I'm modeling a factory with orders arriving independently of each other. I need a series of actual time stamps of arrivals. I've been told that everyone uses the Poisson distribution to model arrivals. Fine. But what I can't understand is how getting Poisson time stamps differs from getting time stamps with a uniform distribution.
Right now I plan all the arrivals before the simulation starts by choosing uniformly distributed random numbers over the total time span. I sort this list and those are my arrival times. I measure the inter-arrival times and I find that they have an exponential distribution. But I am told that inter-arrival times of Poisson arrival times have an exponential distribution, too.
So what's up? If they have the same distribution of inter-arrival times are Poisson and uniform two names for the same distribution? Is this a discrete vs. continuous distinction?
What is the property of a set of arrival times chosen with Poisson that differs from a set chosen with uniform? Are there fewer small gaps? Fewer big gaps? Something I'm not thinking of?