Simulating Poisson Waiting Times

Question

I need to simulate Poisson wait times. I\'ve found many examples of simulating the number of arrivals, but I need to simulate the wait time for one arrival, given an averag

Answer 1

If you want to simulate earthquakes, or lightning or critters appearing on a screen, the usual method is to assume a Poisson Distribution with an average arrival rate λ.

The easier thing to do is to simulate inter-arrivals:

With a Poisson distribution, the arrivals get more likely as time passes. It corresponds to the cumulative distribution for that probability density function. The expected value of a Poisson-distributed random variable is equal to λ and so is its variance. The simplest way is to 'sample' the cumulative distribution which has an exponential form (e)^-λt which gives t = -ln(U)/λ. You choose a uniform random number U and plug in the formula to get the time that should pass before the next event. Unfortunately, because U usually belongs to [0,1[ that could cause issues with the log, so it's easier to avoid it by using t= -ln(1-U)/λ.

Sample code can be found at the link below.

https://stackoverflow.com/a/5615564/1650437

Answer 2

Time between arrivals is an exponential distribution, and you can generate a random variable X~exp(lambda) with the formula:

-ln(U)/lambda` (where U~Uniform[0,1]). 

More info on generating exponential variable.

Note that time between arrival also matches time until first arrival, because exponential distribution is memoryless.