The Poisson process gives you a way to find probabilities for random points in time for a process. A process could be almost anything:
- Accidents at an interchange.
- File requests on a server.
- Customers arriving at a store.
- Battery failure and replacement.
The Poisson process can tell you when one of these random points in time will likely occur. For example, when customers will arrive at a store, or when a battery might need to be replaced. It’s basically a counting process; it counts the number of times an event has occurred since a given point in time, like 1210 customers since 1 p.m., or 543 files since noon. An assumption for the process is that it is only used for independent events.
Poisson Process Example
Let’s say a store is interested in the number of customers per hour. Arrivals per hour has a Poisson 120 arrival rate. This means that 120 customers arrive per hour. This could also be worded as “The expected mean inter-arrival time is 0.5 minutes”, which means that a customer can be expected every 30 seconds.
The negative exponential distribution models this process, so we could write:
Poisson 120 = Exponential 0.5
So the units for the Poisson process are people and the units for the exponential are minutes.
The 120 customers per hour could also be broken down into categories, like men (probability of 0.5), women (probability of 0.3) and children (probability of 0.2). You could calculate the time interval between the arrivals times of women, men, or children.
For example, the time between arrivals of women would be:
A rate 0.3*120 = 36 per hour, or one every 100 seconds.
For men it would be:
A rate of 0.5*120 = 60 per hour, or one every minute.
And for children:
A rate of 0.2*120 = 24 per hour, or one every 150 seconds.
Next: The Poisson Distribution
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