The parameter of arrival-type processes is known as the “arrival rate”.
A Bernoulli process with parameter $p$ is a stochastic process associated with a sequence $\{X_n\}$ where each $X_n$ is iid from $\text{Bernoulli}(p)$ and indicates an arrival at time $n$.
$$ X_1, X_2, \dots \overset{\text{iid}}{\sim}\text{Bernoulli(p)} $$
Moreover, we can equivalently represent a Bernoulli process with a sequence $\{T_n\}$ where each $T_n$ is iid from $\text{Geometric}(p)$ and represents the time between the $n$th and $n-1$th arrival.
$$ T_1, T_2, \dots \overset{\text{iid}}{\sim}\text{Geometric(p)} $$
The future of a Bernoulli process is independent of the past. This is because at any timestep $n$, the sequence of random variables starting time $n+1$ is an identical and independent Bernoulli process.
The number of arrivals $N_n$ in $n$ timesteps is the sum of $n$ iid Bernoulli variables, which is Binomial.
$$ N_n \sim \text{Bin}(n, p) $$
Define $\{Y_n\}$ to be the sequence of arrival times. Then define $\{T_n\}$ as the sequence of interarrival times, i.e. the gaps between arrivals.
$$ T_n := Y_{n} - Y_{n-1} \hspace{15pt} T_1 = Y_1 $$
By the memorylessness property, the process can be “restarted” at each arrival, meaning the time between the $k$th and $k+1$th arrival is identical to the time until the first arrival, should we choose to restart the process at the $k$th arrival. For this reason, we have the following:
$$ T_1, T_2, \dots \overset{\text{iid}}{\sim} \text{Geometric}(p) $$