This queue is a variation of a multiserver system and only maximum customers are allowed to stay in the system. As earlier the number of customers in the system is a birth-deat process with appropriate rates and for the steady-state distribution we have

From the normalizing condition for we have

To simplify this expression let .

Then

Thus

The main performance measures can be obtained as follows

1. Mean queue length

which results

In particular, if then the L'Hopital's rule should be applied twice.

2. Mean number of customers in the system

It is easy to see that

and since

we get

.

3. Mean response and waiting times

The mean times can be obtained by applying the Little's law, that is

In the case of an system these formulas are simplified to

3. Distribution at the arrival instants

By applying the Bayes's rule we have

Obviously in the case of an system since tends to .

4. Distribution of the waiting time

As in the previous parts for the theorem of total probability is applied resulting

Since

applying substitutions , we have

thus

The Laplace-transform of the waiting and response times can be derived similarly, by using the law of total Laplace-transforms.