Poisson Process, Exponential inter-arrival times simulation conundrum

Question

I am trying to simulate a poisson process by using the fact that the inter-arrival times are distributed as an exponential distribution.

I want to generate patient arrival times in (say) 1 hour. So, my basic loop in R notation is

while (nextArrival<60) 

nextArrival <-prevArrival +rexp(1,0.5)
prevArrival <- nextArrival

where 0.5 is the rate parameter of exponential distribution and I am initializing all the variables correctly.

So, I was expecting that the count of patients arrived in 1 hour would be a Poisson with parameter \lambda*t where t=60. But it turns out that this isn't so, as the mean of my simulated counts is no where close to \lambda*t.

For example, for 5 simulations, the counts were 10 7 8 7 7 which are no where close to 0.5*60. Am I missing something? Thanks in advance!

Answer 1

There does not seem to be anything wrong with your interpretation. I took your code and run it in R:

nextArrival <- 0
prevArrival <- 0
k <- 0
while (nextArrival<60) {
  nextArrival <-prevArrival +rexp(1,0.5)
  prevArrival <- nextArrival  
  k <- k+1
}
print(k);

And obtain counts that look like Poisson variable. You may want to double check your implementation.