# Passengers entering a queue

1. Jul 1, 2010

### mikey2322

Hi,

I am trying to assign a distribution to the the rate at which passengers enter a queue over a period of time. The period of time is to remain constant. Passengers start arriving 4 hours before a flight and stop arriving at the scheduled time of departure.

I have been using a Weibull distribution but I find that you cannot lock down the distribution to stay within the time period easily. I am looking for a distribution that I can lock to a certain time period and where I can alter the amplitude and wave length similar to WEIBULL. Is there a distribution curve that will allow this?

Mike

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2. Jul 2, 2010

### mathman

In problems (random events within a fixed time interval) like this the Poisson distribution is usually used. The basic assumption is that the events (arrivals) are independent of each other.

3. Jul 7, 2010

### flux_factor

mathman is correct that Poisson distributions are often used to model arrivals, but I'm confused by your following comment:

"a distribution that I can lock to a certain time period".

By this comment do you mean you want arrivals to completely start/stop at certain points, but that you want arrivals to be random within that timeframe, but at a different mean rate depending on the time (for example, the mean rate of arrival of passengers for a 2pm flight is 0 per hour before 8am, 5 per hour from 8 to 10am, 9 per hour from 10 to 1pm, 3 per hour from 1 to 2pm, and 0 per hour thereafter)?

To do this, you could vary the mean/variance parameter (the lambda parameter) that defines a Poisson r.v. which represents the rate of arrivals within each timeframe, and make the probability of zero arrivals = 1 before 8am and after 2pm (for example).

The actual arrival times will follow a Gamma distribution, and the time between arrivals will follow a exponential distribution, but the rate of arrivals per hour will be Poisson.

If you don't want the mean rate and the rate's variance to be equal, you could use the Negative Binomial distribution or other distributions depending on the shape you believe the rate of arrivals follows in reality.