DESCRIPTION OF THE PROBLEM AND ITS SOLVING
Kella (1989) deals with the M/G/1 queue with server
vacations in which the return of server to service depends on the number of
customers present in the system where the distribution of arrival customers
is Poisson distribution and the queue system contains only one server. Hur
et al. (2003) consider an M/G/1 system with two policies, N and
T policy simultaneously to optimize the operating cost of this system. Tadj
and Choudhury (2005) deal with the three policies (N, T and D) to get the
optimal control of queueing systems. Agnihothri and Kenett
(1995) carried out. The impact of defects on a process with rework. Kennedy
et al. (2002) has done work in “An overview of recent literature
on spare parts inventories”. Buzacott and Shanthikumar
(1992) deal with.
Design of manufacturing systems using queueing models. Ayanzadeh
et al. (2009) have work done on “Determining Optimum Queue Length
in Computer Networks by Using Mimetic Algorithms”. Kharat
and Chaudhari (2005) consider “Parameter Selection Schemes for Web
Proxy Servers and their Performance Evaluation”. Zhou
et al. (2010) carried out “Performance Study of a Network Coded
Nonorthogonal User Cooperation System over Nakagamim Channels”. Wang
et al. (2010) have done work on “Characterization and Evaluation
of EndSystem Performance Aware Transport Schemes for Fast LongDistance Optical
Networks”. Wang et al. (2009) deal with
“Performance Study of Cooperative Diversity System over Nakagamim fading
Channels”.
This study, dealing with queue system which the process of arrival follows Hyper Geometric Distribution and allow to add more than one server and departure process follows Poisson distribution. In Hyper geometric distribution the probability of k defective machines in the sample of size n is given by:
where, M is the number of Machines that need maintenance, N is the number of defective and non defective machines, n is the size of the sample taken from the group size N, k is the number of defective machines in the sample, so nk is the number of non defective machines and the mathematical expression:
is called M combinatory k and it can compute by mathematical program as follow:
Similarly:
Let P is the percentage of defective machines in the sample, so:
then, the average of the number of defective machines in the sample of size
n is given by:
Let λ is the rate of the number of arrival defective machines per unit
time in the queue system that performs maintenance operations, then:
per Unit time, so during the interval time t, M = mλt, where:
Therefore, the Eq. 1 rewritten as follows:
Now, suppose that μt is the departure rate of the number of machines that
have been repaired during the interval time t and the probability distribution
of the number of departure machines follows the Poisson distribution, so the
probability of y departure machines during interval time t can be written as
follows:
In the queue system can add a new server for each machine comes to maintenance to increase the speed of service, this server is lifted in the case of nonarrival of a new machine to the system to minimize the cost, so the Eq. 2 rewritten as follows:
where, j is the number of adding servers in the queue system. Under the condition
that the number of departure machines is independent of the number of machines
coming, so during interval time t the probability distribution of the number
of defective machine in the queue system allowing servers to add or withdrawn
where (j≤n) will take the form:
On the other hand, The average number of the defective machines in the system
can be obtained from the equation:
Application with numerical results: Let N = 50, n = 6, t = 1 h, λ
= 3 defective machines per one hour, μ = 2 repaired machine per one hour.
Then: m = N/n, M = mλt = 25 defective machines.
Apply mathematic program on the above equations P(l) where l
= 1, 2,.., n and entering these data inside the equations, the equations P(l)
will take the forms in mathematic program as follow:
After the implementation of these formulas we can obtain the Fig. 1 of P(l).

Fig. 1: 
The probability distribution of the number of defective machines
in the queue system 
The value of the average of the defective machines in the system in this application
will be:
defective machines in the system during one hour. Therefore, the queue system
needs 2 servers to carry out maintenance on all the defective machines.