Written by on Friday June 6th 2014 in SIMUL8 Tips, Technical Corner

## SIMUL8 Tip: Difference between Poisson and Exponential Distributions for Arrivals

### There is occasional confusion between Poisson and Exponential distributions.  Lets try to clarify: Assuming you are working in minutes (but same applies to other time units), consider what’s often called a “Poisson 5 arrival rate”.

This means “on average I expect 5 people to arrive every minute” or “I expect an average inter-arrival time of 0.2 minutes” (which is the same thing). A Poisson arrival rate means times between individual arrivals are completely random (no upper bound or lower bound, but the average must be 0.2 in this case). The only distribution that creates this is exponential (well, “negative exponential” really but everyone unfortunately shortens it to exponential!)

Exponential gives you many small gaps between arrivals and a few large gaps. You could even get whole minutes with nothing arriving and yet another minute where 15 arrive, but the average will be 5 each minute if the exponential inter-arrival time is 0.2.

• So Poisson 5 = Exponential 0.2
• Poisson 10 = Exponential 0.1
• Poisson 0.5 = Exponential 2

Etc.

#### The units for Poisson are People (or arrivals), the units for Exponential are minutes.

In SIMUL8 Start Points, it asks for minutes of inter-arrival time – so you must use Exponential not Poisson. If you enter Poisson 5 it will still run but will use 5 as the average inter-arrival time and a strange, for the purpose, shape distribution.

#### Poisson is a discrete distribution. Sample values are always integers, another reason it makes no sense to use it for inter-arrival time.

What about other distributions: I’ve never seen a good reason to use normal (or average) distributions for inter-arrival times, but no doubt someone will come up with one.  But if used it is still making sense because it is continuous and can generate meaningful inter-arrival times.  Erlang K distributions are sometimes useful when inter-arrival times are correlated because for example the arrivals tend to arrive in groups (like families arriving at a theme park). Erlang K is related to Exponential, indeed Erlang 1 is identical to Exponential.

Hope this clarifies a little, but ask questions if you need more.