# Arrival,wait and service distributions

1. Jan 18, 2008

### wmac

[SOLVED] Arrival,wait and service distributions

Hello everyone,

I am doing simulation of crowd movements and behaviors. I have developed a very flexible simulator platform (it has taken a year) which can simulate almost 100,000 pedestrians in real time. I wanted to check statistical distributions I use in my simulation.

1- What is the distribution of the people entering a building in a second? (I appreciate if someone can give me the mathematical form). During last week I have seen tens of different distributions referring to Poisson distribution.

I need a function which I give the "average number of people entering in a second" and then I can generate number of people for each second in my program.

2- For some activities people wait in a place for some time (for example wait behind a shop window to look at goods). Is this similar to service time? What distribution we use for this one? (again I tried normal distribution but it sometimes produces negative and sometimes very big values while for example people stand behind a window something between 30s and 60s)

3- What about service time? (for example in the counter of a food shop waiting to get your food) Should I use Poisson distribution? What form of it?

4- The problem after this step is that I have found a Java random number toolkit (colt from cern lab) but it has its own definitions of distributions again!!!

Thank you very much for your time and wish you a good weekend.

Regards,
Mac

2. Jan 18, 2008

### EnumaElish

http://en.wikipedia.org/wiki/Poisson_distribution

Quote: "Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event." So it requires the average (expected) rate as an input (parameter).

http://en.wikipedia.org/wiki/Queueing_theory

3. Jan 18, 2008

### wmac

I have seen both pages (and a few more during last few weeks). I have also looked at a few books (including Jery Banks and Law's simulation books). As I told each of those pages contains several different forms of each distribution and that's my problem.

Thank you for your time anyway.

4. Jan 18, 2008

### EnumaElish

AFA I can see, the Wiki page for Poisson Dist. has 2 formulas for the Poisson density (frequency):
1. f(k, λ)
2. f(k, λt)

The function f is exactly the same in both instances; it is only the second argument that is applied to the function is different. The second argument of f is the mean (= variance) parameter, which is scalable. As the article points out: "λ is a positive real number, equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average every 4 minutes, and you are interested in the number of events occurring in a 10 minute interval, you would use as model a Poisson distribution with λ = 10/4 = 2.5."

5. Jan 20, 2008