# Thread: Multiple poisson versus one poisson for different arrival types

1. ## Multiple poisson versus one poisson for different arrival types

I am trying to model guest arrivals at a restaurant with exponentially distributed times between arrivals. Arrivals can consist of different party sizes from 1 to 10 people, each party size having its own unique rate parameter. I believe arrival behavior to be independent of party size, however, the mean is two person parties.

Here is my question -- Are these two methods mathematically equivolent?:

Method 1. Make model in which each party size has its own unique poisson arrival distribution with rate parameter unique to that party size. Then run simulation with these different distributions in parrallel and not related to each other.

Method 2: Make model in which there is one rate parameter representing the total arrivals of all party size. Then for each arrival, assign that arrival a party size based on the distribution function of party size.

The reason i think these two methods would be equivolent is that I am assuming that party size in no way corresonds to unique arrival behavior. So you can think of arrivals as a process unrelated to party size, then party size is just assigned to each arrival based on its actual frequency of occurance. Does this make sense and am i thinking about the problem the correct way. Any advice would be greatly appreciated. thanks.

2. The models are equivalent. In Method 1 you use the superposition principle of Poisson processes. The sum of all unique arrival rates will be the total arrival rate.
In Method 2 you first look at this total arrival rate and use thinning to get processes with the individual rates.

In both cases the time to next arrival is Exp(lambda), where lambda is total arrival rate. Also, the probability that the next arriving party is of size n is

(rate party size n)/lambda

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