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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?

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    $\begingroup$ For a Poisson process, the times between events follow an exponential distribution. If you pick a time window over which to look at it, the count of events, $n$, follows a Poisson distribution, & given $n$ the time of events follows a uniform distribution. $\endgroup$
    – Scortchi ♦
    Commented Jun 29, 2017 at 16:30

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@scortchi has the right answer. To summarize:

  • The arrival time stamps are uniformly distributed.
  • The inter-arrival times are exponentially distributed.
  • The count of arrivals per uniform time period is Poisson distributed.
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