Exercise 9.1. Find the correlation coefficient of the Poisson process.

Solution:

To get we need the following steps

Thus

After substitution we obtain

Exercise 9.2. Let us consider a service system at which the inter-arrival times of the customers are exponentially distributed with parameter and the service times are also exponentially distributed with parameter Supposing that the involved times are independent of each other find the distribution of the number of customers arrived during a service.

Solution:

By the theorem of total probability

If , then

which is a modified geometric distribution with parameter .

Exercise 9.3. Find the mean number of customers arrived during a service having a general distribution.

Solution:

To solve the problem let us apply the properties of the generating function and the Laplace-transform. So we get

that is

Therefore

Exercise 9.4. Let the service times be Erlang distributed random variables with parameters Similarly to the previous problem find the distribution of the number of customers arrived during a service time if the arrival process remains the same.

Solution:

that is we get the negative binomial ( Pascal ) distribution with parameters.