Monday, June 11, 2007

Poisson Parking

The picture shows an aerial photograph, courtesy of the United States Geological Society, of the main visitor’s parking facility of business in Maryland on a Sunday morning in April. Look at the line of 13 parking spaces at the bottom of the picture. Notice the pattern in the oil stains leaked from the cars that park in those spaces. More oil is leaked in the spaces closest to the building.

Parking lots permit both the dynamic and static viewing of statistical processes. A time lapse film of customers entering and leaving a parking lot could allow us to estimate arrival rates, lengths of stay, or number of parked customers as time progresses. But automobiles, not being the cleanest of vehicles, leave their mark. This process is often modeled by a Poisson distribution. The actions of this process can be seen in static pictures as well.

Being Sunday no one is visiting the company, but the many previous visitors have left their marks. Notice the oil stains in the parking places. Much more oil is leaked and deposited in the places used most often. These places are the ones closest to the building. The one parking space closest to the building is a place for drivers with handicaps. Skip this space and examine the pattern starting with this first non-handicapped parking space. This first space will have oil stains when there are one or more cars in the lot. The second space will have oils stains when there are two or more cars in the lot, and so on. Thus, as the distance from the office front door to the car increases, then the amount of leaked oil decreases. This shows the steady state distribution of a multi-server queue. It takes the form of a truncated Poisson distribution, distributing Poisson probability only among the first few positive integers corresponding to the number of parking spaces.

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