the distribution of registered events in Poisson process Events, occurring according to a Poisson process with rate $\lambda$, are registered by a counter. However, each time an event is registered the counter becomes inoperative for the next  $b$ units of time and does not register any new events that might occur during that interval. Let  $R(t)$ denote the number of events that occur by time $t$ and are registered.
For $t\ge (n-1)b$, find $\mathbb{P}(R(t)\ge n)$
I found a comment from another post, saying that the distribution of $R(t)$ is the identical to $N(t-(n-1)b)$. I don't see the equivalence between these two distributions. Any help would be appreciated!
 A: By the strong Markov property and time homogeneity of $N$ (alternatively, independent and identically distributed interarrival times$^\star$), the jump times $(T_n)_{n\in\mathbb{N}}$ of $R$ are given by
\begin{align*}
T_1 &= S_1&& \\
T_2 &= T_1 + b + S_2&& \\
T_3 &= T_2 + b + S_3&& \\
&\,\,\,\vdots&&
\end{align*}
where $(S_n)_{n\in\mathbb{N}}$ are independent and identically distributed with $S_1 \sim \text{Exp}(\lambda)$.
Note that $T_n = S_1 + \sum_{i=2}^n (S_i + b)$ for $n\geq2$. Now, with $n\in\mathbb{N}$ and $t\geq0$,
\begin{align*}
\mathbb{P}(R(t) \geq n)
&=
\mathbb{P}\!\left(T_n \leq t\right) \\
&=
\mathbb{P}\!\left(\sum_{i=1}^n S_i \leq t - (n-1)b\right)\!.
\end{align*}
Let $(\tau_n)_{n\in\mathbb{N}}$ denote the jump times of $N$. Since $(S_n)_{n\in\mathbb{N}}$ follow the same distribution as the interarrival times $\tau_{n} - \tau_{n-1}$ of $N$ (under the convention $\tau_0 := 0$), we find that
\begin{align*}
\mathbb{P}\!\left(\sum_{i=1}^n S_i \leq t - (n-1)b\right)
=
\mathbb{P}(\tau_n \leq t - (n-1)b)
=
\mathbb{P}(N(t-(n-1)b)\geq n)
\end{align*}
for $t\geq (n-1)b$. Collecting results, we find that
\begin{align*}
\mathbb{P}(R(t) \geq n) = \mathbb{P}(N(t-(n-1)b) \geq n)
\end{align*}
for $n\in\mathbb{N}$ and $t\geq (n-1)b$.
$^\star$Let me know if additional details or explanations are needed; heuristically, you may think of the process starting anew at times $T_n + b$.
