Inter-arrival time of negative binomial distribution

Question

In order to simulate a process I considered two options:

1. number of items received follows a Poisson distribution λ
2. number of items received follows a Negative Binomial distribution (r,p): following Wikipedia definition

I'd like to model both scenarios, customizing the inter-arrival time of the items.

For option 1 (Poisson) the inter-arrival time follows an exponential distribution of λ.

Which distribution (and which is the relation between parameters) should I use?

Thank you very much :)

EDIT: Sorry if I didn't explain myself well. Having a Negative Binomial distribution to model the number of arrivals, I'd like to know if the inter-arrival time of arrivals follows a specific distribution. (the same way as a Poisson distribution is related to an exponential distribution, having both the same parameter λ)

Answer 1

The underlying process is called the Bernoulli process with parameter $p$ in which the inter-arrival time is integer-valued, and has a geometric distribution with parameter $p$. The number of arrivals in the time interval $(0,n]$ is a binomial random variable with parameters $(n,p)$. The time of the $n$-th arrival is a negative binomial random varianble with parameters $(n,p)$. Note that the arrival rate is $p$: on $n$ trials ($n$ very large), we expect to see roughly $np$ arrivals.