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 $t≥(n−1)b$, find $P\{R(t)≥n\}$.
I find some answer using memoryless property of exponential distribution, but I cannot understand, can anyone help me with this here?
Due to the memoryless property of exponential distribution, the distribution of $R(t)$ is identical to $N(t−(n−1)b)$.
I do not know how to use memoryless property of exponential distribution, since not all interarrival time of registered values are greater than b?
For example, $S_1 = s, S_2 = S_1 + \frac{b}{2}, S_3 = S_2 + \frac{2b}{3}$, then $S_1$ and $S_3$ are registered but the interarrival time between 2 and 3 are $\frac{2b}{3} < b$, so how can we use the memoryless property of exponential distribution to get the distribution of $R(t)$ is identical to $N(t−(n−1)b)$?