So far we have been dealing with such queueing systems where arrivals followed a Poisson process, that is the source of customers is infinite. In this chapter we are focusing on the finite-source population models. They are also very important from practical point of view since in many situation the source is finite. Let us investigate the example of the so-called machine interference problem treated by many experts.
Let us consider 
machines that operates independently of each other.
The operation times and service times are supposed to be independent
random variables with given distribution function. After failure the
broken machines are repaired by a single or multiple repairmen according
to a certain discipline. Having been repaired the machine starts
operating again and the whole process is repeated.
This simple model has many applications in various fields, for example in manufacturing, computer science, reliability theory, management science, just to mention some of them. For a detailed references on the finite-source models and their application the interested reader is recommended to visit the following link
Queue, Engset-Loss SystemAs we can see depending on the system capacity r in an
a customer may find the system full. Despite of
the infinite-source model where the customer is lost, in the
finite-source model this request returns to the source and stay there
for a exponentially distributed time. Since all the random variables
are supposed to be exponentially distributed the number of customers
in the system is a birth-death process with the following rates
hence the distribution can be obtained as
which is called a truncated binomial or Engset distribution.
This is the distribution of a finite-source loss or Engset system.
Specially, if 
that is no loss and each customer has its own
server the distribution has a very nice form, namely
that is we have a binomial distribution with success parameter
, where 
is the probability that a given request is in the
system.
It is easy to see that this distribution remains valid even for
a 
system since
where 
, and 
denotes the mean time a customer spends in the
source.
As before it is easy to see that the performance measures are as follows
1. Mean number of customers in the system
2. Mean number of customers in the source
3. Utilization of a source 
thus
This help us to calculate the mean number of retrials of a customer from the source to enter to the system. That it we have
hence the mean number of rejection is 
.
The blocking probability, that is the probability that a customer find the system full at his arrival, by the help of the Bayes's theorem can be calculated as
This can easily be verified by
Let 
denote the blocking probability, that is
, which is called Engset's
loss formula.
In the following we show a recursion for this formula, namely
The initial value is
It is clear that
where
which can be seen formally, too. Moreover, as 
the well-known recursion for 
is obtained which also justifies the correctness of
the recursion for 
.
In particular, if 
then it is easy to see that 
and thus
which was expected.
In general case
Let us consider the distribution of the system at the instant when an arriving customer enters into the system.
By using the Bayes's law we have
which Little's formula for the finite-source loss system.
Queue 
Queue