International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 5, Issue 1, January 2016
ISSN 2319 - 4847
Busy Period Analysis of M/M/2/
Interdependent Queueing Model With
Controllable Arrival Rates And Feedback
G. Rani1, A. Srinivasan2
1
Associate Professor, PG and Research Department of Mathematics,
Seethalakshmi Ramaswami College, Tiruchirappalli, 620002. Tamil Nadu, India.
2
Associate Professor, PG and Research Department of Mathematics,
Bishop Heber College (Autonomous), Tiruchirappalli, 620017. Tamil Nadu, India.
ABSTRACT
In this paper the M/M/2/ queueing model with controllable arrival rates and feedback is considered. Distribution
of busy period and mean length of busy period carried out for this model. The analytical results are numerically
illustrated and the effect of the nodal parameters on the expected length of busy period is studied and relevant
conclusions are presented.
Keywords: Two server Markovian queueing model, Controllable arrival rates, Feedback, infinite capacity, Busy period
Analysis.
1. Introduction
In the earlier work, Thiagarajan and Srinivasan [6] have analysed the busy period analysis of interdependent queueing
model with controllable arrival rates are employed and obtained the average length of the busy period for the model,
Takagi and Tarabia [4] have studied a explicit probability density function for the length of a busy period for a finite
capacity. Artalejo and Lopez Herrero [1] have analysed the busy period of the retrial queue with general service time
distribution. Gopalan [2] has analysed the busy period analysis of a two-stage multi-product system. Soren Asmussen
[7] has studied a busy period analysis, rare Events and Transient behaviour in fluid flow Models. Thangaraj and
Vanitha [5] have analysed on the analysis of M/M/1 feedback queue with catastrophe using continued fractions. In this
Chapter, busy period analysis of two server M/M/2/ interdependent queueing model with controllable arrival rates and
feedback is considered. In section 2, the description of the queueing model is given stating the relevant postulates. In
section 3, Differential-Difference Equations are derived. In section 4, Busy period analysis of the model and average
length of the busy period of the model is obtained. In section 5, the analytical results are numerically illustrated and
relevant conclusion is presented based upon a hypothetical data.
2. Description of the Model
Consider two server infinite capacity queueing system with controllable arrival rates and feedback. Customer arrive at
the service station one by one according to a bivariate Poisson stream with arrival rates (0-), (1-) (>0). There are
two servers which provides service to all the arriving customers. Service times are identically and independently
distributed exponential random variables with mean rate (-) (>0). After the completion of each service, the customers
can either join at the end of the queue with probability p or customers can leave the system with probability q, p+q=1.
The customer both newly arrived and those who opted for feedback are served in the order in which they join the tail of
the original queue. It is assumed that there is no difference between regular arrivals and feedback arrivals. The customers
are served according to the first come first served rule with a busy period as the interval of time from the instant
customers arrive either with feedback or without feedback at an empty system and their service begins to the instant
when the server becomes free for the first time. It is assumed that, The arrival process {X1(t)} and the service process
{X2(t)} of the system are correlated and follow a bivariate Poisson distribution is given by
e
P  X 1  t   x 1 , X 2  t   x 2  
Volume 5, Issue 1, January 2016
  i     t
m in  x 1 ,x 2 
j
   t    
j 0
i
   t 
x1  j
      t 
x2  j
j!  x 1  j !  x 2  j !
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 5, Issue 1, January 2016
ISSN 2319 - 4847
where x1, x2 = 1, 2,..., and 0, 1, >0, 0 < min (i,), (i=0,1) with parameters 0, 1,  and  for mean faster rate of
arrivals, mean slower rate of arrivals either with feedback or without feedback, identical mean service rate for first
server, second server and mean dependence rate respectively Also the state space of the system is
s = {1, 2, 3,..., r–1, r, r+1, ..., R–1, R, R+1,...,k–1, k, k+1,...,}
Postulates of the model are
1. Probability that there is no arrival and no service completion during a small interval of time h, when the
system is in faster rate of arrivals either with feedback or without feedback, is
1    0     2p       2q       h  o  h 
2.
3.
Probability that there is one arrival and no service completion during a small interval of time h, when the
system is in faster rate of arrivals either with feedback or without feedback, is
(0 – )h + o(h)
Probability that there is no arrival and no service completion during a small interval of time h, when the
system is in slower rate of arrivals either with feedback or without feedback is,
1   1     2p       2q       h  o  h 
4.
5.
Probability that there is one arrival and no service completion during a small interval of time h, when the
system is in slower rate of arrivals either with feedback or without feedback is
(1 – ) h + o(h)
Probability that there is no arrival and one service completion during a small interval of time h, when the
system is either in faster or is in slower rate of arrivals either with feedback or without feedback is
 2p       2q       h  o  h 
6.
Probability that there is one arrival and one service completion during a small interval of time h, when the
system is either in faster or slower rate of arrivals either with feedback or without feedback is
  0      1     2p       2q       h  o  h 
3.Differential - Difference Equations
Pn(t0):
the probability that there are n customers in the system at time t0 when the system is in faster
rate of arrivals either with feedback or without feedback.
Pn(t1):
the probability that there are n customers in the system at time t1 when the system is in
slower rate of arrivals either with feedback or without feedback.
we observe that Pn(t0) exists when n = 0, 1, 2, 3,...,r–1; both Pn(t0) and Pn(t1) exist when n = r+1, r+2, r+3, ..., R–2, R–
1; only Pn(t1) exists when n = R, R+1, R+2, ..., Assume that the initial system size when the system is in faster rate of
arrivals is 1 either with feedback or without feedback and that of when the system is in slower rate of arrivals is r+1.
Let
P0'  t 0 
without feedback and
be the busy period density when the system is in faster rate of arrivals either with feedback or
Pr'1  t1  be the busy period
density when the system is in slower rate of arrivals either with
feedback or without feedback.
The differential-difference equations governing the system size with an absorbing barrier imposed at zero
system size when the system is in faster rate of arrivals either with feedback or without feedback and with an obsorbing
barrier imposed at (r+1) system size when the system is in slower rate of arrivals either with feedback or without
feedback, are
P0'  t 0     2       P1  t 0 
(because of the absorbing barrier)
... (3.1)
'
1
P  t 0      0     2q       P1  t 0   2q      P2  t 0 
(because of the absorbing barrier)
... (3.2)
'
n
P  t 0      0     2q       Pn  t 0   2q      Pn 1  t 0 
   0    Pn 1  t 0 
Volume 5, Issue 1, January 2016
n = 2,3,4,...,r–1 ... (3.3)
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Volume 5, Issue 1, January 2016
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Pr'  t 0      0     2q       Pr  t 0     0    Pr 1  t 0  
2q      Pr 1  t 0   2q      Pr 1  t1 
... (3.4)
Pn'  t 0      0     2q       Pn  t 0     0    Pn 1  t 0 
2q      Pn 1  t 0 
n = r+1, r+2,..,R–2 ... (3.5)
PR' 1  t 0      0     2q       PR 1  t 0     0    PR 2  t 0 
(because of the absorbing barrier)
'
r 1
P
... (3.6)
 t1   2q      Pr 2  t1 
(because of the absorbing barrier)
'
r 2
P
... (3.7)
1    1     2q     Pr 2  t1   2q      Pr 3  t1 
(because of the absorbing barrier)
... (3.8)
'
n
P  t1     1     2q      Pn  t1    1    Pn 1  t1 
2q      Pn 1  t1 
n = r+3, r+4,..., R–1
... (3.9)
PR'  t1     1     2q       PR  t1     0    PR 1  t 0 
  1    PR 1  t1   2q      PR 1  t1 
... (3.10)
Pn'  t1     1     2q      Pn  t1    1    Pn 1  t1 
2q      Pn 1  t1 
Let
  0
0  

2
2q     
where
n = R+1, R+2, R+3,... ... (3.11)
and
 1
1  

2
2q     
  0
 1
is faster rate of arrivals intensity and
is slower rate of arrivals intensity.
2
2
4. Busy Period Analysis of the Model
R 1
Define
P  z, t 0    Pn  t 0  z n
... (4.1)
n 0

P  z, t1  
 P t z
n
n
1
... (4.2)
n r 1
Such that the summation is convergent in and on unit circle. When (3.1) to (3.6) are multiplied through by zn
for n = 0, 1, 2,...,R–1 and summing over from n = 0, 1, 2,..., R–1, it is found that
R 1
d
 P  z, t 0     Pn'  t 0  z n
dt 0
n 0
R 1
R 1
R 1


     0     2q        Pn  t 0     0     Pn 1  t 0   2q       Pn 1  0  z n

n 0
n 0
n 0

r 1
r 1
+ 2q      Pr 1  t1  z  2q      Pr 1  t 0  z 

     P 0
    0       0     2q       z  2q
 0 
z 

R 1
R 1
    0     2q        Pn  t 0  z n    0    z  Pn 1  t 0  z n 1
n 0
Volume 5, Issue 1, January 2016
n 1
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Volume 5, Issue 1, January 2016

ISSN 2319 - 4847
2q      R 1
Pn 1  t 0  z n 1  2q     Pr 1  t1  z r 1

z
n 0

    P t
    0    z    0     2q  z     z  2q
 0 0
z 

d
z
 P  z, t 0     0    z 2    0     2q       z  2q      P  z, t 0 
dt 0
     0    z 2    0     2q       z  2q       P0  t 0 
2q      z r 1Pr 1  t1 
... (4.3)
Taking Laplace Transform on both sides of (4.3) we get
z s 0 P  z,s 0   P  z, 0      0    z 2    0     2q       z  2q       P  z,s 0 
     0    z 2    0     2q       z
2q       P 0  s 0   2q      z r 1 P r 1  s1 
P  z,s 0  
z 2  1  z   2q         0    z  P 0  s 0   2q      z r 1 P r 1  s1 
  0     2q       s 0  z  2q         0    z 2
where
L P  z, t 0   P  z,s 0 
... (4.4)
L P0  t 0   P 0  s 0  , L Pr 1  t1   P r 1  s1 
Since the Laplace Transform
and
P  z, 0   z
P  z,s 0  converges in the region z  1,
Real
 s0   0, wherever the
denominator of the right hand side of (4.4) has zeros in that region, so must the numerator, the denominator has two zeros,
that is
  0    z2   0     2q       s0  z  2q       0
2
 0
  0     2q       s 0     0     2q       s 0   4   0    2q     

2  0   
 0
1
  0     2q       s 0     0     2q       s 0   8   0    q     

2   0  
 0
  0     2q       s 0     0     2q       s 0   4   0    2q     

2  0  
 0
  0     2q       s 0     0     2q       s 0   8   0    q     

2  0   
z1
2
z
2
z2
2
z2
and hence
z 1 0  
z 20  
   0     2 q       s 0  
   0     2 q       s 0  
2
   0     2 q       s 0   8   0    q      

2  0   



2
   0     2 q       s 0   8   0    q      

2  0   

... (4.5)
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z1 0  z 20 
From (4.5), It is clear that
 z1 0  z 20 
z1 0 z 20  
ISSN 2319 - 4847
2   0     4q       2s0   0     2q       s0

20  
0  
2q     
0  
and hence
z1 0  z20 
 0  0 
z1 z 2
  0     2q       s0 






0  
2q     

0  
r 1
2
P0 s0  
... (4.6) Using Rouche’s theorem in (4.4) we get
z1 0  2q      z1 0 P r 1  s1 
... (4.7)
s 0 z1 0 
Using the result (4.7) in (4.4) we get
r 1
2
 z 0   2q      z 0 P r 1  s  
1
1 
 1
2

z  1  z   2q         0    z  
0
s 0 z1
 2q      z r 1 P r 1  s1 
P  z,s0  
  0     z20   z  z  z1 0  
...
(4.8)
r 1
2
 z  0   2q      z  0 P r 1  s  
1
1

2q       1
 0
s
z


0 1
P 0  s 0   P  0,s 0  
0
0
   0    z1  z2 

=

r
2q     
s 0   0    z2 
0
2q       2q      

P r 1  s1 
 0 r     
s0z 2
0


r


 2q      


 0    

2q       1
0   




P r 1  s1 
r
 0

0


s0  0    z 2
z2




s 0 P 0  s0  
2q     
  0    z20
r
 2q       Pr 1  s1 
 2q      
r

0
  0    z 2 
... (4.9)
From equation (4.9) we get
r
 2q        P r 1  s1 
d
d  2q     
s 0 P 0  s0  
 ds       z 0  2q             0r
ds 0 
0 
 0
  z2
0
2
2q     
d
s 0 P 0  s0  


 s0 0
ds 0 
  0      0     2q     
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r
 2q      
r
2q      
 P r 1  s1 

  0     2q     
  0    s0 0
r
 1

2q     
 2q      



.r Lt P r 1  s1  
  0     2q         0      0    s0 0

r
 1

2q     
 2q      
d




s0 P0  s0 

r
Lt P r 1  s1  
s0 0
ds 0 
  0     2q        0     0    s0 0

... (4.10)
d
d
s 0 P 0  s0 

L P0'  t 0  s 00


s

0
0
ds 0
ds 0


d   s0 t 0 '

  e P0  t 0  dt 0 
ds 0  0
 S0 0
   s0 t 0

  e
  t 0  P0'  t 0  dt 0 
0
 S0 0

 
 0
   t 0 P0'  t 0  dt 0  E Tbusy
0

d
 0
s 0 P 0  s 0    E Tbusy


S
00
ds0
 
... (4.11)
From equations (4.10) and (4.11) we get
 
 0
E Tbusy
r
 1

 2 
1


 
.r.
Lt
P
s
r

1
 1 

  0    0      0   s0 0

1


2
 The average length of the busy period when the system is in faster rate of arrivals either with feedback or without
feedback is given by
 
 
E Tbusy
0
r
 1

 2 
1



 
.r.
Lt
P
s
r

1



1
  0    0      0   s0 0

1


2
... (4.12)
which is consistent with the corresponding result of the conventional model discussed by Gross and Harris (1974) for
the case  = 0, p=q=1, 0 = 1 =  and is also consistent with the corresponding result of the busy period analysis of
M/M/1/ model discussed by Thiagarajan and srinivasan for the case p=q=1.
From (3.6) to (3.10) are multiplied through by zn,
n = r+1, r+2, r+3,..., R–1, R, R+1,..., and summing on n
from n = r+1, r+2,...R–1, R+1,..., it is found that

d
 P  z, t1     Pn'  t1  z n
dt1
n  r 1
... (4.13)
Using the procedure as in the case when the system is in faster rate of arrivals either with feedback or without
feedback we get
d
 P  z, t1    Pr'1  t1  z r 1  Pr' 2  t1  z r  2  ...  PR' 1  t1  z R 1
dt1
 PR'  t1  z R  PR' 1  t1  z R 1  ...
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
2q     
 
    1     2q      
  1    z   Pn  t1  z n
z

 n  r 1

2q     

   1     2q      
  1    z  Pr 1  t1  z r 1
z


R
   0    PR 1  t 0  z

2q     

    1     2q       
  1    z  P  z, t1 
z



2q     

   1     2q      
  1    z  Pr 1  t1  z r 1
z


   0    PR 1  0  z R
z

d
 p  z, t1     1    z 2   1     2q       z  2q        Pn  t1  z n
dt1
n  r 1
   1     2q       z  2q        1    z 2  Pr 1 1 z r 1
   0    PR 1  t 0  z R 1
Taking Laplace Transform on both side we get
z s1 P  z,s1   P  z, r  1    1    z 2   1     2q       z  2q       P  z,s1 
    1    z 2   1     2q       z
2q       P r 1  s1  z r 1    0    P R 1  s 0  z R 1


 z r  2   1    z 2   1     2q      z  2q      P r 1  s1  z r 1



   0    P R 1  s0  z R 1 
  1    z 2   1     2q       s1  z  2q       P  z,s1 
P  z,s1  
z r  2  1  z   2q        1     z r 1 P r 1  s1     0    P R 1  s 0  z R 1
 1     2q       s1  z   1    z 2  2q     
... (4.14)
where


L P  z, t1   P  z,s1  , L Pr 1  t1   P r 1  s1  and P  z, r  1  z r 1
Since the Laplace Transform
P  z,s1  Converges in the region z  1,
Real (s1)0, wherever the
denominator of the right hand side of (4.14) has zeros in that region, so must the numerator, the denominator has two
zeros.
Using Rouehe’s theorem in (4.14) we get
1r 1
1R
P r 1  s1  
P  z,s1  
z1
   0    P R 1  s 0  z1
s1
... (4.15)
z r  2  1  z   2q        1    z  z r 1 P r 1  s1     0    z R 1PR 1  s 0 
 1     z  z11   z21  z 
... (4.16)
Volume 5, Issue 1, January 2016
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Volume 5, Issue 1, January 2016
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where
2
 1     2q       s1    1     2q       s1   8  1    q      
z 

2  1   



2





2q




s







2q




s


8



q
















1
1
1
1
1




1
z2  

2  1   

1
1
... (4.17)
L  P  z, t1    P  z,s1  ,L  PR 1  t 0    P R 1  s0 
It is clear that
z11  z 21
, from equation (4.17)
 1     2q       s1 

 1   


2q     
1 1

z1 z 2 

 1   

z1   z 2  
1
1
Using (4.15) in (4.16) we get
1r  2
1 
z1

 1  z1

... (4.18)
r 1
2q       1    z11  z11 P r 1  s1     0    P R 1  s0  z11
 

z1 
1
P r 1  s1  
1
1
   0    P R 1  s 0  z1R 1
 2q        1    z11  z1r 1


r 1
z1     0    P R 1  s 0  z1R 1
P r 1  s1  
1  z11  2q            z11  z1r

0
r2
1  z  
1
R 1
... (4.19)

From equations (4.15), (4.17), (4.18) and (4.19) we get
r 1
R 1
 z1       P R 1  s  z 1  z r 1
0
0
1
 1

z r  2  1  z   2q        1    z  
1
1
1  z1   2q        1    z1   z1r




   0    z R 1 P R 1  s0 
P  z,s1  
 1     z  z11   z21  z 
r 1
R 1
 z 1       P R 1  s  z 1  z r 1
0
0
1
 1
 1
z r  2  1  z   2q        1    z  
r 1
sz
1 1
   0    z R 1 P R 1  s 0 
P  z,s1  
 1     z  z11   z21  z 
r 1
R
 z1      P R 1  s  z 1 
0
0
1
 1

z r  2  1  z   2q        1    z  
s
1
 P  z,s1  
Volume 5, Issue 1, January 2016
   0    z R 1 P R 1  s 0 
 1     z  z11   z21  z 
Page 124
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 5, Issue 1, January 2016
r 1
ISSN 2319 - 4847
R
1 
 1
2q       z1    0    PR 1  s0  z1 
P  0,s1   
1  
 z11 z 21s1
r 1
z1     0    P R 1  s 0  z1 
P  0,s1  
s1
1
1
r 1
R
... (4.20)
s1 P  0,s1   z1     0    P R 1  s 0  z1 
1
1
 1 2q      
  1

 z 2 1   
r 1
 1 2q      
s1 P  0,s1    1

 z 2 1   
R
 2q      

   0    P R 1  s 0   1
 z 2  1    
R
r 1
1  2q      
   0    P R 1  s 0  R 

z 21  1   
r 1

 2q      
d
1
s1 P  0,s1   
       2q         r  1     
ds1 
1

1


R
  2q       
 Lt   0    P R 1  s 0  R 

S1 0
 1   
r 1

 2q      
d
1
s1 P  0,s1  

 S1 0        2q       r  1     
ds1 
1
 1


R
 2q       
 Lt   0    P R 1  s 0  R 
 
S1  0
 1    


d
d  S1t1 '
s1 P1  s1  

e
P
t
dt




r

1
1
1

S1  0 ds
ds t 
1 0
S1 0
R




... (4.21)
   S1t1

    e t1Pr 1  t1  dt1 
 0
S1  0

 
1
   t1d  Pr 1  t1     E Tbusy
0

d
1
s1 P1  s1    E Tbusy


ds1
 
... (4.22)
From (4.21) and (4.22) we get
r 1

 2q      
1
 r  1 
E Tbusy 

 1     2q      
 1   
R
 2q       
 Lt   0    P R 1  s 0  R 
 
S1  0
 1    
 
1
and hence
Volume 5, Issue 1, January 2016
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
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Volume 5, Issue 1, January 2016
ISSN 2319 - 4847
The average length of the busy period when the system is in the slower rate of arrivals either with feedback or
without feedback, is
 
1
E Tbusy
r 1
R

 2 
 2  
1
r  1 

  0    P R 1  s0 .R   
  SLt
1 0
2q      
1

1


1







  1  


... (4.23)
which is consistent with the corresponding result of the conventional model discussed by Gross and Harries (1974) for
case  = 0, 0 =1 =  and p=q=1, the present model reduces to busy period analysis of M/M/1/ interdependent model
with controllable arrival rates discussed by Thiagarajan and Srinivasan [6] for the case p=q=1, Two server = single
server (2q=q).
5. Numerical Illustrations
For various values of r, R, 0, 1,  and ,
p=q=
  and ET   are computed and tabulated by taking
0 
E Tbusy
1
busy
1
.
2
Table 5.1
R
R
0
1


5
5
5
5
5
5
5
5
6
6
6
8
8
8
10
10
10
10
10
10
10
10
10
14
14
14
13
13
5
5
5
5
5
5
8
5
8
8
8
9
9
8
4
4
4
4
4
4
4
4
4
5
5
6
6
6
9
9
9
11
11
11
11
11
12
12
12
12
12
12
0.5
0.0
1.0
0.5
0.0
0.5
0.5
1.0
0.5
0.5
1.0
0.5
0.0
1.0
 
 0
E Tbusy
1.888
1.800
2.000
1.555
1.460
1.666
1.866
1.666
1.916
1.916
2.054
3.156
3.111
2.750
 
 1
E Tbusy
164.10
103.79
287.67
416.57
247.15
937.29
416.57
783.85
305.08
913.86
1189.40
509.24
341.33
804.85
i. It is observed from the Table 5.1 that the average length of the busy period is increase when the system is either in
faster or in slower rate of arrivals by increasing the value of r while the other parameters were fixed.
ii. The average length of the busy period rapidly increases when the system is in faster rate of arrivals by increasing
the value of 0 and keeping the other parameters were fixed.
iii. The average length of busy period rapidly decreases by increasing the value of  and keeping the other parameters
were fixed while the system is in faster rate of arrivals.
iv. The average length of busy period increases by increasing the value of  and keeping the other parameters were
fixed while the system is in slower rate of arrivals.
v. The average length of busy period increases while the system is in faster rate of arrivals by increasing the value of
 keeping the other parameters were fixed.
vi. The average length of busy period increases while the system is in slower rate of arrivals by increasing the value of
 and keeping the other parameters were fixed.
vii. The average length of busy period decreases while the system is in slower rate arrival by decreasing the value of R
keeping the other parameters were fixed.
viii. The average length of busy period is remain unchanged while the system is in faster rate of arrivals by increasing
the value of R keeping the other parameters were fixed.
References
[1] Artalejo, J.R. and Lopez-Herrero, M.J. On the Busy Period of the M/G/1 Retrail Queue, Naval Res. Logist, 47,
115-127, (2000).
Volume 5, Issue 1, January 2016
Page 126
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org
Volume 5, Issue 1, January 2016
ISSN 2319 - 4847
[2] Gopalan, M.N. On the busy period analysis of the two stage multi-product system, International Journal of
Management and Systems, 11, 249-266. (1995).
[3] Srinivasan, A. and Thiagarajan, A Finite Capacity Multi Server Poisson Input Queue with Interdependent
Inter-arrival and Service Times and Controllable Arrival Rates, Bulletin of Calcutta Mathematical Society, 99(2),
173-182, (2007).
[4] Takagi, H. and Tarabia, H. Explicit Probability Density Function for the Length of a Busy Period in an M/M/1/k
Queue, Int. Advances in Queueing Theory and Network Applications, 213-226, (2009).
[5] Thangaraj, V. and Vanitha, S. On the Analysis of M/M/1 Feedback Queue with Catastrophes using continued
Fractions, Int. Jr. of Pure and App. Math., 53(1), 133-151, (2009).
[6] Thiagarajan, M. and Srinivasan, A. Busy Period Analysis of M/M/1/ Interdependent Queueing Model with
Controllable Arrival Rates, Acta Ciencia India, 33M(1), 19-24, (2007).
[7] Soren Asmussen,”Busy period analysis, rare events and transient behavior in Fluid Flow models”, Journal
of Applied Mathematics and Stochastic Analysis,Vol.3,269-299,(1994).
AUTHOR
Dr.G.Rani is an associate professor in Seethalakshmi Ramasami College (Autonomous),Trichy.She
received Ph.D. degree in Mathematics from Bharathidasan University. Her research focuses on
Stochastic models in Queueing Theory with feedback.
Volume 5, Issue 1, January 2016
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