Exercise 13.1. Solve the following system of equations by the help of difference equations.

Solution:

It is easy to see that it can be rewritten as

which is a 2-nd order difference equation with constant coefficient. Its general solution can be obtained in the form

where are the solutions to

It can easily be verified that and thus

However , and because , thus and .

Exercise 13.2. Find the generating function of the number of customers in the system for an queueing system by using the steady-state balance equations. Then derive the corresponding distribution.

Solution:

Starting with the set of equations

by multiplying both sides by and then adding the terms we obtain

Thus we can calculate as

Since , therefore

That is

which is exactly the generating function of a modified geometric distribution with parameter . It can be proved as follows, if

then its generating function is

Exercise 13.3. Find the generating function of the number of customers waiting in a queue for an queueing system.

Solution:

Clearly

For verification let us calculate the mean queue length, thus

Exercise 13.4. Find the Laplace-transform of and for an queueing system.

Solution:

It is easy to see that

which was expected, since follows an exponential distribution with parameter .

To get the Laplace-transform of we have

which should be

since

To show this it can be calculated that

Let us verify the result by deriving the mean values and .

thus

which was obtained earlier.

Exercise 13.5. Show that for an queueing system

Solution: It is well-known if then

Since it is enough to show that

This can be proved by the L'Hospital's rule, namely

Exercise 13.6. Show that for an queueing system the Laplace-transform

satisfies .

Solution:

Exercise 13.7. Find by the help of the Laplace-transform for an queueing system.

Solution:

Since

then

that is

which was obtained earlier.

The higher moments can be calculated, too.

Exercise 13.8. Consider a closed queueing network with nodes containing customers. Assume that at each node the service times are exponentially distributed with parameter and , respectively. Find the mean performance measures at each node.

Solution:

It is easy to see that the nodes operate the same way and they can be considered as an queueing system. Hence the performance measures can be computed by using the formulas with and , respectively.

Furthermore, one can easily verify that

where , is the utilization of the server.

Exercise 13.9. Find the generating function for an queueing system.

Solution:

To verify the formula let us calculate .

Since , therefore take the derivative, that is we get

hence

which was obtained earlier.

Exercise 13.10. Find for an queueing system.

Solution:

Since , let us calculate first . That is

Since , therefore

Exercise 13.11. Show that is a monotone decreasing sequence and its limit is .

Solution:

and thus it tends to as increasing. The sequence is monotone decreasing if

that is

which is satisfied if . Since therefore , and thus , , that is . It means that is monotone decreasing for which was expected since as the number of servers increases the probability of loss should decrease.

Exercise 13.12. Find a recursion for .

Solution:

Let , then by the help of

we should write a recursion for since can be obtained recursively. First we show how can be expressed by the help of and then substituting into the recursion

we get the desired formula. So let us express via that is

which is positive since is the stability condition for an queueing system.

This shows that

which was expected because of the nature of the problem.

Consequently

and is also valid due to the stability condition. Let us first express by the help of then substitute. To do so

Now let us substitute into here. Let us express the numerator and denominator in a simpler form, namely

Thus

and the initial value is . Thus the probability of waiting can be computed recursively. It is important because the main performance measures depends on this value.

Now, let us show that is a monotone decreasing sequence and tends to as , increases which is expected. It is not difficult to see that

and if we show that

then we have

To do so it is easy to see that

that is if then the values of the parabola are positive. However, this condition is satisfied since the stability condition is .

Furthermore, since

therefore , which was expected.

This can be proved by direct calculations, since

and from

the limit is . It is clear because there is no waiting in an infinite-server system.

Exercise 13.13. Verify that the distribution function of the response time for a queueing system in the case of reduces to the formula obtained for an system.

Solution:

Thus

Exercise 13.14. Show that for an queueing system.

Solution:

therefor the L'Hospital's rule is applied. It is easy to see that

and thus

Exercise 13.15. Show that if the residual service time in an queueing system is denoted by then its Laplace-transform can be obtained as .

Solution:

Using integration by parts we have

Verify the limit .

It is easy to see that

therefore apply the L'Hospital's rule. Thus

Exercise 13.16. By the help of prove that if , then !

Solution:

így .

Exercise 13.17. By the help of the formulas for an system derive the corresponding formulas for an system.

Solution:

In this case

therefore the Laplace-transform of the response time is

that is , as we have seen earlier.

For the generating function of the number of customers in the system we have

as we proved in the case of an system.

For the mean waiting and response times we get

To calculate the variance we need

thus

as we have seen earlier.

Furthermore

The variance of the number of customers in the system is

as we have seen earlier.

Finally

These verifications help us to see if these complicated formulas reduces to the simple ones.

Exercise 13.18. Based on the transform equation

find -t!

Solution:

It is well-known that , that is why we have to calculate the derivative at the right hand side. However, the term takes an indetermined value at hence the L'Hospital's rule is used. Let us first define a function

Hence one can see that

Applying the expansion procedure

we have

Thus and .

After these calculations we get

and hence

which was obtained in a different way.

Exercise 13.19. Find by the help of .

Solution:

Let us define a function

which is after expansion can be written as

Therefore

Hence

Consequently, because

we have

Thus

Similarly

Thus

Finally

Exercise 13.20. By using the Laplace-transform show that

Solution:

As we have seen earlier

and it is well-known that

Thus for we get

Therefore

Consequently

Exercise 13.21. Find the generating function of the number of customers arrived during a service time for an system.

Solution:

By applying the theorem of total probability we have

Hence its generating function can be written as