, and follows the exponential distribution afterwards.
Question: Calculate the probability of observing $k$ events within a time interval $\Delta t$. If it helps, you may assume that the beginning of the interval is not affected by compulsory interval due to events that may have happened before the interval.