DOI : 10.17577/IJERTV2IS70255
- Open Access
- Total Downloads : 439
- Authors : J. Ganeshsiva Kumar, M. Revathy
- Paper ID : IJERTV2IS70255
- Volume & Issue : Volume 02, Issue 07 (July 2013)
- Published (First Online): 12-07-2013
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Steady State Analysis of A Non-Markovian Bulk Queueing Model with Multiple Vacation , Accessible Batches, Setup Times with N-Policy and Closedown Times
J. Ganeshsivakumar 1, M. Revathy 2
1( Department of Mathematics, CMS college of Science and Commerce, Coimbatore)
2( Department of Maths(CA), Dr.N.G.P. Arts and Science College,Coimbatore)
Abstract
In this paper, a generalized non-Markovian bulk arrival service queueing system is considered with multiple vacations, setup times and closedown time. The service starts only if minimum of a customers are available in the queue. At the service completion epoch,
if the number of customers is ,where a d-1 (db)
then the server takes the entire queue for batch service and admits the subsequent arrivals for service till the service of the current batch is over, or the accessible limit d is reached, whichever occurs first. At the service initiation epoch, if the number of customers waiting in
the queue is atleast d (adb), then the server
takes min( ,b) customers for service and does not allow further arrival into the batch. On completion of a service, if the queue length is less than a, then the server performs a closedown work such as, shutting down the machine, removing the tools etc. Following closedown work, the server leaves for a vacation of random length irrespective of queue length. When the server returns for a vacation and if the queue length is still less than a, he leaves for another vacation and so
on until he finds atleast a customers waiting for service in the queue. That is, if the server finds atleast N customers waiting for service, then he requires a setup time R to start the service. After the setup he serves a batch of b customers, where ba. Various characteristics of queueing system and a cost model are presented.
Keywords: Markovian Bulk, Multiple Vacation, Setup time, Accessible Batches and Closedown times.
-
Introduction
Many researchers have concentrated on bulk service queueing models, in which once the service is started arriving customers, can not enter the service station though enough space is available to accommodate them. It can be observed in many practical situations that arriving customers will be considered for service with current batch in service with some restriction. The concept of non accessibility while receiving service, has been studied by Weiss[6], Sivasamy[7] analyzed a Markovian single arrival bulk service queue with accessible and non accessible batches. R. Arumuganathan and S. Jeyakumar[1] had given results for setup times with N-Policy. Sharma and Jain[8] obtained results for average queue length and waiting time distribution for state dependent Markovian single arrivl bulk service queueing system with accessible and non accessible batches. Sharma et al.[9] established the expression for average queue length for state dependent Mx/M(a,d,b)/1 queue with accessible and non accessible batches without vacations. In the literature, only less attention is given for general because of the complexity in getting a closed form solution.
In this paper, a generalized non-Markovian bulk arrival service queueing system is considered with multiple vacations, setup times and closedown time. The service starts only if minimum of a customers are available in the queue. At the service completion
epoch, if the number of customers is ,where a d-
1 (db) then the server takes the entire queue for batch
service and admits the subsequent arrivals for service till the service of the current batch is over, or the accessible limit d is reached, whichever occurs first. At the service initiation epoch, if the number of customers
waiting in the queue is atleast d (adb), then the
server takes min( ,b) customers for service and does not allow further arrival into the batch. On completion of a service, if the queue length is less than a, then the server performs a closedown work such as, shutting down the machine, removing the tools etc. Following closedown work, the server leaves for a vacation of random length irrespective of queue length. When the server returns for a vacation and if the queue length is still less than a, he leaves for another vacation and so on until he finds atleast a customers waiting for service in the queue. That is, if the server finds atleast
N customers waiting for service, then he requires a setup time R to start the service. After the setup he serves a batch of b customers, where ba.
Q a
Service Accessibility a n d-1
Non-Accessibility d n b
Q < a
Closedown work
Closedown work
-
Mathematical model
Let X be the group size random variable, be the Poisson arrival rate, gk be the probability that k customers arrive in a batch and X(z) be its probability generating function. Let S(.),V(.), R(.) and C(.) be the cumulative distributions of the service time, vacation time, setup time and closedown time, respectively. Let s(x), v(x), r(x) and c(x) be the probability density functions of service time, vacation time, setup time and closedown time respectively. S0(t), V0(t), R0(t) and C0(t) denote the remaining service tine of a batch, the remaining vacation time, setup time and closedown time of a server at an arbitrary time t, respectively. Let us denote the Laplace transform (LT)of s(x), v(x), r(x)
~ ~ ~ ~
and c(x) as S , V , R and C respectively.
The number of customers in the queue and the number of customers in service are denoted by Ns(t), Nq(t), respectively. The different states of the server at time t are defined as follows:
Y(t) = 0, if the server is busy with bulk service
= 1, if the server is doing closedown work
= 2, if the server is on vacation
and define Z(t) = j, if the server is on jth vacation starting from the idle period.
To obtain system equations, the following probabilities are defined. Let
Pij(x,t)dt = P{Ns(t)=i, Nq(t)=j, xS0(t)x+dt, Y(t)=0},
aib, j0,
which means that there are i customers under service, j
Multiple Vacations Q a
Setup job
Q < a
customers in the queue, the server is busy with remaining service time of x.
In a similar manner, it is defined,
Cn(x,t)dt = P{Nq(t)=n, xC0(t)x+dt, Y(t)=1},
n0,
Qjn(x,t)dt = P{Nq(t)=n, xV0(t)x+dt, Y(t)=2, Z(t)=j},
Fig 1: Schematic Representation of Queueing Model
For the proposed model, the probability generating function of the number of customers in the queue at an arbitrary time epoch is obtained using supplementary variable technique. The complexity of general service accessible batch queueing system involving LST of unknown probability functions is overcome by providing a recursive epoch. Expression
for expected queue length, expected length of idle
n0, j1, Rn(x,t)dt = P{Nq(t)=n, xR0(t)x+dt, Y(t)=1},
nN
-
Analysis
b
b
The steady state queue size equations are obtained as
P' (x) P (x) P (0)s(x) Q (0)s(x)
period, expected length of busy period and expected waiting time are derived. A cost model of the queueing system is discussed.
i0 i0
ia
-
Pik ,0
k 1
mi
md
(x)gk ,
li
l 1
a i d
(1)
P' (x) P
(x) P
(0)s(x)
~ ( ) P (0) ~ ( ) P
(0) ~( ) Q (0) ~( )
b
b
i 0 i 0
p>
mi
md
Pi0
b
b
io
ia
Pi 0
mi S
md
li S
l 1
Qli (0)s(x) ,
l 1
d 1 i b
(2)
Pik ,0 ( )gk ,
k 1
a i d
(15)
P' (x) P
-
P
(x)g
~ ( ) P
(0) ~ ( ) P
(0) ~( )
n
n
dn dn
k 1
d ,nk k
Pi0 i 0
Pi0
mi S
b
b
md
d a
-
P
-
(x)g
, n 1
(3)
Q (0) ~( ) ,
d 1 i b
(16)
d k ,0
k 1
k n
li S
l 1
n
n
P' (x) P (x) P
(x)g ,
~
~
~
in in
k 1
i,nk k
Pdn ( )
Pdn (0)
Pdn ( )
n
n
k 1
Pd ,nk (
) gk
d i b 1, n 1
b
(4)
d a
~
~
Pd k ,0
( )g
k n ,
n 1
(17)
P' (x) P
(x) P (0)s(x) Q
(0)s(x)
k 1
bn bn
m,b n
l ,b n ~
~ n ~
md
n
n
l 1
P ( ) P (0) P ( ) P
( )g ,
P (x)g ,
n 1
(5)
in in
in i,nk k k 1
b,nk k
k 1
d i b 1, n 1
(18)
b
b
C ' (x) C (x) P
(0)C(x)
(6)
~
~
~ ~
0 0 m0
ma
Pbn ( )
Pbn
Pbn ( )
b
b
md
Pm,bn (0)S ( ) Ql ,bn (0)S ( )
l 1
b
b
C ' (x) C (x) P
(0)C(x)
~
n n
n
n
-
C
mn
md
(x)g ,
1 n a 1
(7)
~
Pb,nk (
n
n
k 1
~
) gk , n 1
b
b
~
(19)
k 1
nk k
C0 ( )
C0 (0)
C0 ( ) Pm0 (0)C( )
ma
(20)
n
n
C ' (x) C (x) C
(x)g ,
n a
(8)
~
~ ~
n n
k 1
nk k
Cn ( )
Cn (0)
Cn ( )
b
b
md
Pmn (0)C( )
Q' (x) Q (x) C (0)v(x)
(9) n ~
10 10 0
C ( )g ,
1 n a 1
(21)
Q' (x) Q (x) C (0)v(x)
nk k
1n 1n n
n
n
k 1
n
n
~ ( ) C (0) ~ ( ) ~
( )g ,
k 1
Q1,nk (x)gk ,
n 1
(10)
Cn n
Cn
k 1
Cnk k
-
Q
-
Q
j 0
j 0
j 0
j 0
' (x) Q
-
Q
j1,0
(0)v(x),
j 2
(11)
~
n a
~ ~
(22)
Q' (x) Q
-
Q
(0)v(x)
Q10 ( ) Q10 (0) Q10 ( ) C0 (0)V ( ),
(23)
jn jn
n
n
j 1,n
~ ( ) Q (0) ~ ( ) C (0) ~( )
-
Qj ,nk (x)gk ,
-
-
1 n a 1, j 2
(12)
Q1n
1n Q1n n V
n ~
k 1
Q1,nk ( )gk ,
n 1
(24)
Q' (x) Q
(x) Q
(x)g ,
k 1
~ ~ ~
n
n
jn jn
k 1
j,nk k
Qj 0 ( ) Qj 0 (0) Qj 0 ( ) Qj 1,0 (0)V ( ),
n a, j 2
n n N
(13)
~ ( ) Q
(0) ~
( ) Q
j 2
(0) ~( )
(25)
R' (x) R (x) Q (x)r(x) R
(x)g ,
Qjn
jn Qjn
j 1,n V
n n
k 1
j ,n
n N
k 1
nk
k
(14)
n
n
k 1
~
Qj ,nk
( )gk ,
1 n a 1, j 2 (26)
Qjn
Qjn
Taking LT on both sides of the equations (1)- (14), we get
~
Qjn
( ) Qjn
(0) ~
n
n
( )
k 1
~
Qj ,nk
( )gk ,
n a, j 2 (27)
~
~
n N
~
~
( X (z)) ~(z, ) C(z,0)
Rn ( )
Rn (0)
Rn ( )
n
n
l 1
Ql ,n
(0)R( )
k 1
Rnk (
) gk ,
C
~ b
a1 b
n n
-
-
-
Queue Size Distribution
n N
(28)
C( ) Pm0 (0)z
ma
Pmn (0)z
n1 md
(32)
To obtain the probability generating function of the queue size at an arbitrary time epoch, the following probability generating functions are defined.
Multiplying the equation (15) by z0 with i=d, (17) by zn (n 1) and taking summation from n=0 to and using (29), we get
~ ~ b
~ ij
( X (z))Pd (z, ) Pd (z,0) S ( ) Pmd (0) Qld (0)
Pi (z, ) Pij ( )z
j0
d 1 ~
md
i
l 1
ij
ij
Pi (z,0)
j0
P (0)z j ,
d i b
z d
d 1i
ia
Pi0 ( )z
* X (z) Gi (z)
(33)
~ ~ n
where Gi(z)= g z k
k
k
Qj (z, ) Qjn ( )z k
n0
Qj (z,0) Qjn
n0
(0)zn ,
j 1
(29)
Multiplying the equation (16) by z0, (18) by zn (n 1) and taking summation from n=0 to and using (29), we get
~ ~ n
C (z, ) Cn ( )z
~ ~ b
n0
( X (z))Pi (z, ) Pi (z,0) S ( ) Pmi (0) Qli (0)
C(z,0) C (0)z
C(z,0) C (0)z
n
n
n0
md l 1
d 1 i b 1 (34)
~ ~ n 0 n
R(z, ) Rn ( )z
Multiplying the equation (16) by z with i=b, (19) by z
n N
(n 1) and taking summation from n=0 to and using
R(z,0)
Rn
(0)zn
(29), we get
~
n N
( X (z)) ~ (z, ) P (z,0) S ( )
Multiplying the equation (23) by z0 and (24) by zn
Pb b
b
zb
b1
(n 1) taking summation from n=0 to and using
P (z,0) P
m
m
(0)zn
(29), we get
md
mn
n0
(35)
b1
~ ~ Q (z,0) Q
(0)zn
( X (z))Q (z, ) Q (z,0) V ( )C(z,0)
(30)
l
ln
1 1
l 1
n0
Multiplying the equation (25) by z0 , (26) by zn (1 n a-1) and (27) by zn (n a) taking summation
Multiplying the equation (28) by z0 (n 1) and taking summation from n=0 to and using (29), we get
from n=0 to and using (29), we get ~ ~
( X (z)) ~ (z, ) Q (z,0)
( X (z))R(z, ) R(z,0) R( )
Qj j
~ a n
b Q (z,0) N Q (0)zn
(36)
V ( )Qj 1,n (0)z
n0
(31)
l
l 1
ln
n0
Multiplying the equation (20) by z0 , (21) by zn (1 n a-1) and (22) by zn (n a) taking summation from n=0 to and using (29), we get
Substituting X (z) in (30) through (36), we get
Q (z,0) ~( X (z))C(z,0)
(37)
1 V
~ ~ a1 n
( ~( X (z)) ~( ))
Q (z,0) V ( X (z))Q (0)z
(38) V V
j
n0
j 1,n
a1
Q
~
j 1,n
(0)z n
d 1 P
(0)z 0
Qj (z, )
n0 ,
X (z)
j 2
(45)
~ m0
C(z,0) C( X (z))ma
a1 b
n0 md
Pmn
-
zn
(39)
From the equations (32) and (39), we get
P (z,0) ~( X (z))b P
(0) Q
(0)
~(z, ) ~( X (z)) ~(z, )
d S
d ~
md
md
ld
l 1
C C
d 1 P
C
1
1
-
z 0
z
z
-
d
-
-
ia
Pi 0
( X (z))z i *[ X (z) G (z)] (40)
m0
i
i
ma
a1 b
*
X (z)
(46)
k
k
d 1i
where Gi(z)= gk z
Pmn
n0 md
(0)zn
k From the equations (33) and (40), we get
P (z,0) S~( X (z))b P (0) Q (0) ~ ~
i mi
md
li
l 1
(S ( X (z)) S ( ))
d 1 i b 1
(41)
b
~
Pmd
(0) Qld (0)
d
d
P (z, ) md
l 1
d ~
i
i
b1
b b1
n
n
Pi0 ( X (z))z
Pm (z,0) Pmn (0)z z ia
d
P (z,0) ~( X (z)) md
md n0
*[ X (z) Gi (z)]
b S
b1
Q (z,0) Q
(0)z n 1
l
l 1
ln
l 1 n0
* X (z)
(47)
*
*
1
S
S
zb ~( X (z))
(42)
From the equations (34) and (41), we get
~
N 1
n
(S~( X (z)) S~( ))b P
(0) Q (0)
R(z,0) R( X (z))(Ql (z,0) Qln (0)z
(43) ~
mi
li
l 1
n0
P (z, ) md l 1
From the equations (30) and (37), we get
i X (z)
d 1 i b 1
(48)
~
~
Q1 (z, )
(V~( X (z)) ~( ))C(z,0)
V
V
X (z)
(44)
From the equations (35) and (42), we get
From the equations (31) and (38), we get
( ~( X (z)) ~( )) (~( X (z)) 1)
S S S
m P (z,0)
m P (z,0)
P (0)z
P (0)z
n
n
b1 P (z,0) b
b
b1
mn
b
~
Pmd
(0) Qld
(0)
md
md n0
P (z,0) md
l 1
d
d
b1
d ~
Q (z,0) Q
(0)z n
P ( X (z))z i
l
l 1
ln
l 1 n0
d i 0
z
z
ia
1
S
S
zb ~( X (z))( X (z))
(49)
*[ X (z) Gi (z)]
* 1
X (z)
(54)
From the equations (36) and (43), we get
~ b
~ (S ( X (z)) 1) Pmi (0) Qli (0)
P (z,0) md l 1
R
R
[R~( X (z)) ~( )]i X (z)
(Q (z,0) N 1 Q (0)zn
d 1 i b 1
(55)
~ l
ln
S
S
R(z, ) l 1 n 0
( X (z))
(50)
( ~( X (z)) 1)
b1 P (z,0) b b 1 P (0)zn
Substituting =0, in equations (44) through (50), we get
P (z,0) m
mn *
b md md n 0
b 1
n
~ Ql (z,0) Qln (0)z
~ (V ( X (z)) 1)C(z,0)
l 1
l 1 n 0
Q1 (z,0)
X (z)
(51)
1
zb S~( X (z))( X (z))
(56)
V
V
( ~( X (z)) 1)
[ ~( X (z)) 1]R
R
a1
Q
(0)z n
(Q (z,0) N 1
n
~
~
j 1,n
l
Qln (0)z
Q (z,0) n0 ,
j 2
(52)
~ l 1 n0
j X (z)
R(z,0)
( X (z))
(57)
d 1 P
(0)z 0
Let P(z) be the probability generating function of the queue size at an arbitrary time epoch. Then,
~ ~
m0
C (z,0) C ( X (z)) 1 ma
a 1 b P
(0)z n
d 1 ~ ~
b1 ~ ~
mn
n0 md
P (z) Pi (0) Pd (z,0) Pi (z,0) Pb (z,0)
ia id 1
* 1
(53)
~(z,0) ~ (z,0) ~(z,0)
(58)
X (z)
C Qj R
j 1
Using the equation (51) through (57) in (58), we get
S
S
( ~( X (z)) 1)
b
b
md
Pmd
-
Qld
l 1
(0)
pi Pmi (0),
md
d ~
P
( X (z))z i
qi Qli (0),
z
z
d i 0
i a
l 1
c p q ,
i
i
d 1 ~
*[ X (z) G (z)]
i i i
P (z) Pi 0 (0)
~
~
~
~
d 1
i a
X (z)
(S , z)
~
i a d 1
Pi 0 (0),
~
~ i d
~ b
(S , z) (Pi 0 ( X (z)) Pi 0 (0))z
( X (z) Gi (z))
b1
(S ( X (z)) 1) Pmi (0) Qli (0)
S
S
d
d
i a
~
md l 1
X (z)
f (S , z) ~( X (z))c
i d 1
d 1
~
~
-
P
-
( X (z))z i ( X (z) G (z))
(60)
~ z d
i 0 i
i a
(S ( X (z)) 1)
b1 ~
b1
b b1
S ( X (z))cm
P (z,0) P
(0)z n
md 1
m
md
mn
md n0
V~( X (z)) ~( X (z)) a 1 P z n
b1
n
C n
Ql (z,0) Qln (0)z
n0
l 1 l 1 n0
V~( X (z)) a1 q
b1
z n c z n
S
S
S
S
z b ~( X (z))( X (z))
n n0
n n0
d 1 P
(0)z 0
u(z,0) (z b ~( X (z)))R(z,0)
~ m0
Using the equation (60) and (59) is simplified as
C ( X (z)) 1 ma
a1 b P
(0)z n ~ ~
mn
(zb S ( X (z)))( X (z)) (S , z)
n0 md
~ ~
X (z)
(Z b S ( X (z))) (S , z)
~ ( ~( X (z)) 1) f ( ~, z) ( ~( X (z))
(V ( X (z)) 1)C(z,0)
X (z)
S S V
~ a1 n
C ( X (z)) 1)( Pn z )
n0
( ~( X (z)) 1)
(zb ~( X (z))) ( ~( X (z)) 1)
V
a1
n
S
b ~
V
a1
n
Qj 1,n (0)z
(z
S ( X (z)))qn z
n0
P(z)
n0
~
X (z)
( X (z))(zb S ( X (z)))
[R~( X (z)) 1]U (z,0) (61)
(Q (z,0) N 1 Q (0)z n
In P(z) functions (~, z) , (~, z) and
l
ln ~ S S
Let
l 1 n0
( X (z))
(59)
f (S , z) which involve LST of the unknown functions ~ (0) are present. While modelling a
Pi0
Pi0
non- accessible batch service queue, such complexity will not occur. In order to resolve
this complexity,
~ ( ) are expressed in terms of
n n1
Pi0
~
Qln (0)
j ni j pn
known function S ( ) .
-
Expected length of Idle Period.
i0 j0
a1
P(U 0) 1 Qln (0)
n0
Expected Idle Period can be defined as the
a1 n n1
mean time gap between the completion epoch of a
1 j ni j pn
(63)
service and initiation epoch of the next service, inclusive of multiple vacations and closedown period. Let I be the random variable idle period. The expected length of idle period is given by E(I)=E(I1)+E(C) where I1 is the random variable denoting idle period due to
multiple vacation process and E(C) is the expected
n0 i0 j0
where i, i are the probabilities of i customers arrive during vacation and closedown time. Using (62) and (63), the expected idle period E(I) is obtained as
E(V )
closedown time.
E(I )
a1
n n
E(C)
Let U be the random variable defined by
1 j ni j
U = 0, if the server finds at least a customers after the first vacation
1, if the server finds less that a customers after
n0 i0 j0
E(R) (64)
the first vacation
then, using conditional expectation, the expected length of idle period E(I1) is given by
E(I1) = E(I1/U=0)P(U=0)+ E(I1/U=1)P(U=1)
= E(V)P(U=0)+(E(V)+E(I1))P(U=1)
Where E(V) is the mean vacation time. Solving for E(I1), we get
E(I1) = E(V)/[1-P(U=1)] = E(V)/[P(U=0)] (62)
To find P(U=0), we do some algebra in the equations
(29) and (37) we get
Q (z,0) Q (0)zn
-
Expected length of Busy Period
In this section, the expected length of busy period which is useful to find the overall cost of the system is derived. Using the conditional expectation concept, the expected length of busy period is derived as follows:
Busy period is defined as the time interval from the moment when the server starts serving the queue, after returning from a vacation until the server leaves the system for another vacation.
Let B be the random variable for busy period. Define another random variable J as
J=0, if the server finds less than a customers in the queue
after first service
1, if the server finds atleast a customers in the
1 ln
n0
~ ~
a1
queue
after first service
V ( X (Z )) C ( X (z)) p zn
n
n
n0
zn
j0
a1
j
j
n
n
z j p
n0
zn
n
n0
Now the expected length of busy period E(B) is
E(B) = E(B/J=0)P(J=0)+E(B/J=1)P(J=1)
= E(B/J=0)P(J=0)+(E(B)+E(S))P(J=1)
= E(B/J=0)P(J=0)+(E(B)+E(S))(1-P(J=0))
equating the coefficient of zn (n=0,1,2,….,a-1) on both sides, we get
Where E(S) is expected service time. Solving for E(B), E(B) = E(S)/P(J=0)
is obtained.
a1
=E(S)/ Pi (65)
i0
C2= X2E(C)+2X2 E(C2)
1
1
7. Expected waiting time
The expected waiting time is obtained by using the Littles formula as E(W)=E(Q)/E(X), where E(Q) is
6. Expected Queue Length
The expected queue length E(Q) (i.e., average number of customers waiting in the queue) at an arbitrary time epoch is obtained by differentiating p(z) at z=1 and is
expected queue length as in (65).
-
Cost Model
given by E(Q)= npn
n0
p1 (1) . From the equation
Cost Analysis is an important phenomenon in any system. In this section, the total average coast of the
queueing system is derived with the following
(61) using LHospitals rules and evaluating the limit,
assumptions.
lim
d p (z) , we get
z1 dz
Cs : Start up cost per cycle.
Ch : Holding cost per customer per unit time C0 : Operating cost per unit time
E(Q)
'' 4 X
[k112 * k 2] 82 X 2 (b S )2
Cr : Reward due to vacation per unit time Cu : Closedown cost per unit time.
1 1 1
1
1
[ (X1 )k3 k 4] 2(2 X 2 )
The length of cycle is the sum of the idle period and
busy period. From the equations(63) and (64), the expected length of cycle, E(Tc) is obtained as
a1 a1
[X1 (2V1 nqn V2 qn )E(Tc ) E(I ) E(B)
(X
n0
a1
)(V q )]
n0
a1 n
E(V )
n
E(C)
2 1 n
n0
1
j ni j pn
where
2(2 X 2 )
(65)
n0 i0 j0
1
1
E(R)
a1
a1
E(S )
pi
n0
(66)
k1 = {4X1(b-S1)}{2S2.f+3S1f}
k2 = (S1.f)[( X1)(b(b-1)-S2)+( C1X2)(b-S1)]
a1
k3={(2V1C1+V2+C2)( pn )+(V1+C1)
The total average cost per unit is given by Total average cost = start up cost per cycle
+ holding cost of number of
a1
n0
a1
customers in
( npn )+(V1+C1)( pn )}
the queue
n0
a1
n0
+ operating cost*
+ closedown time cost
k4={(V1+C1)( pn )(X2)}
n0
S1=X1E(S)
1
1
S2= X2E(S)+ 2X2 E(S2)
+ setup cost per cycle
-
reward due to vacation per unit Time
X1=E(X) X2=X(1) V1= X1E(V)
V2= X2E(V)+ 2X21E(V2)
C1= X1E(C)
C
C
-
C
-
C
E(V )
s r P(U 0)
Total average cos t Cu E(C) Cr E(R)
E(Tc )
-
Gross, D and Harris, C.M.(1998): Fundamentals of queueing theory, John Wiley & Sons, New York.
-
Kelinrock, L.(1975): Queueing systems.Vol.1 Theory, John Wiley, New York.
-
Ch E(Q) C0
(67)
-
-
Medhi, J.(2002):Stochastic Models in Queueing
-
Where = E(X)E(S)/b and E(Tc),E(Q) are given in
(66) and (65) respectively
-
-
References
-
R.Arumuganathan and S.Jeyakumar (2005): Steady state analysis of a bulk queueing with multiple vacations, setup times with N-Poilcy and closedown times, Applied Mathematical Modelling 29 pp 972-986.
-
Kendal, D G (1951):Some problems in the theory of queues, J.Roy.Statist.Soc B13,pp 1511-185.
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Cox, D R (1955):The analysis of non-markovian stochastic process by the inclusion of supplementary variables, Proc.Camphil.S0C.51 pp 433-441.
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Cox, D R and Miller H D (1965): The theory of stochastic processes, section 4, wiley, New York.
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Lee, H W and Lee, S S(1991): A batch arrival queue with different vacations, Comput and oper. Res., Vol.18, No.1,pp51-58.
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Weiss, H J (1979): The computation of optimal control limits for a queue with batch service, Mgmt. Sci., Vol.25, pp 137-142.
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Sivasamy, R.(1990): A bulk service queue with accessible and non accessible batchs, opsearch, Vol.27, pp 957-967.
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Sharma, G C and Jain, M.(1991): State dependent bulk queueing system with accessible and non accessible batches, Proc. Of ORSI, New Delhi,pp113.
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Sharma, G C., Sharma, R and Jain(2002):Mx/M(a,d,b)/1 queue with state dependent bulk service of accessible and non accessible batches, Operations Researach, IT and Industry, Y K Publishers, New Delhi.
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Arumuganathan,R and Jeykumar,S.(2004): Analysis of a bulk queue with multiple vacations and closedown times, Int.J.Informatiion and Mgmt. Sci, Vol.15, No.1,pp 45-60.
Theory, Second edition, Academic press, USA
-
Ross,Sheldon,M(2001): Introduction to probabilistic models, Seventh edition, Academic Press.
-
Saaty, T.L.(1961):Elements of queueing theory with applications, Mc Graw Hill Book CO., New York.
-
Kishore S Trivedi, Probability and Statistics with reliability, Queueing, and Computer Science Applications, Prentice-Hall of India, 1982.