) / 1 Retrial Queueing System with Server Breakdown, Delayed Repair and Reserved Time

(1)

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 3, Issue 8, August 2013)

587

M / (

G1

,

G2

, …,

Gk

) / 1 Retrial Queueing System with Server

Breakdown, Delayed Repair and Reserved Time

J. Ebenesar Anna Bagyam1, Dr. K. Udaya Chandrika2

1

Assistant Professor, Department of Mathematics, SNS College of Technology, Coimbatore. 2

Professor, Department of Mathematics, Avinashilingam Deemed University For Women, Coimbatore.

Abstract--A single server retrial queue with server breakdown is considered. Primary customers get into the system according to Poisson process. The server provides k stages of heterogeneous service in succession. If the server is free, an arriving customer receives first stage service immediately. Otherwise, he enters a retrial orbit. After first stage service completion, the customer may opt for second stage service or depart the system. After the completion of second stage service, the customer may opt for third stage service or depart and so on up to k stages. The server is subject to breakdown while it is working. The repair is not immediate and it starts after a random amount of time. Upon failure of the server, interrupted customer either remains in the service position until the server is up or enters retrial orbit. After repair the server waits for the interrupted customer to provide the remaining service. This waiting time of the server is referred as reserved time. The probability generating function is employed to obtain the joint distributions of the server state and queue length. Expected system size, orbit size and failure frequency of the server are derived. Stochastic decomposition law is verified and special cases are derived.

Keywords- Retrial, Multi Phase, Breakdown, Delayed Repair and Stochastic Decomposition

I. INTRODUCTION

System with unreliable server is worth investigating from the queueing theory as well as reliability view point, since the performance may be heavily affected by the breakdowns. Queueing system with random breakdowns have been studied by numerous researchers including Choudhury and ChuanKe [4], Ebenesar Anna Bagyam and Udaya Chandrika [5], Nathan and Jeffery [6] and Rehab et. al [7].

Considerable attention has been devoted to the queueig system with two or more stages of heterogeneous service. Shakar and Badamchizadeh [9] have studied a single server general service queue with k stages of service and vacation. Salehirad and Badamchizadeh [8] have analyzed k stages of queueing system with feedback.

The retrial queueing system has been studied extensively due to its wide applicability in telephone switching system, telecommunication and computer networks.

These systems are characterized by the feature that arrivals who find the server busy leave the service area immediately or join the retrial queue (orbit) to try again their requests. Recent works on retrial queue includes Aissani [1], Arivudainambi and Godhandaraman [2] and Artalejo and Li [3].

In this paper multi stage unreliable queueing model is analyzed by including the repeated attempts of orbital customers, delay time and reserved time. Many queueing models can be deduced as a particular case of this model.

II. MODEL DESCRIPTION

Assume that the customers arrive at the system in accordance with a Poisson process with rate λ. If an arriving customer finds the server idle, the customer enters immediately for first stage service. If the server is found to be blocked the arriving customer enters a retrial group. The retrial time is generally distributed with distribution function A(x), Laplace Stieltjes transform A*(s) and conditional completion rate η(x) = d A(x) /(1−A(x)).

The server provides k stages of heterogeneous service in succession. The first stage service is followed by second stage, second stage service is followed by third stage and so on up to k stages. The service discipline is assumed to be first come first served. Service time of ith stage is denoted by the random variable Bi having distribution function

) x (

B_i , Laplace Stieltjes transform B (s), first two _i moments i1 and i2 and conditional completion rate

) x ( i

 = dB_i(x)/(1−B_i(x)), i = 1, 2, . . . k.

After completion of ith stage service the customer may move to (i+1)th stage with probability1q_i or depart the

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588 It is assumed that the server is subject to breakdown when it is busy. The life time of the server in ith stage is exponentially distributed with rate _i ( i = 1, 2, …, k). Once the system fails, the repair does not start immediately.

There is some delay to start the repair. The delay time distribution is general with distribution function D_i(x),

conditional completion rate _i(x)= d D_i(x)/(1 −D_i(x)) and nth moment vin, i = 1, 2, …, k. The repair time is also

generally distributed with distribution functionH_i(x),

conditional completion rate _i(x)= d H_i(x)/(1 −H_i(x)) and nth moment hin, i = 1, 2, …, k.

Upon failure of the server, the interrupted customer in the ith stage ( i = 1, 2, …, k) either remains in the service position with probability p_ior enters retrial orbit with

probability 1p_i and keeps returning at times

exponentially distributed with mean 1/ _iuntil the server is up. If the interrupted customer enters the retrial orbit, the server after repair must wait for the same customer to return for completing service. This waiting time of the server is referred as reserved time.

The server is not allowed to accept new customer until the interrupted customer leaves the system.

III. JOINT DISTRIBUTION OF THE SERVER STATE The state of the system at time t can be described by the Markov Processes



N

(

t

)

:

t



0 

=

}

k

,...,

2 ,

1 i

,

0 t

);

t

(

),

t

(

),

t

(

),

t

(

),

t

(

),

t

(

X

),

t

(

*

J

),

t

(

J

{

where J(t) denotes the server state 0, Bi, Dei, Rei or Rsi

according as the server is idle, busy in ith stage, waiting for repair in ith stage, under repair in ith stage or in reserved in ith stage respectively. J*(t)denotes the interrupted customers position 0 or 1 according as the customer is in service position or in retrial queue. X(t) denotes the number of customer in the retrial queue at time t. If J(t) = 0 and X(t) > 0 then ₀(t)represents the elapsed retrial time. If J(t) =

Bi, Dei, Rei or Rsi ,



_i

(

t

)

represents the elapsed service time

in ith stage (i=1, 2, …, k). If J(t) = Dei, 2_i(t)represents the

elapsed delayed time in ith stage, if J(t) = Rei, 3_i(t)

represents the elapsed repair time in ith stage and if J(t) =

Rsi, i4(t)represents the elapsed reserved time in ith stage.

Define the probabilities,

)

t

(

I

₀ = P{J(t) = 0, X(t) = 0}

)

x

,

t

k

1 i

1 i iQ

q i _j₁[1 _j(v_j₁ h_j₁ (1 p_j)/ _j)] 1

j

        

 <

) ( *

A 

(3)

589 Proof:

Let S be the generalized service time of a customer in _i

ith stage service. Then {S } are independently and _i identically distributed with Laplace transform

                    

 F (s)D (s)

s s p s

B _i* *_i

i i i i i * i

and expected value

E(S )=_i

μ

_i₁

                  

 _i₁ _i₁

i i

i v h

p 1

1 , i =1, 2, …,k

The expected service time of a customer is

E(S) = 

 

k

1 j j j 1

Q q   j 1 i i ) S ( E

Let P(S) and P(I) denote respectively the probabilities that the system is blocked and idle.

Let E(I) be the expected idle time then

P(S) =

) I ( E ) S ( E ) S ( E

 and P(I) =` E(S) E(I) ) I ( E



The arrival rate at the retrial queue when the system is blocked = λP(S).

The exit rate from the retrial queue by entering service

=

)

I

(

E

)

I

(

P

)

λ

(

*

A

.

The system is stable if and only if the total arrival rate is less than the exit rate.

Hence, λ E(S) <A*() is necessary and sufficient condition for the system to be stable.

IV. THE STEADY STATE SOLUTION

The steady state equations that govern the model under consideration are 0 I  =    1 k 1 i i

q  



0

i 0 ,

i (x) (x)dx

W +  



0

k 0 ,

k (x) (x)dx

W (1)

dx ) x ( dI_n

= − ( λ + η(x)) I_n(x), n ≥ 1 (2)

dx ) x ( dW_i_,_n

= − ( λ + _i(x)+ _i)W_i_,_n(x) +    0 i n , 0 ,

i (x,y) (y)dy

F +  _i



0 n ,

i (x,y)dy R

+ λ ( 1− ₀_n) W_i_,_n_₁(x), i = 1, 2, …, k, n ≥ 0 (3)

           y

x Di,j,n(x,y) = − ( λ + i(y))Di,j,n(x,y) + λ ( 1− 0n) Di,j,n1(x,y),

j = 0, 1; i = 1, 2, …, k, n ≥ 0 (4)

           y

x Fi,j,n(x,y)= − ( λ + i(y))Fi,j,n(x,y) + λ ( 1− 0n) Fi,j,n1(x,y),

j = 0, 1; i = 1, 2, …, k, n ≥ 0 (5)

           y

x Ri,n(x,y) = − ( λ + i)Ri,n(x,y) + λ ( 1− 0n) Ri,n1(x,y),

(4)

590 with boundary conditions

) 0 (

I_n =  

 1 k

1

i i

q  



0

i n ,

i (x) (x)dx

W +  



0

k n ,

k (x) (x)dx

W , n ≥ 1 (7)

) 0 (

W₁_,₀ = I₀ +   

0

1(x) (x)dx I

(8)

) 0 (

W₁_,_n =  

0

n(x) dx

I +  

  0

1

n (x) (x)dx

I , n ≥ 1 (9)

) 0 (

W_i_,_n = (1q_i₁)   

 

0

1 i n , 1

i (x) (x)dx

W , i = 2, 3, …, k, n ≥ 0 (10)

) 0 , x (

D_i_,₀_,_n = p_i _i W_i_,_n(x) , i = 1, 2, …, k, n ≥ 0 (11)

) 0 , x (

D_i_,₁_,_n = (1p_i)_iW_i_,_n(x) , i = 1, 2, …, k, n ≥ 0 (12)

) 0 , x (

F_i_,_j_,_n =  



0

i n

, j ,

i (x,y) (y)dy

D , i = 1, 2, …, k, j = 0,1; n ≥ 0 (13)

) 0 , x (

R_i_,_n =  



0

i n

, 1 ,

i (x,y) (y)dy

F , i = 1, 2, …, k, n ≥ 0 (14)

Define Probability generating functions for i = 1, 2, …, k and j = 0,1 as follows

I(z, x) =  

1 n

n n(x)z

I W_i(z,x) =  

0 n

n n , i (x)z W

) y , x , z (

D_i_,_j =  

0 n

n n

, j ,

i (x,y)z

D F_i_,_j(z,x,y) =  

0 n

n n

, j ,

i (x,y)z

F and R_i(z,x,y) =  

0 n

n n

,

i (x,y)z R

Multiplying equation (2) by znand summing over n, we obtain

I(z, x) = I(z, 0) ex[1 – A(x) ] (15)

Performing similar operations on equations (3) – (14), we get

dx ) x , z ( dW_i

+ [λ ( 1 – z) + _i(x)+_i]W_i(z,x) =   

0

i 0

,

i (z.x,y) (y)dy

F +  _i



0

i(z,x,y)dy

R ,

i = 1, 2, …, k (16)

) y , x , z (

D_i_,_j = D_i_,_j(z,x,0)e(1z)y [1 – D_i(y)], i = 1, 2, …, k ; j =0,1 (17)

) y , x , z (

(5)

591 )

y , x , z (

R_i = R_i(z,x,0) _e[(1z)i]y_{, i = 1, 2, …, k (19)}

I(z, 0) =  

 1 k

1

i i

q  



0

i i(z,x) (x)dx

W +  



0

k k(z,x) (x)dx

W − I₀ (20)

) 0 , z (

W₁ ₌ I₀+ z

) 0 , z ( I

] ) ( * A ) z 1 ( z

[    (21)

) 0 , z (

W_i ₌ (1q_i_₁)   

 

0

1 i 1

i (z,x) (x)dx

W , i = 2, …, k (22)

) 0 , x , z (

D_i_,₀ = p_i _i W_i(z,x) , i = 1, 2, …, k (23)

) 0 , x , z (

D_i_,₁ = (1p_i)_iW_i(z,x) , i = 1, 2, …, k (24)

) 0 , x , z (

F_i_,_j =  



0

i j

,

i (z,x,y) (y)dy

D , i = 1, 2, …, k; j = 1, 2 (25)

) 0 , x , z (

R_i =  



0

i 1

,

i (z,x,y) (y)dy

F , i = 1, 2, …, k (26)

Using the equations (17), (18), (23), (24) and (25), the expressions of F_i_,_j(z,x,0)andR_i(z,x,0)for i = 1, 2, …, k and j = 0,

1 become

) 0 , x , z (

F_i_,₀ = p_i _i W_i(z,x) D*_i((1z)), i = 1, 2, …, k (27)

) 0 , x , z (

F_i_,₁ = (1p_i)_i W_i(z,x) D*_i((1z)) , i = 1, 2, …, k (28)

) 0 , x , z (

R_i = (1p_i)_i W_i(z,x) D*_i((1z))F_i*((1z)), i = 1, 2, …, k (29)

Now F_i_,₀(z,x,y) and R_i(z,x,y) can be expressed

interms of W_i(z,x)by using equations (27) and (29).

Substituting the resultant expression of F_i_,₀(z,x,y)and

) y , x , z (

R_i in equation (16) and simplifying we obtain

) x , z (

W_i = W_i(z,0)_eGi((1z))x_{[1 –}_B ₍_x₎

i ], i = 1, 2, …, k (30)

where G_i(x) = x + _i− _i D_i*(x) F_i*(x)

i i i x

] x p [

 

Using the expression of W_i(z,x), the equations (20) to (26) yield the following results

I(z, 0) = I₀z 

 

k

1 i i i 1

Q q 1

[ Λ*_i(G_i((1z)))]/ D(z) (31)

) 0 , z (

(6)

592 )

0 , x , z (

D_i_,₀ = p_i _iI₀ (1 – z) A*() Q_i_₁ Λ*_i_₁(G_i_₁((1z)))_eGi((1z))x

[1– B_i(x)] /D(z), i = 1, 2, …, k (33)

) 0 , x , z (

D_i_,₁ = (1p_i)_iI₀ (1 – z) A*()Q_i_₁ Λ_i*_₁(G_i_₁((1z)))

x )) z 1 ( ( Gi

e   [1– B_i(x)] /D(z), i = 1, 2, …, k (34)

) 0 , x , z (

F_i_,₀ = p_i _iI₀ (1 – z) A*() Q_i_₁Λ*_i_₁(G_i_₁((1z)))D*_i((1z))

x )) z 1 ( ( Gi

e   [1– B_i(x)] /D(z), i = 1, 2, …, k (35)

) 0 , x , z (

F_i_,₁ = (1p_i)_iI₀ (1 – z) A*()Q_i_₁Λ*_i_₁(G_i_₁((1z)))

)) z 1 ( (

D*_i   _eGi((1z))x_[1–_B ₍_x₎

i ] /D(z), i = 1, 2, …, k (36)

) 0 , x , z (

R_i = (1p_i)_iI₀(1–z)A*() Q_i_₁Λ*_i_₁(G_i_₁((1z))) D*_i((1z)) F_i*((1z)) _eGi((1z))x

[1– B_i(x)] /D(z), i = 1, 2, …, k (37)

where D(z) = [z(1z)A*()] 

 

k

1 i

iQ_i ₁

q Λ*_i(G_i((1z))) − z (38)

and Λ_i =

B

₁

B

₂ . . . .

B

_i with Λ₀ = 0. Theorem 2

In equilibrium state, the joint distribution of the server state has the following partial generating functions

I(z) = I₀(1A*())z  

 

k

1 i

iQ_i ₁ q 1

[ Λ*_i(G_i((1z)))]/ D(z) (39)

) z (

W_i = I₀ (1–z)A*()Q_i_₁ Λ*_i_₁(G_i_₁((1z)))

[1–B_i*(G_i((1z)))] / {D(z) G_i((1z))}, i = 1, 2, …, k (40)

) z (

D_i_,₀ = p_i _i I₀ A*()Q_i_₁ Λ*_i_₁(G_i_₁((1z)))[1–B_i*(G_i((1z)))]

[1–D*_i((1z))]/ {D(z) G_i((1z))}, i = 1, 2, …, k (41)

) z (

D_i_,₁ = (1p_i)_i I₀A*()Q_i_₁ Λ*_i_₁(G_i_₁((1z)))[1–B*_i(G_i((1z)))]

[1–D*_i((1z))]/ {D(z) G_i((1z))}, i = 1, 2, …, k (42)

) z (

(7)

593

[1–B_i*(G_i((1z)))][1–F_i*((1z))]/ {D(z)G_i((1z))},i= 1, 2, …, k (43)

) z (

F_i_,₁ = (1p_i)_i I₀A*()Q_i_₁ Λ*_i_₁(G_i_₁((1z))) D*_i((1z))

[1–B_i*(G_i((1z)))][1–F_i*((1z))]/{D(z)G_i((1z))},i = 1, 2, …, k (44)

) z (

R_i = (1p_i)_iI₀(1–z)A*() Q_i_₁ Λ*_i_₁(G_i_₁((1z))) D*_i((1z))

)) z 1 ( (

F_i*   [1–B*_i(G_i((1z)))]/ {D(z)G_i((1z))[(1z)_i]},

i = 1, 2, …, k (45)

0

I = [A*()E(S)] / A*()

(46)

Proof:

Define the partial generating function (z)as

) z (

 =  



0

dx ) x , z

( (47)

and (z) =   

0 0

dy dx ) y , x , z (

(48)

Using the equations (31) - (37) and the definition in equations (47) and (48), the results in equation (39) – (45) can be obtained from equation (15), (30) and (17) – (19). The expression of I₀can be derived from the normalizing

conditionI₀+ I(1) +

    

 

k

1 i

1

0 j

j , i j , i i

i(1) R (1) [D (1) F (1)]} W

{ = 1.(49)

V. SYSTEM MEASURES

In this section, performance measures are derived.

Corollary 1

If the system is in steady state, then The probability that the system is idle is

0

I = [A*()E(S)] / A*() (50)

The probability that the server is idle in the non empty system is

I

= [1A*()]E(S)/ A*()

(51)

The probability that the server is busy is

W =  

 

k

1

i i 1 i1

Q

(52)

The probability that the server is in delayed mode is

D =  

 

k

1

i i i 1 i1 i1 v

Q

(53)

The probability that the server is in failure mode is

F =  

 

k

1 i

1 i 1 i 1 i

iQ h

(54)

The probability that the server is in reserve mode is

R =   

 

k

1

i i i i 1 i1 Q ) p 1

( /



_i

(55)

Proof:

Taking limit as z →1 on equations (39) to (45), the stated formulas followed by direct calculations.

Corollary 2

(8)

594

q

L = {[1−A*()]E(S)+2E(S2)}/{[A*()E(S)] (56)

The mean system size is

s

L = L_q + E(S)

(57)

where E(S2) = 

 

k

1 i i i 1

Q

q            

 i

1

j j2 j2

2 j j j

j 1 j 1 j 1 j 1 j 2 j

j [v h (v h )(1 p )/ (1 p )/ (v h )/2]

Proof:

Partial generating function of the number of customer in the orbit is given by

) z (

P_q = I₀+ I(z) +   k

1 i

[W_i(z)+   1

0 j

[D_i_,_j(z)+ F_i_,_j(z)+ R_i(z)]]

= I0 A*()(1–z) / D(z) (58)

Partial generating function of the number of customer in the system is given by

) z (

P_s = I₀+ I(z) + z  k

1 i

[W_i(z)+  

1

0 j

[D_i_,_j(z)+ F_i_,_j(z)+ R_i(z)]]

= I₀ A*()(1–z) 

 

k

1 i i i 1

Q

q Λ*_i(G_i((1z)))/ D(z) (59)

The results in equations (56) and (57) are obtained by differentiating both the equations (58) and (59) and taking limit as z → 1.

Corollary 3

The probability that the orbit is empty while the server is busy

0

W =   k

1

i 0

I

 Q_i_₁ Λ*_i_₁(G_i_₁())[1–B*_i(G_i())] /

[   k

1

i i

q

1 i

Q_ Λ_i*(G_i())G_i()] (60)

The probability that the orbit is empty while the server is under delay mode

0

D = 

 k

1

i i

 I₀Q_i_₁Λ*_i_₁(G_i_₁())[1–B_i*(G_i())][1–D*_i()]/

[   k

1

i i

q Q_i_₁Λ*_i(G_i())G_i()_{] (61)}

(9)

595

0

F = 

 k

1

i i

 I₀Q_i_₁Λ_i*₁(G_i_₁()) D*_i()[1–B*_i(G_i())]

[1–F_i*()]/ [   k

1

i i

q

1 i

Q _ Λ*_i(G_i())G_i()_{] (62)}

The probability that the orbit is empty while the server is under reserve mode

0

R = 

 k

1 i

) p 1

(  _i _iI₀ Q_i_₁ Λ*_i_₁(G_i_₁()) D*_i()

) (

F_i*  [1–B*_i(G_i())]/ [  k

1

i i

q

1 i

Q_ Λ*_i(G_i())G_i()[_i]] (63)

The probability that the orbit being empty as

0

E = I₀{1+  k

1 i

[Q_i_₁Λ*_i₁(G_i_₁())[1–B*_i(G_i())]] /

  k

1

i i

q

1 i

Q_ Λ*_i(G_i())} (64)

Proof:

Define W₀ = lim k W_i(z)

1 i 0

z  ; D0 = lim Di,j(z)

k

1 i

1

0 j 0

z  _{ } ;

0

F = lim k F_i_,_j(z)

1 i

1

0 j 0

z  _{ } ; R0 = lim Ri(z) k

1 i 0

z  and

0

E = I₀+W₀+D₀+F₀+R₀ The results are easily obtained by direct calculation.

VI. RELIABILITY MEASURES

Now, we consider some reliability measure of the queueing system under consideration.

Let A(t) be the system availability at time t, that is, the probability that the server is either working on a customer or in an idle period.

Theorem 3

The steady state availability of the server in equilibrium state is

A = 1 −



E

(

S

)

+







 

k

1

i i 1 i1

Q

(65)

Proof:

A = I₀+  

1 n

 

0

n(x) dx

I + 



0 n

  k

1 i

 

0 n ,

i (x) dx W

= I₀+ 1 z lim

 



0

dx ) x , z ( I

{ + 

 k

1 i

 

0

i(z,x) dx W

= I₀+ 1 z lim

 [I(z) + _ k

1 i

) z (

W_i ] (66)

The result in equation (65) is obtained by substituting the corresponding expressions in equation (66).

Theorem 4

(10)

596 ₣ =  

 

k

1

i i i 1 i1 Q

(67)

Proof:

₣ =  0 n

  k

1 i

  

0

n n , i

iW (x)z dx

= 1 z lim

  k

1 i

  

0 i

iW (z,x) dx

=   k

1

i i

 W_i(1)

Using the equation (40), by direct calculation the result (67) can be obtained.

VII. STOCHASTIC DECOMPOSITION

Stochastic decomposition has been widely observed among M/G/1 type queues.

The decomposition property states that the number of customers in the system in steady state at a random point of

time is distributed as the sum of two independent random variables, one of which is the number of customers in the corresponding standard queueing system in steady state at a random point in time, the other random variable may have different probabilistic interpretation in specific cases depending on the vacation scheduled.

If the server is idle either due to retrial of customers from the orbit due to empty system, then it is assumed that the server is vacation.

Theorem 5

The stochastic decomposition law for system size is given by P_s(z) = φ (z) (z) (68)

and for orbit size is P_q(z) = ϕ(z) (z) (69) Proof:

The probability generating function of the number of customers in the system for the classical M/G/1 queueing system with k-stages of service facility with server breakdown, delayed repair and reserved time is

φ (z) = [kqQ (G ( (1 z)))] 1

i i

* i 1 i i

   

  [1E(S)][1− z] / [ qQ (G ( (1 z))) z]

k 1

i i

* i 1 i

i 

   

  (70)

The probability generating function of the number of customers in the system at a random point of time given that the server is on vacation is

(z) =

(1) (z)

0 0

  

= I₀ A*()[kqQ (G ( (1 z))) z] 1

i i

* i 1 i

i 

   

  /{D(z) [1E(S)]} (71)

The probability generating function of the number of customers in the queue for the classical M/G/1 queueing system with k-stages of service facility with server breakdown, delayed repair and reserved time is

ϕ(z) = [1E(S)][1−z] / [kqQ (G ( (1 z))) z] 1

i i

* i 1 i

i 

   

  (72)

From equation (58) and (59) it is observed P_s(z) = φ(z) (z) and P_q(z) = ϕ(z) (z). Corollary 4

The expected number of customers in the orbit during the retrial time, i.e., mean of additional system size due to retrials is



L = 1 /

[

A

*

(



)





E

(

S

)

] −2E(S2)/ {2A*() (1E(S))} (73)

(11)

597 Taking limit as z → 1 on both sides of equation (71) we get the result in equation (73).

VIII. THE MEAN BUSY PERIOD

We derive the mean busy period and expected length of a busy cycle under the steady state condition.

Theorem 6

Let T_b and T_c respectively be the length of a busy period and length of a busy cycle. Under the steady state conditions, we have

E(T_b) = E(S) / [A*()E(S)]

(74) E(T_c) = A*()/ {λ[A*()E(S)]} (75) Proof:

The results follow directly by applying the argument of alternating renewal process which lead to well known result

E(T_b) = (I₀1− 1) / λ (76)

and E(T_c) = (λI )₀ 1 (77)

Inserting the equation (50) in the equations (76) and (77) we get the equations (74) and (75) respectively.

IX. PARTICULAR CASES

1.If A*(λ)  1, then the model reduces to multi stage queueing system with server breakdown, delayed repair and reserved time.

2.If p_i= 1, i= 1, 2, …, k, then the model reduces to multi stage retrial queueing system with server break down and delayed repairs.

3.If k = 2 and p₁= p₂= 1, then the system reduces to a two phase single server retrial queue with server breakdown and delayed repair.

4.If k = 2 and ₁= ₂= 0, then this model reduces to two phase single server retrial queueing system. 5.If k = 2 and A*(λ)  1, then this model reduces to

two phase queueing system with server breakdown , delayed repair and reserved time.

6. If k = 1,p₁= 1 and D₁*(s)  1, then the results obtained in our model coincide with the results of Wang and Cao [10].

7. If k = 1, ₁= 0 and A*(λ)  1, then we have classical M/G/1 queueing system.

X. CONCLUSION

A multi stage retrial queueing system with server breakdown is studied. The repair of the failed server starts after a random of time as in real situation. After return to working condition the server completes the service of the interrupted customer without accepting new customer. Some of the system performance measures and reliability indexes are derived. This queueing system is a generalization of many existing queueing systems of this type.

REFERENCES

[1 ] Aissani, A., An Mx_{/G/1 Energetic Retrial Queue with Vacations and}

Control, IMA Journal of Management Mathematics, 2011,22, 13-32. [2 ] Arivudainambi, D. and Godhandaraman, P., A Batch Arrival Retrial Queue with Two Phases of Service, Feedback and K Optional Vacations, Applied Mathematical Sciences, 2012, 6(22), 1071-1087. [3 ] Artalejo, J.R. and Li, Q., Performance Analysis of a Block Structured Discrete Time Retrial Queue with State Dependent Arrivals, Discrete Event Dynamic Systems, 2011,20(3), 325-347. [4 ] Choudhury, G. and Ke, C.J., A batch Arrival Retrial Queue with

General Retrial Times Under Bernoulli Vacation Schedule for Unreliable Server and Delaying Repair, Applied Mathematical Modelling,, 2012, 36, 255-269.

[5 ] Ebenesar Anna Bagyam. J. and Udaya Chandrika.K., Two Phase Retrial Queueing System With Impatient Customers, Additional Optional Service, Server Breakdown, Delayed Repair And Reserved Time , International Journal Of Mathematical Archive, 2012,3(6), 2283-2290.

[6 ] Nathan, P.S. and Jeffery, P.K., Optimal Bernoulli Routing in an Unreliable M/G/1 Retrial Queue, Probability in the Engineering and Informational Sciences,2011, 25, 1-20.

[7 ] Rehab, F. K., Madan, K.C. and Lukas, C.A., An M[x]_{/G/1 Queue}

with Bernoulli Schedule, General Vacation Times, Random Breakdowns, General Delay Times and General Repair Times, Applied Mathematical Sciences, 2011,5 (1), 25-51.

[8 ] Salehirad, M.R. and Badamchizadeh, A., On the multi-phase M/G/1 queueing system with random feedback, EJOR, 2009, 17, 131-139. [9 ] Shahkar, G.H. and Badamchizadeh, A., On M/(G1, G2, . . . ,

Gk)/V/1(BS), Far East J. Theor. Stat., 2006, 20 (2), 151-162.