# Arrival Rates

Living in New York you spend a fair amount of sitting and thinking while you wait for the subway. When I’m waiting I get a lot of reading done or check twitter, but a lot of the time I just think. One natural thing to think about is of course arrival times. When will my train come? How often do trains arrive? What is the average arrival interval?

These questions make me think back to my operations class in which we learned about various processes that depend on random distribution of arrivals such as shipping and logistics, call centers, networks, receivables and deaths by horse kick… It turns out that the first person to discover this naturally occurring and fairly common distribution was measuring how many people died by horse kick in the Prussian army in a given year. Ladislaus Bortkiewicz built up on the work of Simeon Poisson who had discovered this distribution in the extreme case of a binomial distribution.

Think of a case like waiting for the train (or getting kicked in the head) as the train is either there or not there. This is like flipping a coin and getting heads or tails. Poisson showed that, in a case when the probability of the train being there approaches 0 at any given instant and the number of observations approaches infinity, there is a pattern in the distribution of train arrivals. All processes that occur across time roughly fit these constraints.

Simeon Poisson & Ladislaus Bortkiewicz

Since I’ve been playing with R shiny, I thought I’d see if I could embed a simple shiny app here that you can play with on my site. Below is an interactive histogram that displays a Poisson distribution given an average arrival time you can choose on the slider. It allows you to change the lambda value, also known as the event rate. If you know that on average you get one email every 60 minutes at work, you can see the distribution of the emails for 100 emails. Play around with it for a minute and see what happens!

At a rate of an email an hour, it looks like there’s less than a 10% chance that you’ll get an email at any given 5 minute block with the most likely time block (60 minutes) to only have a chance of about 5%. Phew, now I can relax about getting up for a cup of coffee at the office! Ok, so that’s not the most useful thing, but imagine predicting the arrival of your next amazon package, the stock order being submitted or rocket propelled grenades in a war zone. Bortkiewicz could help you manage time, make money or save your life.