I am building a simulation. I want to generate arrivals according to a negative binomial process. The data will show the minute of each "customer" arrival and look something like the following:
Arrival Time 1 1/1/2015 0:12 2 1/1/2015 0:56 3 1/1/2015 1:27 ... ... 83465 1/10/2015 22:12
I have historical count data that I am modeling in hopes of estimate the parameters of the negative binomial distribution. It looks like this:
Hour Arrivals 10/1/2014 0:00 2 10/1/2014 1:00 3 10/1/2014 2:00 1 ... ... 10/20/2014 22:00 1
I know that if I was using Poisson regression, I would have an estimate for a parameter $λ$ returned by the regression. This parameter would represent the expected number of arrivals during a given period (in my case an hour).
I could then simulate this process by from time $t=0$ by sampling from an exponential distribution with mean $μ=1/λ$. This would return the an interarrival time $t1$. I could then schedule an arrival to occur at $t0+t1$. I could repeat this process for the next arrival which would occur at $t0+t1 + t2$ and so on...
What is the analogous process for a negative binomial process? NB regression returns an estimate for the mean but also the dispersion $θ$.