This system is the oldest and thus the most famous system in queueing theory. The origin of the traffic theory or congestion theory started by the investigation of this system and Erlang was the first who obtained his well-reputed formulas, see for example Erlang [ 21 ],[ 22 ].

By assumptions customers arrive according to a Poisson process and the service times are exponentially distributed. However, if servers all busy when a new customer arrives it will be lost because the system is full. The most important question is what proportion of the customers is lost.

The process is said to be in state if servers are busy, which is the same as customers are in the system. It is easy to see that is a birth-death process with rates

Clearly the steady-state distribution exists since the process has a finite state space. The stationary distribution can be obtained as

Due to the normalizing condition we have

and thus the distribution is

The most important measure of the system is

which was introduced by Erlang and it is referred to as Erlang's B-formula, or loss formula and generally denoted by .

By using the Bayes's rule it is easy to see that is the probability that an arriving customer is lost. For moderate the probability can easily be computed. For large and small , and thus

that is the Poisson distribution. For large and large

However, in this case the central limit theorem can be used, since the denominator is the sum of the first terms of a Poisson distribution with mean . Thus by the central limit theorem this Poisson distribution can be approximated by a normal law with mean and variance that is

where

and

Another way to calculate is to find a recursion. This can be obtained as follows

Using as an initial value the probabilities can be computed for any . It is important since the direct calculation can cause a problem due to the value of the factorial.

For example for the exact formula cannot be computed but the approximation and the recursion gives the value .

Due to the great importance of in practical problems so-called calculators have been developed which can be found at

To compare the approximations and the exact values we also have developed our own Java script which can be used at

Now determine the main performance measures of this system

1. Mean number of customers in the systems, mean number of busy servers

thus the mean number of requests for a given server is

2. Utilization of a server

As we have seen

This case

3. The mean idle period for a given server

By applying the well-known relation

where is the mean idle time of the server.Thus

hence

4. The mean busy period of the system

Clearly

thus

Example 11.4. In busy parking lot cars arrive according to a Poisson process one in seconds and stay there in the average of minutes. How many parking places are required if the probability of a loss is no to exceed 1%?

Solution:

Following a normal approximation

Thus

It is not difficult to verify by using the Table for the standard normal distribution that .

Thus the approximation value of is ,

and the exact value is

Example 11.5. A telephone exchange consists of lines and calls arrive according to a Poisson process, the mean interarrival time is minutes. The mean service time is minutes.

Find the main performance measures.

Solution:

Using Poisson approximation where

, event for

. This means that a call is almost never lost.

Mean number of busy lines can be obtain as

The utilization of a line is

The utilization of the system is

.

The mean busy period of the system can be obtained as

Mean idle period of a line is

Heterogeneous Servers

In the case of an ystem the service time distribution depends on the index of the server. That is the service time is exponentially distributed with parameter for server . An arriving customer choose randomly among the idle servers, that is each idle server is chosen with the same probability. Since the servers are heterogeneous it is not enough to to the number of busy servers but we have to identify them by their index. It means that we have to deal with general Markov-processes.

Let denote the indexes of the busy servers, which are the combinations of objects taken at a time without replacement. Thus the state space of the Markov-chain is the set of these combinations, that is .

Let us denote by

the steady-state distribution of the chain which exists since the chain has a finite state space and it is irreducible. The set of steady-state balance equations can be written as

where denotes the ordered set , and are not defined. Despite of the large number of unknowns, which is , the solution is quite simple, namely

where , , , which can be determined by the help of the normalizing condition

Let us check the first equation . By substitution we have

Lets us check now the third equation

Finally let us check the most complicated one, the second set of equations , namely

which shows the equality.

Thus the usual performance measures can be obtained as

1. the utilization of the th server can be calculated as

and thus

where is the mean idle period of the th server. Hence

2.

3. The probability of loss is .

It should be noted that in this case the following relation also holds .

In homogeneous case, that is when , after substitution we have

that is it reduces to the Erlang's formula derived earlier.

It should be noted that these formulas remains valid under generally distributed service times with finite means with . In other words the Erlang's loss formula is robust to the distribution of the service time, it does not depend on the distribution itself but only on its mean.