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Batch Arrival Queueing System Under J - Working Vacation
Policy
M. I, Afthab Begum
1, Fijy Jose P
2 1Professor, 2Research Scholar, Department of Mathematics, Avinashilingam University, Coimbatore, Tamil Nadu, India
Abstract-This paper analyses an MX/G/1 queueing system in which the server takes at most J working vacations during the idle period. We derive the Probability Generating Functions of the system using supplementary variable technique in which the remaining service times are introduced as supplementary variables. The model under consideration is a generalization of the single and multiple working vacation queueing models. The steady state results of the single and multiple working vacation models are derived from the present model as particular cases. Moreover, some important performance measures of this model are also presented with some numerical examples.
Keywords- J working vacation, MX/G/1, Probability generating functions, Remaining service time Supplementary variables.
I. INTRODUCTION
The queueing systems with vacations have been well studied because of their applications in modeling the computer networks, communication and manufacturing service systems. In classical vacation queues server stops serving customers during vacation. Servi and Finn (2002) extended the Basic Vacation Queueing system (BVQ) to the Working Vacation Queueing system (WVQ), where the server serves customers at a lower service rate instead of completely stopping service. Obviously WVQ is a generalization of BVQ and is motivated by real waiting time system. The introductory paper by Servi and Finn(2002) and a survey paper by Tian et al. (2009) contain respectively, Internet Protocol Access Network and Internet Protocol Router as applications of the working vacation queueing models in the field of computer science. Jemila Parveen and Afthab Begum (2011) first analysed M / G / 1 /WV queues using supplementary variable technique. Due to the wide applications in the performance analysis of communication and computer systems, the discrete-time queues with various working vacation policies have been considered in the works of several authors mentioned in Tian et al. (2009) . M / G / 1 queue with exponential working vacation is studied by Ji-hong Li et al. (2009) using matrix-analytic approach.
Julia Rose Mary and Afthab Begum (2009, 2010) have analysed the batch arrival queueing system MX / M/ 1 under both multiple and single working vacation and bulk service queueing system M / M(a, b) / 1 under multiple working vacation and derived the steady-state results. Recently Shan Gao and Yunfei Yao (2013) analysed MX / G/ 1queue with randomized working vacations and at most J vacations.
The aim of the present chapter is to analyze a more general bulk arrival queueing system MX/G/1 under J-working vacation. The server works at normal service rate when the customers present in the system and leaves the system for vacation when the system empties. After returning from vacation, if the server finds at least one customer in the system, he immediately starts the normal service. Otherwise he repeats at most J- vacations with probability until the system contains at least one customer. At the end of the Jth vacation if the server finds the system still empty, he joins the system and stays idle until a customer arrives. During vacation, the server works at a lower rate. The system is studied in steady-state and the steady state system size probabilities are derived for the model. The system performance measures are also derived and through numerical analysis the influence of the parameters on various performance measures is illustrated. The results of MX/G/1 queueing model with server‟s multiple and single vacation are derived as particular cases.
II. MODEL DESCRIPTION
A.Arrival Pattern
Consider a batch arrival MX/G/1 queue with multiple vacations under exhaustive service rule, such that the server works with different service rates rather than completely stopping service during a vacation period. The arrival
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B.Regular Busy PeriodThe server serves the customers one at a time and the normal service time Sb during the regular busy period follows a general distribution. The normal service continues until the system becomes empty.
C.Working Vacation Period
When the system becomes empty at a normal service completion instant, the server begins at most J randomized working vacations. If customer arrive during vacation,the server serves the customers at a lower rate. At a vacation completion instant, if there are customers in the system, the server will come back to the normal working mode and restart to serve the interruptive customer and a regular busy period begins. Otherwise, if no customers are in the system at the end of the vacation,the server either joins the system and remains idle with probability p to serve the arriving customers in regular mode or leaves for another working vacation with probability
p
1
p
.This pattern continues until the number of working vacations reaches J.If the system is empty by the end of the Jth vacation, the server joins the regular service channel and remains idle in the system. Whenever a new batch of customers arrives at the server idle state, the server immediately starts normal service for the arrivals. The vacation times are assumed to be independently and identically distributed randomvariables and follows exponential distribution
1
e
ηt. It is assumed that inter-arrival times, service times and working vacation times are mutually independent of another. The service discipline assumed to follow is first come first served.The general distribution of service times during regular busy period and working vacation period are respectively denoted by Sb(x)=Pr(Sb<x) and Sv(x)=Pr(Sv<x).The Laplace Stieltjes Transform (LST) and kth moment about the origin of the distributions are given by
d
e
)
θ
(
S
0 t θ *
b
)
t
(
S
b
S
(
θ
)
e
d
0 t θ *
v
)
t
(
S
vd
t
)
S
(
E
0 k k
b
)
t
(
S
bE
(
S
)
t
d
0 k k
v
)
t
(
S
vThis model is denoted by MX/G/1/JWV.
The state system size equations under the steady-state condition are analyzed using supplementary variable technique. The remaining service times are introduced as supplementary variables and the following notations are used to discuss the model.
III. MATHEMATICAL ANALYSIS OF THE MODEL
Notations
N (t) : The system size at time t
: Group arrival rateX : Group size random variable gk : Pr(X = k), k = 1, 2, 3, …
X(z) : Probability generating function of X.
Let
S
0v(
t
)
andS
0b(
t
)
denote the remaining service time during working vacation period and regular busy period respectively at time t.Further the server states are denoted by Y(t) at time t.Let Y(t) = 0 if the server is in idle state in system 1 if the server is in regular busy state
i+1,i=1,2…J if the server is on ith working vacation state.
At an arbitrary time t, the state of the system can be
described by the Markov process {N(t),Y(t),
S
0b(
t
)
,)
t
(
S
0v ,t≥0}.Now the system state probabilities at time t are defined as follows:
Let
R
0(
t
)
Pr(
N
(
t
)
0
,
Y
(
t
)
0
)
)
1
i
)
t
(
Y
,
dt
x
)
t
(
S
x
,
n
)
t
(
N
Pr(
)
t
,
x
(
Q
0 v i
, n
n≥1
)
1
)
t
(
Y
,
dt
x
)
t
(
S
x
,
n
)
t
(
N
Pr(
)
t
,
x
(
P
0 b n
n≥1
),
1
i
)
t
(
J
,
0
)
t
(
N
Pr(
)
t
(
Q
o,i
i
1
toJ
Thus
Q
n,i(
x
,
t
)
denotes the probability that there are n(≥1) customers in the system at an arbitrary epoch with the remaining service time during ith working vacation lies between x and x+dt ,andQ
o,i(
t
)
denotes the probability that the server is idle in ith vacation at time t.)
t
,
x
(
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Further,Q
n,i(
0
)
(
P
n(
0
)))
represents the probability that there are n customers in the system at the departure epoch during ith working vacation period (service completion instant during regular service).IV. THE SYSTEM SIZE DISTRIBUTION
Assuming the steady-state probabilities
)
t
,
x
(
Q
lim
)
x
(
Q
n,it i , n
)
t
,
x
(
P
lim
)
x
(
P
n t n
andQ
lim
Q
0,i(
t
),
t i , 0
ThusJ
i
1
,n
1
andR
lim
R
0(
t
)
t 0
exist andindependent of time t, the following steady-state equations are obtained for the queueing system using supplementary variable technique.
A. Idle In System
J 0 1
J
1
i 0,i
0
η
p
Q
η
Q
R
λ
(1)
B. Idle During Vacation)
0
(
Q
)
0
(
P
Q
)
η
λ
(
01
1
11(2)
)
0
(
Q
Q
η
p
Q
)
η
λ
(
0i
0,i1
1i(
2
i
J
)
where
p
1
p
(3)
C.Working Vacation State
1 n 1k n k,i k
1 , n v i , 1 n v n i 0 ni ni
g
)
x
(
Q
)
δ
1
(
λ
)
x
(
s
)
0
(
Q
)
x
(
s
g
Q
λ
)
x
(
Q
)
η
λ
(
)
x
(
Q
dx
d
(4)
for
n
1
,1
i
J
andδ
n,1 denotes the kroneckerDelta which represents1 ifn=1 and 0 if
n
2
D. Regular Busy State
k 1
n
1
k n k
1 , n b 1 , 1 n b n 0 J 1 i 0 b ni n n
g
)
x
(
P
)
δ
1
(
λ
)
x
(
s
)
0
(
P
)
x
(
s
g
R
λ
)
x
(
dys
)
y
(
Q
η
)
x
(
P
λ
)
x
(
P
dx
d
n
1
(5)
Taking the LST of equations (1), (4) and (5) the following are obtained as
J 0 1
J
1
i 0i
0
η
p
Q
η
Q
R
λ
(6)
k 1 n 1 k * k n 1 , n * v i , 1 n * v n i , 0 * i , n i , n * i , n
g
)
θ
(
Q
)
δ
1
(
λ
)
θ
(
S
)
0
(
Q
)
θ
(
S
g
Q
λ
)
θ
(
Q
)
η
λ
(
)
0
(
Q
)
θ
(
Q
θ
1
n
,
1
i
J
(7)
k 1 n 1 k * k n 1 , n * b 1 , 1 n * b n 0 * b J 1 i 0 i , n * n n * n
g
)
θ
(
P
)
δ
1
(
λ
)
θ
(
S
)
0
(
P
)
θ
(
S
g
R
λ
)
θ
(
dyS
)
y
(
Q
η
)
θ
(
P
λ
)
0
(
P
)
θ
(
P
θ
1
n
(8)
where
P
(
θ
)
e
P
n(
t
)
dt
0t θ *
n
and
dt
)
t
(
Q
e
)
θ
(
Q
n,i0 t θ * i ,
n
V. STEADY –STATE SOLUTIONS
The following probability generating functions are defined for |z|≤1 to derive the system size distribution of the model n 1 n * i , n *
i
(
z
,
θ
)
Q
(
θ
)
z
Q
n
1
n n,i
i
(
z
,
0
)
Q
(
0
)
z
Q
n 1 n * n *
z
)
θ
(
P
)
θ
,
z
(
P
n 1 n n
z
)
0
(
P
)
0
,
z
(
P
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Following the algebraic manipulation, the expressions for the partial PGFs of the system size for the present J-working vacation model are listed below)
Hz
1
(
)
Hz
1
(
Hz
Q
1 1 J 1 1 J 1i 0i
p
η
)
0
(
P
1 where))
z
(
w
(
h
η
P
Hz
1 x 1
(9)
)
Hz
1
(
)
Hz
Hz
(
))))
z
(
w
(
h
(
S
z
)))(
z
(
w
(
h
(
η
p
))))
z
(
w
(
h
(
S
1
))(
z
(
X
)
z
(
X
(
z
λ
)
0
(
P
)
0
,
z
(
Q
)
0
,
z
(
Q
1 1 J 1 1 x * V x x * v 1 1 J 1 i * i *
(10)
where
1 k k kz
g
)
z
(
X
,
w
x(
z
)
=
λ
(
1
X
(
z
))
and))
z
(
X
1
(
λ
η
))
z
(
w
(
h
x
λ
)
0
(
P
)
z
(
ψ
R
0
1 1 where)
Hz
1
(
p
)
Hz
Hz
(
p
)
Hz
1
(
Hz
)
z
(
ψ
1 J 1 1 J 1 1 1
(11)
))
z
(
w
(
S
z
))(
z
(
w
(
)))
z
(
w
(
S
1
(
z
)
0
,
z
(
P
x * b x x * b *
ψ
(
z
)
X
(
z
)
1
)
)
0
(
P
1 1
))
z
(
w
(
h
(
S
z
))(
z
(
w
(
h
(
p
))
z
(
X
)
z
(
X
))(
z
(
w
(
h
(
S
1
(
z
λ
x * v x 1 x * v
)
Hz
1
(
Hz
Hz
1 1 J 1 1(12)
Therefore, the total PGF of the system size probabilities
)
z
(
P
JWV of the model can be obtained. By adding equations (9 to 12)(ie)
P
(
z
)
R
JQ
Q
(
z
,
0
)
P
(
z
,
0
)
1 i * * oi 0
JWV
Equations (9 to 12) show that the partial generating function are expressed in terms of unknown P1 (0). And P1 (0) can be calculated from the normalizingcondition.
1
)
z
(
P
lt
JWV 1 z
))
0
,
z
(
P
)
0
,
z
(
Q
(
lt
Q
R
1
J 1 i * * 1 z oi0
(13)
To obtain P1(0) ,the following measures are derived.
VI. PERFORMANCE MEASURES
A. Steady State System Size Probabilities
In this section, the steady-state size probabilities at various states are derived.
(i) The probability that the server is idle in vacation (
J
IV
P
)is given by
)
Hz
1
(
)
Hz
1
(
Hz
Q
P
1 J 1 1 J 1i 0i
J IV
p
η
)
0
(
P
1(14)
(ii) The probability that the server is idle in the system
(
J
0
R
) is given byP
(
0
)
λ
)
z
(
ψ
R
J0
1 1(15)
(iii) The probability that the server is on regular busy
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where
b
ρ
=λ
E
(
X
)
E
(
S
b)
(iv) The probability that the server is on working vacation (
Jwv
P
) is given by)
0
,
z
(
Q
lt
P
* 1 z Jwv
)
0
(
P
η
p
Hz
Hz
1
Hz
1
η
))
z
(
X
1
(
λ
1 1 1 J 1 1
(17)
By adding equations (14) to (17),
P
1(
0
)
can be calculated. ThusP
1(
0
)
is given byJwv b 1
d
)
ρ
1
(
)
0
(
P
where
)
Hz
1
(
λ
Hz
Hz
)
η
(
S
1
)(
X
(
E
η
)
η
(
S
)
z
(
X
1
(
ρ
λ
1
η
))
z
(
w
(
h
)
Hz
1
(
p
)
Hz
1
(
Hz
d
1 J 1 1 * V * V 1 b 2 1 x 1 J 1 1 JwvB. Mean System Size
The average number of customers waiting in the system
when the server is in working vacation
(
L
Jwv)
and in system(
L
Jbusy)
are calculated below:(i)
1 z *
Jwv
Q
(
z
,
0
)
dz
d
L
)
Hz
1
(
η
p
)
Hz
1
(
Hz
))
η
(
S
1
(
η
)
η
(
S
)
z
(
X
1
(
η
)
z
(
w
(
h
)
X
(
E
)
0
(
P
λ
1 J 1 1 * V * V 1 2 x 1(18)
(ii)
1 z * Jbusy
Q
(
z
,
0
)
dz
d
L
)
X
(
E
λ
)
1
(
y
L
ρ
1
1
1 1 / G / M b X
b 11 2 1))
X
(
E
λ
(
2
)
1
(
y
))
1
X
(
X
(
E
λ
)
1
(
y
)
X
(
E
λ
ρ
(19)
where ) X ( E ) z ( ψ ) η ( S 1 ) η ( S η ) z ( X 1 η ) z ( w ( h ) X ( E ) Hz 1 ( p ) Hz 1 ( Hz λ ) 1 ( y 1 * V * V 1 2 1 x 1 J 1 1 1 and
)
1
(
y
11
))
η
(
S
1
(
η
))
z
(
w
(
h
)
X
(
E
2
))))
1
X
(
X
(
E
η
)
X
(
E
λ
η
)(
X
(
E
2
(
η
)
z
(
h
)
Hz
1
(
p
)
Hz
1
(
Hz
λ
* V 2 x 3 1 x 1 J 1 1
))
η
(
S
1
(
)
η
(
S
)
X
(
E
λ
1
η
))
z
(
X
1
(
2
* V * V 1 1where
(
S
(
η
))
θ
d
d
)
η
(
S
*V1
*V atθ
η
The mean system size of the model
(
L
Jwv)
can be obtained.From
L
Jwv
L
JV
L
Jbusy(20)
VII. PARTICULAR CASES
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A. MX/G/1 Single Working Vacation Model (SWV)When J=1 and
p
0
,0
Hz
1
,))
z
(
w
(
h
η
p
Hz
x1
,))
z
(
w
(
h
η
)
z
(
ψ
x 1
and))
z
(
w
(
h
)
0
(
P
Q
x 1 0
Substituting these in (14 to 17),the system size probabilities of single workingvacation model are obtained and listed:
0
0
η
Q
R
λ
(i.e)
0Q
0λ
η
R
)
X
(
E
))
η
(
S
1
(
)
η
(
S
))
z
(
X
1
(
λ
η
η
))
z
(
w
(
h
Q
ρ
1
ρ
P
* V * V 1 x 0 b b S busy,
Swv b 0d
ρ
1
P
where
)
X
(
E
))
η
(
S
1
(
)
η
(
S
))
z
(
X
1
(
ρ
λ
η
η
))
z
(
w
(
h
d
* V * V 1 b x Swv
andη
Q
))
z
(
X
1
(
P
wvS
1 0 , (from (15) to (17))The mean system size for the SWV model when the server is on working vacation is
))
η
(
S
1
(
η
)
η
(
S
))
z
(
X
1
(
η
))
z
(
w
(
h
)
X
(
E
Q
λ
L
* V * V 1 2 x 0 S V (from equation(18))The total expected system size for SWV model
))
η
(
S
1
(
d
)
η
(
S
))
η
(
S
1
(
d
S
)
z
(
w
ρ
))
η
(
S
1
(
)
η
(
S
η
)
X
(
E
λ
L
L
* V SWV * V 2 * V SWV ) 1 ( * V 1 x b * V * V 1 / G / M S WV X
)
2
)
S
(
E
))
S
(
E
)((
X
(
E
λ
η
ρ
)(
z
(
w
λ
η
ρ
1
d
)
X
(
E
2 b 2 b b 1 x b SWV(21)
follows from equation (20) , where
2
)
S
(
E
))
1
X
(
X
(
E
λ
)
S
(
E
)
X
(
E
λ
ρ
L
2 b b b 1 / G / MX
is the mean system size of the MX/G/1 model without vacation.
B. MX/G/1 Multiple Working Vacation Model (MWV) As
J
,p
1
,it is found that the measures of J working vacation model exactly coincide with the multiple working vacation model.0
p
,η
)
z
(
w
Hz
Hz
1
x 1 1
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The total expected system sizeR WV
L
))
η
(
S
1
(
d
)
η
(
S
))
η
(
S
1
(
d
S
)
z
(
w
ρ
))
η
(
S
1
(
)
η
(
S
η
)
X
(
E
λ
L
* V MWV
* V 2
* V MWV
) 1 ( * V x b
* V * V 1
/ G / MX
)
2
)
S
(
E
))
S
(
E
)((
X
(
E
λ
η
ρ
)(
z
(
w
ρ
1
2 b 2
b b
1 x b
(25)
Equations (22) to (25) follow from equations (14,16 ,18 and 20).
VIII. NUMERICAL ANALYSIS
In the present section the graphical representation of the mean queue length and the probability that the server is busy for different values of working vacation service rate „muv‟ and vacation parameter „ita‟ corresponding to two values of
J (=1,10) for the model (M/(E3,M)/1/JWV) is presented. The values of the parameters considered are „la‟ =.31 (.48) when J=1(10) and „mub‟= 2 respectively.
Figures1(a) (b) (corresponding to J=1(10) ) illustrate that the expected queue length is decreasing in „ita‟ and the effect of „ita‟ becomes smaller and turns to zero when „muv‟ =‟mub‟ =2.Another extreme case is when „muv‟ = 0 (no service during the vacation period or the classical vacation model) the effect of „ita‟ is largest. The graph figure(2) gives the probability that the server is busy(Probability that the server is busy) for different values of „muv‟ and „ita‟ when J=1.
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 12.5
13 13.5 14 14.5 15 15.5 16 16.5 17 17.5
muv when la=.31 mub=2.0 J=1 figure 1(a)
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ita=.6 ita=1 ita=1.5
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
150 200 250 300 350 400 450
muv when la=.48 mub=2.0 J=10
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figure 1(b)
ita=.6 ita=1 ita=1.5
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.58 0.585 0.59 0.595 0.6 0.605 0.61 0.615 0.62 0.625
muv when la=.31 mub=2.0 J=1 figure (2)
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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 4, Issue 9, September 2014)
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IX. CONCLUSION
The present model is a generalization of both classical single and multiple working vacation models,since the results of the single and multiple working vacation models can be derived as a special case when J=1 and J
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