It is a variation of the classical queue assuming that the service is provided by servers operating independently of each other. This modification is natural since if the mean arrival rate is greater than the service rate the system will not be stable, that is why the number of servers should be increased. However, in this situation we have parallel services and we are interested in the distribution of first service completion.

That is why we need the following observation.

Let be exponentially distributed random variables with parameter , () and denote by their minimum. It is not difficult to see that is also exponentially distributed with since

Similarly to the earlier investigations, it can easily be verified that the number of customers in the system is a birth-death process with the following transition probabilities

where

It is understandable that the stability condition is .

To obtain the distribution we have to distinguish two cases according to as depends on . Thus if , then we get

Similarly, if , then we have

In summary

where

This is exactly the utilization of a given server. Furthermore

and thus

Since the arrivals follow a Poisson law the the distribution of the system at arrival instants equals to the distribution at random moments, hence the probability that an arriving customer has to wait is

that is it can be written as

This probability is frequently used in different practical problems, for example in telephone systems, call centers, just to mention some of them. It is also a very famous formula which is referred to as Erlang's C formula, or Erlang's delay formula and it is denoted by .

The main performance measures of the systems can be obtained as follows

1. For the mean queue length we have

2. For the mean number of busy servers we obtain

3. For the mean number of customers in the system we get

which is understandable since a customer is either in the queue or in service. Let us denote by -gal the mean number of idle servers. Then it is easy to see that

thus

hence

4. Distibution of the waiting time

An arriving customer has to wait if at his arrival the number of customers in the system is at least . In this case the time while a customer is serviced is exponentially distributed with parameter consequently if there customers in the system the waiting time is Erlang distributed with parameters , . By applying the theorem of total probability for the density function of the waiting time we have

Substituting the distribution we get

Hence for the complement of the distribution function we obtain

Therefore the distribution function can be written as

Consequently the mean waiting time can be calculated as

5. Distribution of the response time

The service immediately starts if at arrival the number of customer in the system is than . However, if the arriving customer has to wait then the response time is the sum of this waiting and service times. By applying the law of total probability for the density function of the response time we get

As we have proved

Thus

Therefore

Consequently for the complement of the distribution function of the response time we have

Thus the distribution function can be written as

In addition for the mean response time we obtain

as it was expected.

In stationary case the mean number of arriving customer should be equal to the mean number of departing customers, so the mean number of customer in the system is equal to the mumber of customers arrived during a mean response time. That is

in addition

These are the Little's formulas, that can be proved by simple calculations. As we have seen

Since

thus

that is

because .

Furthermore

since

6. Overall utilization of the servers can be obtained as

The utilization of a single server is

Hence the overall utilization can be written as

7. The mean busy period of the system can be computed as

The system is said to be idle if the is no customer in the system, otherwise the system is busy. Let denote the mean busy period of the system. Then the utilization of the system is

thus

If the individual servers are considered then we assume that a given server becomes busy earlier if it became idle earlier. Hence if customers are in the system then the number of idle servers is .

Let as consider a given server. On the condition that at the instant when it became idle the number of customers in the system was its mean idle time is

The probability of this situation is

Then applying the law of total expectations for its mean idle period we have

where denotes the probability that an arriving customer find an idle server.

Since

thus

where denotes it busy period.

Hence

In this case of it reduces to

thus

which was obtained earlier.

In the following we are going to show what is the connection between these two famous Erlang's formulas. Namely, first we prove how the delay formula can be expressed by the help of loss formula, that is

As we have seen in the previous investigations the delay probability , plays an important role in determining the main performance measures. Notice that the above formula can be rewritten as

moreover it can be proved that there exists a recursion for it, namely

starting with the value .

If the quality of service parameter is then it is easy to see that there exists an , for which . This can easily be calculated by a computer using the above recursion.

Let us show another method for calculating this value. As we have seen earlier the probability of loss can be approximated as

Let , thus . Hence

That is if we would like to find such an for which , then we have to solve the following equation

which can be rewritten as

If is given then .

It should be noted that the search for is independent of the value of and thus it can be calculated for various values of .

For example, if ,

then the corresponding are .

The formula is called as square-root staffing rule. As we can see in the following Table it gives a very good approximation, see Tijms [ 91 ]

Figure 11.1. Exact and approximated values of

Let us see an example for illustration.

Let us consider two service centers which operate separately. Then using this rule overall we have to use servers. However, if we have a joint queue to get the same service level we should use servers. The reduction is

that is the reason that the joint queue is used in practice.

is of great importance in practical problems hence so-called calculators have been developed and can be used at the link

Example 11.6. Consider a service center with servers where , .

Find the performance measures of the system.

Solution:

Example 11.7. Find the number of runways in an airport such a way the the probability of waiting of an airplane should not exceed 0.1. The arrivals are supposed to be Poisson distribuetd with rate per hour and the service times are exponentially distributed with a mean of minutes.

Solution:

First use the same time unit for the rates, let us compute in hours. Hence and for stability we need which results .

Denote by the probability of waiting for runways. By applying the corresponding formulas we get

Hence the solution is . In this case and

Example 11.8. Consider a fast food shop where to the customers arrive according to a Poisson law one customer in 6 seconds on the average. The service time is exponentially distributed with seconds mean. Assuming that the maintenance cost of a server is Hungarian Forint and the waiting cost is the same find the optimal value of the server which minimizes the mean cost per hour.

Solution:

Computing for the values we have found that the minimum is achieved at . This case the performance measures are