### Modeling the Arrival Rate of Cars to a Parking Garage

I am interested in finding out more about Poisson processes, like the arrival of cars at a toll booth. I do not have much experience with discrete event systems, so this is a learning experience for me.

There is a parking garage near an office building with an electronic sign outside it that displays how many spaces are available. I went there at 6:00 on a Friday and saw there were 635 empty spaces.

I recorded the number of space available every 5 minutes until 9:40. This does not give an exact count of the number of cars arriving, since some cars arrive, drop someone off and then leave, and similar other exceptions. But, the parking garage and building are at the end of a dead-end road, so there is little other traffic at that time of day except people going to that garage.

The graph below shows the observed number of spaces available over time, as well as a piecewise function that attempts to model the net arrival rate of cars to the parking garage.

Taking the difference in number of spaces available at each 5 minute interval gives the net arrival rate of cars. You can see this in the graph below.

I have attempted to represent the arrival rate using the Erlang distribution, which is related to the expected time between events.

The specific parameter values I used were k=79 and lambda=9. I then multiplied this entire distribution by a factor of 70 to fit it to the net total number of cars that arrived.

This does not seem to work well after 9:00 am, so after that point I have simply modeled the arrival rate with a downward sloping line with a slope of -1, since I did not have enough data to make anything meaningful from this time interval.

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