Finite Controllable Markovian Model with Balking and

IJSTE - International Journal of Science Technology & Engineering | Volume 2 | Issue 08 | February 2016 ISSN (Online): 2349-784X

Finite Controllable Markovian Model with Balking and Reneging M. R. Dhakad Directed of Technical Education Bhopal

Madhu Jain Indian Institute of Technology Roorkee

Abstract This paper presents a study of controllable Markovian queueing system with interdependent rates. The customer’s behavior is incorporated according to which balking and reneging with certain probability is taken into account. The models with finite capacity (FCM) and finite population (FPM) are developed. We derive queue size distribution, which is further employed to derive formulae for average number of customers in the system and the expected waiting times for both models. Some earlier results are deduced by setting suitable parameters. Keywords: Controllable Queue, Interdependent rates, Balking, Reneging, Multi- servers, Queue size ________________________________________________________________________________________________________

I. INTRODUCTION Markovian analysis is a way of analyzing the current movement of some variables in an effort to forecast its future movement. Multi-server Markovian models can play a significant role in day-to-day queueing situations encountered, marketing, production, transportation, computer, communication and manufacturing systems etc. In many real life queueing situations due to long queue of customers, the arriving customers may be discouraged. The balking and reneging phenomena arise in the queueing system when the customers leave the system before joining the queue and depart after joining the queue without getting service due to impatience, respectively. Many researchers have done a lot of work on queueing models with balking and reneging in different frame-works. Haghighi et al. [6] obtained the steady-state distribution for multi-server Markovian queueing system with balking and reneging. Abou-El-alta and Hariri [1] discussed M/M/C/N queue with balking and reneging. Various aspects of balking and reneging can be found in the textbook by Hillier and Lieberman [7]. The single-server Markovian overflow queue with balking, reneging and an additional server for longer queues was also analysed by Abou-El-alta and Shawky [2]. Jain and singh [10] derived steady state probability distribution and various characteristics for M/M/m queue with balking, reneging and additional servers. Abou-El-alta and Kote [3] included the concept of linearly dependent service rate for the M/M/1/N queue with genral balk function, reflecting barrier, reneging and an additional server for longer queues. The controllable queue with balking and reneging. To reduce the balking behavior of the customers in the controllable queue, the provision of additional removable servers, considered by Jain and Sharma (14). Singh et al. (15) have examined a Single server interdependent queueing model with controllable arrival rates and reneging. Jain et al. (16) described the controllable and interdependent rates for the machine re-pair problem (MRP) with additional repairman and mixed standbys. Yang et al.(17) considered the optimization and sensitivity analysis for controlling the arrivals in the queueing system with single working vacation. A Balking and reneging in multi-server Markovian queuing system was examined by Choudhury and Medhi (18). Mandelblaum and Momcilovic (19) described a queues with many Servers and Impatient Customers. Kumar and Sharma (20) have present a multi-server Markovian Feedback Queue with Balking Reneging and Retention of Reneged Customers. In this investigation we develop finite controllable Markovian queueing model with balking and reneging by applying birthdeath process. We obtain queue size distribution, the average number of customers in the system and waiting time. We also deduce some particular cases by setting suitable parameters.

II. MATHEMATICAL MODEL Consider a finite controllable Markovian queueing system with balking and reneging. To formulate the problem mathematically, the following postulates are taken into consideration;  The pattern of arrival is Poisson with mean rate  and the service is provided in exponential fashion with mean rate  .  The system is operated by controllable arrival rates with prescribed forward and backward threshold values; when the queue size reaches the forward threshold level (say R), the arrival rate reduces from  to  1 . However as soon as the



queue size reduces to backward threshold level (say r), the arrival rate changes to  and the same process continues to be repeated. The system said to level 0(zero) and 1 when customers arrive with rate  and  1 respectively. The customers may balk depending upon the queue size. The probability of joining the queue is a non-linear function of the number of servers per customer.

All rights reserved by www.ijste.org

36

Finite Controllable Markovian Model with Balking and Reneging (IJSTE/ Volume 2 / Issue 08 / 006)

The state-dependent arrival rate for finite capacity model (FCM) is given by

    e  , n  C    C   (n)    C  n  R 1 for level 0    e  , n  1     C   r  R  n  K, for level 1   1  e  ,   n  1  where K is the finite capacity of the system. The state-dependent arrival rate for finite population model (FPM) is given as follows    M  n   e  ,     C   ( n )    M  n  C     e  ,  n 1     C    M  n r     1  e  ,   n  1  where M is the finite population size. The customers may also renege due to impatience while they are according to exponential distribution with parameter  .



 (n)  

(1)

n  C n  R 1

for level 0

R  n  M,

(2)

for level 1

waiting in the queue. The reneging phenomenon occurs

n   e 

 C    e    n  C 

nC C n L

The state-dependent service rate for finite capacity model (FCM) and finite population model (FPM) are given as follows

(3)

where L takes value K and M for FCM and FPM models, respectively.

III. QUEUE SIZE DISTRIBUTION Let us denote  Pn 0  as the steady state probability at zero level when that there are n customers in the system, and the customers arrive 

in the system with faster rate  and exists for states 0  n  R  1 . Pn 1 as the steady state probability at level 1 when there are n customers in the system, and the system is operating with

slower arrivals rate  1 and exists for states r  1  n  L . where L takes value K(M) for finite capacity (population) model. Now we obtain steady -state queue size distribution, average queue length and expected waiting time for both finite capacity and finite population models by using the respective arrival and service rates for the construction of governing ChapmanKolmogorov equation at equilibrium. Finite Capacity Model (FCM) The steady-state equations governing the FCM model are given by 0     e P0 0      e P1 0 

…(4)

0     e   n    e Pn 0     e Pn 1 0   n  1   e Pn 1 0  , 1  n  C

0    n   n Pn 0    n 1 Pn 1 0    n 1 Pn 1 0  , C  n  r  1

0    r   r Pr 0    r 1 Pr 1 0    r 1 Pr 1 0    r 1 Pr 1 1

…(5) …(6) …(7)

0    n   n Pn 0    n 1 Pn 1 0    n 1 Pn 1 0  , r  1  n  R  2

…(8)


37


0    R 1   R 1 PR 1 0    R  2 PR  2 0 

…(9)

0    r 1   r 1 Pr 1 1   r  2 Pr  2 1

…(10)

  ' ' 0     n   n Pn 1   n 1 Pn 1 1   n 1 Pn 1 1 , r  2  n  R  1 0       P 1   P 1   P 0    P 1 ' ' R  n  K 0     n   n Pn 1   n 1 Pn 1 1   n 1 Pn 1 1 , '

'

where  n

 C      n 1

'

R



R



R

 e ;  n '

R  1 R 1

 C      n 1

R 1 R 1



 1  e 

…(11) …(12)

R 1 R 1

…(13)

and  n  C    e   n  C  .

The expressions for Pn ( 0 ) ( 0  n  C ) is derived using equations (4) & (5) and is given by, n

1  e   P0 0  , 0  n  C Pn ( 0 )  n!    e 

…(14)

The expression for Pn ( 0 ) ( C  n  r ) are recursively obtained from equation (6) as, n  n  c    1   1    e  (C !) .C  Pn ( 0 )   P0 0  , n   e ( n! )  Bi

Bi  1 

where

C  n  r

…(15)

i  C 1

i  C   C   e 

Equations (7) and (8) yield, n       e  ( C ! )   1 .C n  c   1  Pn ( 0 )   P0  0    n  r Pr  1 1  , r  1  n  R  1 n   e ( n! )  Bi

…(16)

i  C 1

where  1  1 ;  2   2 ;  i   i i 1   i i  2 , for i  3 ;  i 

 r  i 1   r  i 1 r i

and  i 

r i 2

Using equation (9), the expression for Pr 1 (1) in terms of P0 0  is given by,

r i

Pr 1 (1)  W . P0 0   e  W    e

where

R

.

…(17) (C !)

 1

( R!)



 R  c    1 

.C R

 B i . R  r

i  C 1

Putting the value of Pr 1 (1) from equation (17) in (16), we get, n  n  c    1   1    e  (C !) .C  Pn ( 0 )   P0 0    n  r W . P0 0  , r  1  n  R  1 n   e ( n! )  Bi

…(18)

i  C 1

From equations (10) and (11), we obtain Pn (1) as,

Pn (1)   n  rW .P0 ( 0 ) , '

r 1  n  R

…(19)

 r  i 1   r  i 1

'

'

'

'

'

'

'

'

r i

 r i2 '

'

where  2   2 ;  i   i i 1   i i  2 , for i  3 ;  i 

and 

'

i



r i

Equation (13) gives

Pn 1  [ n  rW  D n ] P0 0  , '

R  n  M

…(20)


38






where D R 1   R 1 R 1 r .W   R 1S ; D n   n D n 1   n D n  2 , for n  R  2 ; '

 e  and S    e

'

R 1

(C !)

 1

.C

'

'

 R  1  c    1 

( R  1! )

.

R 1



 Bi

i  C 1

Probability of System Being Empty The probability that system is in empty state i.e. P0 ( 0 ) can be calculated by using normalizing condition, R 1



K

 Pn (1)  1

Pn ( 0 ) 

n0

…(21)

n  r 1

so that we obtain P0 ( 0 ) 



1 e    n  0 n!    e  C

n

 e     n  C 1    e  r

n

(C !)

 1

( n! )

.C

 n  c    1  n



 Bi

i  C 1

 e   n  r 1    e  R 1



n



(C !)

 1

( n! )

.C

 n  c    1  n



…(22)

P0  0    n  r W

 Bi

i  C 1 R



 n  rW 

K

'

 n  r 1



  n  rW  D n '

n  R 1



1

Mean Queue Length We find the expected number of customers in the system as L





   e  ( n  1)!    e    n0 C

1

n

r





n

1 

n  C 1

   e      e

n

(C !)

 1

.C

(( n  1)! )

 n  c   

1 

n

 Bi

i  C 1

   1    n n  r 1   

n    e  ( C ! )  1 .C  n  c    1     n  n  rW n   e (( n  1)! )  Bi

R 1

i  C 1

R



 n  n  r W K



n  n  rW  '

n  r 1

'

 Dn

      

  P0 (0 )

…(23)

n  R 1

Expected Waiting Time By using the Little’s formula, we calculate the expected waiting time of the customer in the system as L E (w)  …(24)



where  is the effective mean arrival rate of the system and is given by R 1

C

 



n0

 Pn ( 0 ) 



 C    n  1   Pn ( 0 )   n  C  1

K



 C    n  1  1 Pn (1)  n  r 1 

Finite Population Model (FPM) The steady-state equations governing the FPM model are given by 0   M   e P0 0      e P1 0 

…(25)

0    M  n   e   n    e Pn 0    M  ( n  1)   e Pn 1 0   n  1   e Pn 1 0  1 n  C

0    n   n Pn 0    n 1 Pn 1 0    n 1 Pn 1 0  , C  n  r  1

,

…(26) …(27)


39


0    r   r Pr 0    r 1 Pr 1 0    r 1 Pr 1 0    r 1 Pr 1 1

…(28)

0    n   n Pn 0    n 1 Pn 1 0    n 1 Pn 1 0  , r  1  n  R  2

…(29)

0    R 1   R 1 PR 1 0    R  2 PR  2 0 

…(30)

0    r 1   r 1 Pr 1 1   r  2 Pr  2 1

…(31)

  ' ' 0     n   n Pn 1   n 1 Pn 1 1   n 1 Pn 1 1 , r  2  n  R  1 0       P 1   P 1   P 0    P 1 ' ' R  n  M 0     n   n Pn 1   n 1 Pn 1 1   n 1 Pn 1 1 , '

'

'

R

R

R

 C  where  n   M  n    n 1

R  1 R 1





R 1 R 1

…(32) …(33)

R 1 R 1

 C  '  e  ;  n   M  n    n 1



…(34)

 1  e 

and  n  C    e   n  C  Equation (25) provides, M   e  P1 0   P 0    e  0

…(35)

From equation (26), the expression for Pn ( 0 ) , ( 1  n  C ) is given by, n

 e   P 0  , 1  n  C Pn ( 0 )   M  n ! n!    e  0 M!

…(36)

The expression for Pn ( 0 ) ( C  n  r ) is recursively derived from equation (27) as follow, n  n  c    1   1    e  (C !) .C   Pn ( 0 )  P0 0  , C  n  r n  M  n !    e   ( n! )  Bi

M!

i  C 1

i  C  . C   e 

where B i  1 

…(37)

From equations (28) and (29), the expression for Pn ( 0 ) ( r  1  n  R  1 ) is obtained as, Pn ( 0 ) 

n  n  c    1   1    e  (C !) .C   P0 0    n  r Pr  1 1  , r  1  n  R  1 …(38) n  M  n !    e   ( n! )  Bi

M!

i  C 1

where 1  1 ;  2   2 ;  i   i i 1   i i  2 , for i  3 ;  i  and  i 

 r  i 1   r  i 1 r i

r  i 2 r i

Equation (30) gives the expression for Pr 1 (1) in terms of P0 0  as,

Pr 1 (1)  U .P0 0 

 e   where U   M  n !    e  M!

R

(C !)

…(39)  1

( R!)



.C

 R  c    1 

R

.

 B i . R  r

i  C 1

Substituting the value of Pr 1 (1) from (39) in equation (40), Pn ( 0 ) is obtained as,


40

Finite Controllable Markovian Model with Balking and Reneging (IJSTE/ Volume 2 / Issue 08 / 006) n  n  c    1   1    e  (C !) .C   Pn ( 0 )  P0 0    n  r U . P0 0  , r  1  n  R  1 n  M  n !    e   ( n! ) B  i

M!

…(40)

i  C 1

From equations (31) and (32), we obtain Pn (1) as,

Pn (1)   n  rU .P0 ( 0 ) , r  1  n  R '

…(41)

 r  i 1   r  i 1 '

where  2   2 ;  i   i i 1   i i  2 , for i  3 ;  i  '

'

'

'

'

'

'

'

r i

 r i2 '

and



'



i

r i

From equation (34), the value of Pn 1 is obtained as,

Pn 1  [ n  rU  H n ] P0 0  , '

R  n  M



…(42)

where H n   n H n 1   n H n  2 , n  R  2 ; H R 1   R 1 R 1 r .U   R 1G '

and

G 

'

 e    M  ( R  1) !    e  M!

R 1

(C !)

 1

'

'



 R  1  c    1 

.C

( R  1! )

.

R 1



 Bi

i  C 1

Probability of system being in empty state ( P0 ( 0 ) ) The probability P0 ( 0 ) can be calculated by using normalizing condition as

P0 ( 0 ) 



n

 e     M  n ! n!    e 

C

M!

 n0

n       e  ( C ! )   1 .C n  c   1    n  n  C  1  M  n !    e  ( n! )  Bi r

M!

i  C 1



R 1

   e     n !    e 

M!



n  r 1  M

n

(C !)

 1

( n! )

.C

 n  c  

1

  n  rU

n



 Bi

i  C 1

R

  n  rU 



'

n  r 1

K

 ( n  r U  H n ) '

1

…(43)

n  R 1

Mean Queue Length The expected number of customers in the system is given by L 



C

   e     M  n ! ( n  1)!    e  M!

 n0

n



r

 n  C 1

1 

M!   e   M  n !    e n

   

n

(C !)

 1

.C

(( n  1)! )

 n  c   

1

n

 Bi

i  C 1

 n  n  c    1  1   1  M !    e  (C !) .C  n     n n  rU     n  n  r  1   M  n !    e  (( n  1)! )  Bi  i  C 1  R 1



R

 n  r 1

n n  rU  '

K

      

 n ( n  r U  H n ) P0 ( 0 ) '

…(44)

n  R 1

IV. SOME PARTICULAR CASES In this section we discuss some special cases, which match with the earlier existing results by setting appropriate parameters.


41


1) Case I: When   0 i.e. there is no balking, the performance indices for finite capacity and finite population models reduce to the following forms; ForFP Mmodel:

  n  1  e    P0 0  ,  n!    e   n 1  e  Pn ( 0 )    P0 0  , n     e  C  n  c C  B i  i  C 1 n  1     e  P0 0   Q n  r V . P0 0  ,   n   e n  c  C C !  Bi  i  C 1 

1 n  C

C  n  r

r 1  n  R 1

Where Q1  1 ; Q 2   2 ; Q i   i Q i 1   i Q i  2 , for i  3 ;  i 

i 

r i 2

 e  and V    e

r i

R

1 R R  c  C C !  B i .Q R  r

…(45)

 r  i 1   r  i 1 r i

;

.

i  C 1

'   Q n  rV

. P0 ( 0 ) , r 1 n  R Pn (1)   '   ( Q n  r V  A n ) P0 0  , R  n  K

…(46)

 r  i 1   r  i 1 '

where Q 2   2 ; Q i   i Q i 1   i Q i  2 , for i  3 ;  i  '

'

'

'

'

'

'

'

r i

 r i2 '

and  also and

'

i



r i





An   n An 1   n An  2 ; AR 1   R 1Q R 1 r .V   R 1T , for n  R  2 R 1 .    e  1 '

  T       e 

'

C

 R 1 c  C!

'

'

R 1

 Bi

i  C 1

For FPM Model: In this case we obtain   n     e  M!     P0  0  ,   M  n ! n!     e   n    e  M! 1    Pn ( 0 )   P0  0  ,    e  n   M  n !  n  c    C !C  Bi  i  C 1 n    M !   e 1    P0  0    n  r H . P0  0   n   M  n !     e  C !C  n  c   B i  i  C 1 

1  n  C

…(47)

C  n  r

,r  1  n  R  1


42


 e   H   M  n !    e  M!

R

1 R R  c  C C !  B i . R  r

 r  i 1   r  i 1

i 

 i   i i 1   i i  2 , for i  3 ;

2  2 ;

1  1 ;

where

r i

;

i 

r i 2 r i

and

.

i  C 1

And  H . P0 ( 0 ) , r 1 n  R Pn (1)   n  r '  ( n  r H  E n ) P0  0  , R  n  M '

…(48)

 r  i 1   r  i 1

'

'

'

'

'

'

'

'



r i

; 

'

i

Also E n   n E n 1   n E n  2 , n  R  2 ; E R 1   R 1 R 1 r H   R 1 Z '

'

 e  and Z   M  ( R  1) !    e M!

  

R 1

'

1 R 1  R 1  c  C C !  Bi

 r i2 '

'

where  2   2 ;  i   i i 1   i i  2 , i  3 ;  i 

'



r i



,.

.

i  C 1

2) Case II: When C=1,   0 ,   0 , i. e. for single server queue without discouragement, We set i   for i  R

 λ1 for r  i  R and , i  R  L In this case our model reduces to the M/M/1 interdependent queueing model with controllable arrival rates. The same case for infinite capacity was discussed in [9]. We get results for FCM and FPM models as follow; FCM Model:      e   P0  0  , 1 n  1    e   n 1    e   Pn ( 0 )    P0  0  , 1 n  r n     e  ( n! )   B i  i2 n  1     e  P0  0   q n  r a . P0  0  , r  1  n  R  1   n     e   ( n ! ) B  i  i2 

where

 e  a     e  

R

1 ( R!)



…(49)

.

R

 B i .q R  r

i2

And  q ' a . P ( 0 ) , 0 Pn (1)   n  r '  ( q n  r a  d n ) P0  0  ,

r 1 n  R

…(50)

R  n  M

 r  i 1   r  i 1

'

'

'

'

'

'

'

'



r i

d n   n d n 1   n d n  2 , n  R  2 ; d R 1   R 1q R 1 r .a   R 1 f '

'

'

'



 r i2 '

'

where q 2   2 ; q i   i q i 1   i q i  2 , for i  3 ;  i 

; 

'

i



r i

;


43


 e  and f     e  

R 1

1 (( R  1)! )



.

R 1

 Bi

i2

FPM Model: In this model, we obtain     M   e  P 0  , 0    e   n    e  M! 1    Pn ( 0 )   P0  0  , n   M  n !    e  ( n! )   B i  i2 n     e  M! 1    P0  0    n   M  n !    e   ( n! )  B i  i2 

where

1  1;



2  2 ; '

'

 e   l   M  n !    e  M!

R

'

i

  i '

1 ( R!)



'

i 1

  i '

'

n 1

'

n  r l . P0



for i  3 ;

i2,

…(51)

1 n  r

0  , r

'

i



1  n  R 1

 r  i 1   r  i 1 r i

;

i  '

r i 2 r i

and

.

R

 B i .

'

Rr

i2

Also  l .P (0 ) ,  Pn (1)   n  r 0 "   ( n  r l  h n ) P0 0  , ''

r 1 n  R

…(52)

R  n  M

 r  i 1   r  i 1 '

where  2   2 ;  i   i  i 1   i  i  2 , for i  3 ;  i "

''

"

'' ''

'' ''

''



hn   n hn 1   n hn  2 , n  R  2 ; hR 1   R 1 "

"

 e   and p   M  ( R  1) !    e  M!

R 1

"

'

R 1 r .l

1 (( R  1)! )





R 1

r i   R 1 p "



 r i2 '

; 

"

i



r i

;

.

 Bi

i  C 1

3) Case III: When e=0,   0 ,   0 and   1 , then our model reduces to the classical finite population M/M/C model. 4) Case IV: When   0 ,   0 i.e. the customers behavior is not taken account and N   , our results of FCM model tally with those of Aftab and Maheswari [11].

V. DISCUSSION In this investigation we have developed finite controllable Markovian models with balking and reneging. The arrival rate influenced by the queue size is included which is further controlled by forward and backward preassigned threshold levels. The explicit formulae for queue size distribution, average number of customers in the system and expected waiting times are established. The concept of balking and reneging used in the system modeling makes our models closer to the real life situation.

REFERENCES [1]

Abou-El-alta, M.O. and Hariri, A.M.A. (1992): The M/M/C/N queue with balking and reneging. Computers and Operations Research, Vol. 19(8), pp. 713716.


44

Finite Controllable Markovian Model with Balking and Reneging (IJSTE/ Volume 2 / Issue 08 / 006) [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

Abou-El-alta, M.O. and Shawky, A.I. (1992): The single-server Markovian overflow queue with balking, reneging and an additional server for longer queues. Microelectron. Reliab., Vol. 32, pp. 1389-1394. Abou-El-alta, M.O. and Kote, K.A.M. (1992): A linearly dependent service rate for the queue M/M/1/N with genral balk function reflecting barrier, reneging and an additional server for longer queues. Microelectron. Reliab., Vol. 32(12). Aftab Begum, M.I. and Maheswari, D. (2002): The M/M/C interdependent queueing model with controllable arrival rates, OPSEARCH, Vol. 39(2) pp. 89110. Gross and Harris, C.M. (1974): Fundamental of Queueing Theory, John Wiley, New York. HAGHIGHI, A.M., Medhi, J. and Mohanty, S.G. (1986): On multi-server Markovian queueing system with balking and reneging. Computers and Operations Research, Vol. 13, pp. 421-425. Hillier, F. S. and Lieberman, G.J. (1992): Operations Research. Holden-Day Inc: San Francisco. Jain, M. (1998): Finite population loss and delay queueing system with nopassing. OPSEARCH, Vol. 35(3), pp. 261-276. Jain, M. (1998): M/M/m queue with discouragement and additional servers. Gujarat Statistical Review, Vol. 25(1-2), pp. 31-42. Jain, M. and singh, P. (2002): M/M/m queue with balking and reneging and additional servers. International Journal of Engineering. Srinivasa Rao, K., Shobha, T. and Srinivasa Rao, P. (2000): The M/M/1 interdependent queueing model with controllable arrival rates, OPSEARCH, Vol. 37(1). Washurn, A. (1974): A multi-server queue with nopassing. Oper. Res., Vol. 16, pp. 428-834. Ma Z. Y et al.,(2007): discrete time queueing system with two classes of customers and priorities. Sys. Engg. Theo. Pra. Vol. 27, pp.91-98. Jain M. and Sharma G.C.(2004): The controllable queue with balking and reneging. To reduce the balking behavior of the customers in the controllable queue, the provision of additional removable servers, Nepali Math. Sci. Report Vol.22, pp.113-120. Singh C.J et al. (2007): Single server interdependent queueing model with controllable arrival rates and reneging Pak. J. Statist. Vol.23, pp. 171-178. Jain M.et al. (2009): The controllable and interdependent rates for the machine re-pair problem (MRP) with additional repairman and mixed standbys Raj. Acad. Phy. Sci. Vol.8, pp. 447-456. Yang D.Y. et al.(2010): The optimization and sensitivity analysis for controlling the arrivals in the queueing system with single working vacation J. Comput. App. Math. Vol.234,pp. 545-556. Choudhury, A. and Medhi, P. (2011), Balking and reneging in multiserverMarkovian queuing system, International Journal of Mathematics in Oper. Res., Vol. 3, pp. 377-394. Mandelblaum, A. and Momcilovic, P. (2012) Queues with Many Servers and Impatient Customers, Mathematics of Operations Research, Vol. 37,pp. 14165. 20. Kumar,R.and Sharma, S.K.(2014): A Multi-Server Markovian Feedback Queue with Balking Reneging and Retention of Reneged Customers, Advanced Modeling and Optimization (AMO), Vol.16(2), pp.395-406.


45