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Elevator Stochastic Problem

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Elevator Stochastic Problem

Hi I need help with this problem:

The number of people entering an elevator on the first floor of the building is a Poisson random variable with mean lambda.
If the building has N floors, excluding the first floor, and the probability that one person get down on the floor x was the same for each floor, determine the expected value of stops where the elevator is empty. The lift takes people only on the first floor.

I came here for the same hahahaha :)

Hi guys,

Here is a possible way to attack the problem.
Let n be the number of persons in the elevator (Poisson probability P(n)). Each one choose a number between 1 and N, the number of floors. Let Q(n, m) be the probability to have m different numbers (number of stops) from the n choices. The number n of persons may be greater thant the number N of floors. So, let u be the minimum of the two numbers n and N:

u = min(n, N)

Then the mean value M of of m is:

M = sum{m=1..u}{n=1..infnty} [mP(n)Q(n,m)]

Since P(n) is known,you need only Q(n,m).

In order to have m different numbers, there are two possibilities: the n-1 first persons have already choosen m different numbers and the last one must chose one of these numbers, or, the n-1 first persons have choosen m-1 different numbers and the last one must choose a new number. So:

Q(n, m) = (m/N)Q(n-1, m) + [(n-m+1)/N]Q(n-1, m-1)

Now you have a recursive linear equation. Characteristic functions technique is the obvious method to solve it. Since you are only asked for the mean value, you get it by differentiating the characteristic function.

I hope that this helps.


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