Consider arrivals that occur randomly but independently in time. Let the average arrival rate be equal to per unit of time and let the time interval be t. Then one would expect the number of arrivals during the interval to be = . The actual number of arrivals occurring in the interval is a random variable governed by the Poisson distribution.

The parameter of the distribution is the dimensionless quantity , which is the mean number of arrivals in the interval. We call an arrival process that gives rise to this kind of distribution a Poisson process. The distribution is very important in queuing theory.

Example: A traffic engineer is interested in the traffic intensity at a particular street corner during the 1 – 2 a.m. time period. Using a mechanical counting device, the number of vehicles passing the corner is recorded during the one hour interval for several days of the week. Although the numbers observed are highly variable, the average is 50 vehicles. The engineer wants a probability model to answer a variety of questions regarding the traffic.

To use the distribution for the example, we must assume that vehicles arrive independently and that the average arrival rate is constant. This does not mean that the vehicles pass in a steady stream with a fixed interval between cars. Rather, with the assumption of randomness, vehicle arrivals are extremely variable; the rate of 50 per hour is an average.

Using the distribution, we can model the probabilities for any interval of time, however consider a one minute period. The random variable is the number of vehicles passing during the one minute period. The parameter of the distribution is

= (50/hour) (1/60 hour) = 5/6 = 0.833.

The first six values of the probability distribution for this example are computed by the add-in with the formulas given for the Poisson distribution and shown in the table above. Although the average is nearly 1, there is over a 40% probability that none will be observed.