INTRODUCTION
Queueing systems in which arriving customers who find all servers and waiting
positions (if any) occupied may retry for service after a period of time is
called retrial queues. For detailed survey of retrial queues and bibliographical
information obtained from Falin (1990), Artalejo
(1999a, b, 2010), monograph
by Falin and Templeton (1997) and Artalejo
and GomezCorral (2008). Retrial queues with unreliable servers have been
studied by Kulkarni and Choi (1990) and Aissani
and Artalejo (1998). Because of the complexity of the retrial queueing models,
analytic results are generally difficult to obtain. There are a great number
of numerical and approximations methods available, in this study we will place
more emphasis on the solutions by direct t runcation method. The direct truncation
method was studied by Subramanian et al. (2009)
for retrial queueing system with priority services under negative arrival. Multi
server retrial queueing model is studied by Falin and Templeton
(1997) and Neuts and Rao (1990) and multiserver
retrial queueing with priority services is studied by Ayyappan
et al. (2010). This study has its own importance in real models like
reservation in Railway system, call centre etc., Many computational methods
are available to solve Retrial queueing models. In this study, Direct Truncation
Method (DTM) is used. This is one of the most reliable methods to find the steady
state probability vector. By using this steady state probability vector, the
system measures can be obtained. The main objective of this study was to study
the unreliable nature of the servers under Retrial queueing system and also
to find the system performance measures like mean number of customers in the
orbit, Mean number of busy servers etc.,
MODEL DESCRIPTION
Consider a multi server retrial queueing system with breakdown and repair of services in which customers arrive in a Poisson process with arrival rate λ. These customers are identified as primary calls. Further it is assume that the service time follows an exponential distribution with parameter μ. The breakdown of service follows an exponential distribution with parameter α and repair of service follows an exponential distribution with parameter β. Let c be the number of servers in the system. If any one of the server is free at the time of a primary call arrival, the arriving call begins to be served immediately by free server and customer leaves the system after service completion. Otherwise, if c servers are busy or in breakdown then arriving customer goes to orbit and becomes a source of repeated calls. The pool of sources of repeated calls may be viewed as a sort of queue. Every such source produces a Poisson process of repeated calls with intensity σ. If an incoming repeated call finds any one of the servers is free, it is served and leaves the system after service while the source which produced this repeated call disappears. If there is a breakdown in service for a customer (active breakdown), then the server goes to the state of breakdown and the customer with his incomplete service goes to orbit.
Most of the queueing system with repeated attempts assume that each customer in the retrial group seeks service independently of each other after a random time exponentially distributed with rate σ so that the probability of repeated attempt during the interval (t, t +Δt) given that there were n customers in orbit at time t is nσ Δt + O (Δt). This discipline of access to the server from the retrial group is called classical retrial policy. The input flow of primary calls, interval between repeated calls, times between breakdown and repair of service are mutually independent.
Let N (t) be the random variable which represents the number of customers in the orbit at time t, H (t) be the random variable which represents the number of servers in breakdown at time t and S (t) represents the number of busy servers at time t.
The random process X is described as:
H (t) 
= 
0 and S (t) = 0 means all servers are idle in the system 
H (t) 
= 
0 means no servers in breakdown 
H (t) 
= 
j means j servers are in breakdown 
S (t) 
= 
i means i servers are busy 
The possible state space is:
The infinitesimal generator matrix Q is given below:
The matrices described in the Infinitesimal generator matrix Q can be obtained from the following infinitesimal transition rates of process X as follows:
If the capacity of the origin is finite say M then:
DESCRIPTION OF COMPUTATIONAL PROCEDURES
The above infinitesimal generator matrix is Level Dependent Quasi Birth and Death Process (LDQBD). This type of Retrial queueing models can be solved computationally by one of the following techniques:
• 
Direct truncation method 
• 
Generalized truncation method 
• 
Truncation method using Level Dependent Quasi Birth and Death Process
(LDQBD) 
• 
Matrix geometric approximation 
DIRECT TRUNCATION METHOD
In this method, one can truncate the system of equations for sufficiently large value of the number of customers in the orbit, say M. That is, the orbit size is restricted to M such that any arriving customer finding the orbit full is considered lost. The value of M can be chosen so that the loss probability is small. Due to the intrinsic nature of the system, the only choice available for studying M is through algorithmic methods. While a number of approaches are available for determining the cutoff point, M, the one that seems to perform well is to increase M until the largest individual change in the elements of X for successive values is less than ε a predetermined infinitesimal value.
ANALYSIS OF STEADY STATE PROBABILITIES
Let X be a steadystate probability vector of Q partitioned as X = (x (0), x (1), x (2),...) where X satisfies:
where, x (i) = (P_{i00}, P_{i01}, P_{i02}, P_{i03},..., P_{i0c}, P_{i10}, P_{i11}, P_{i12}, P_{i13},..., P_{i1c1}, P_{i20}, P_{i21}, P_{i22}, P_{i23},..., P_{i2c2},..., P_{ic10}, P_{ic11}, P_{ic0}) for i = 0, 1, 2,...
If M denotes the cutoff point or Truncation level, then the steady state probability vector X (M) is partitioned as X (M) = (x (0), x (1), x (2),..., x (M)), where X (M) satisfies:
where, x (i) = (P_{i00}, P_{i01}, P_{i02}, P_{i03},..., P_{i0c}, P_{i10}, P_{i11}, P_{i12}, P_{i13},..., P_{i1c1}, P_{i20}, P_{i21}, P_{i22}, P_{i23},..., P_{i2c2},..., P_{ic10}, P_{ic11}, P_{ic0}) for i = 0, 1, 2,... M
If the capacity of the orbit is finite say M then Eq. 1 becomes 2. The system of Eq. 2 is solved exploiting the special structure of the coefficient matrix. It is solved by Numerical methods. Since there is no clear cut choice for M, we may start the iterative process by taking, say M = 1 and increase it until the individual elements of X do not change significantly. That is, if M* denotes the truncation point then:
where, ε is an infinitesimal quantity.
STABILITY CONDITION
The necessary and sufficient condition for the system to be stable is:
SPECIAL CASES
• 
Model becomes multi servers retrial queueing system if α→0 
• 
Model becomes multi servers classical queueing system if α→0
and σ→∞ 
• 
Model becomes single server retrial queueing system with unreliable servers
if c = 1 
• 
Model becomes single server retrial queueing system if c = 1 and α→0 
SYSTEM MEASURES
The system measures are used to bring out the qualitative behavior of the queueing model under study. Numerical study has been dealt to find the following measures. The following system measures can be study with steady state probability vectors for various values of λ, μ, σ, α, β and c.
• 
The probability mass function of number of busy servers 
• 
The probability mass function of number of servers in breakdown 
• 
The probability mass function of number of customers in the
orbit 
• 
Mean number of busy servers 
• 
Mean number of servers in breakdown 
• 
Mean number of customers in the orbit 
• 
The probability that the orbiting customer is blocked 
• 
The probability that an arriving customer enter into service
station immediately 
NUMERICAL STUDY
The stability condition is most important for every queueing system and λ,
μ, c, α and β are chosen so that they satisfy the stability condition.
System performance measures of this model have been done and expressed in the
form of tables which are shown below by finding the steady state probability
vector X for various values of λ, μ, σ, c, α and β.
If λ = 30,μ = 10, σ = 100, c = 5, α = 10 and β = 100 , then
the steady state probability vector is X = (x [0], x [1], x [2] ,…, x [M])
where:
Similarly, we can find x (n) for n≥6 and it is noticed that x (n)→0
as n→∞. For the numerical parameters chosen above, x (n)→0 for n≥12
and the sum of the steady state probabilities becomes 0.9999999999. In the same
manner, we can find the steady state probability vector X for all values λ,
μ, σ, c, α and β.
SYSTEM PERFORMANCE MEASURES
The system performance measures are most important for every queueing system.
These measures are calculated numerically by using the steady state probability
vectors x [0], x [1], x [2],... and using formulas described under system measures
and presented in Table 1 to 6.
Table 1 represents the for probability mass function of number of busy servers in the system for λ = 30, μ = 10, σ = 100, c = 5, α = 10 and β = 100. From this table we can find the Mean number of busy servers.
Mean No. of busy servers = 2.999999
Table 2 represents the for probability mass function of number of servers in breakdown for λ = 30, μ = 10, σ = 100, c = 5, α = 10 and β = 100. From this table we can find the Mean number of servers in breakdown.
Mean No. of servers in breakdown = 0.300000
Table 3 represents the for probability mass function of number of customers in the orbit λ = 30, μ = 10, σ = 100, c = 5, α = 10 and β = 100. From this table we can find the Mean number of customers in the orbit.
Table 1: 
Number of busy servers with probabilities 

Table 2: 
Number of servers in vacation with probabilities 

Table 3: 
Number of customers in the orbit with probabilities 

Table 4: 
Breakdown rate (α) and c against mean number of customers
in the orbit for λ =8, μ = 10, β = 100 and σ = 100 

c: Number of servers, α: Rate of breakdown, MNBS: Mean
number of busy servers, MNSB: Mean number of servers in breakdown 
Mean number of customers in the orbit = 1.193782
Table 4 shows the impact of α and c over Mean number of customers in the orbit. Further, the following information can be obtained:
• 
Mean number of customers in the orbit decreases as α
decreases 
• 
Mean number of customers in the orbit decreases as number of servers increases 
• 
Mean Number of Busy Servers (MNBS) is independent of α 
• 
This model becomes multi server retrial queueing system if α→0 
Table 5 shows the impact of σ and c over Mean number of customers in the orbit. Further, the following information can be obtained:
• 
Mean number of customers in the orbit decreases as retrial
rate σ increases 
• 
Mean number of customers in the orbit decreases as number of servers increases

• 
Mean Number of Busy Servers (MNBS) and Mean Number of Servers in Breakdowns
(MNSB) are independent of retrial rate σ 
• 
Model becomes Multi server classical queueing system with unreliable servers
if σ→∞ 
Table 5: 
Retrial rate (σ) and c over mean number of customers
in the orbit for λ =30, μ = 40, α = 10 and β = 100 

c: No. of servers, σ: Retrial rate, MNBS: Mean number
of busy servers, MNSB: Mean number of servers in breakdown 
Table 6: 
Repair of service rate (β) and c against mean number
of customers in the orbit for λ = 8, μ = 10, α = 10 and σ
= 100 

c: Number of servers, β: Repair rate, MNBS: Mean number
of busy servers, MNSB: Mean number of servers in breakdown 
Table 6 shows the impact of β and c over Mean number of customers in the orbit. Further, the following information can be obtained:
• 
Mean number of customers in the orbit decreases as β
increases 
• 
Mean number of customers in the orbit decreases as number of servers increases 
• 
Mean Number of Busy Servers (MNBS) is independent of β 
• 
This model becomes Multi server retrial queueing system if β is large 
CONCLUSION
It is observed from the numerical study that means number of customers in the orbit decreases as the retrial rate increases and means number of busy servers and mean number of breakdown independent of retrial rate σ. The various cases which have been discussed under special cases are particular cases of this study. This study can be further extended by introducing various concepts like second optional service, loss and feedback and vacation policies etc.