Arrival,wait and service distributions

[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

 

[EnumaElish]

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.

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).

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?

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

 

[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.

 

[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."

 

[wmac]

Thank you for your help.

My problem was that the notations and formulas being used in programming libraries for example are different from what is mentioned in books.

Anyway, with your help I was able to use cern.colt library to generate Poisson randoms (and also exponential and other distributions). I investigated the data generated for wait, arrival, service times and they make sense. (though I was forced to put a max and min limit on exponential distribution because some types of waiting for example eating lunch, have a min and max boundary).

Thank you again for your time.