iBusiness, 2010, 2, 363-369
doi:10.4236/ib.2010.24047 Published Online December 2010 (http://www.scirp.org/journal/ib)
Copyright © 2010 SciRes. iB
The System Size Distribution for M/G/1 Queueing
System under N-Policy with Startup/Closedown
Mingwu Liu1, Yongkai Ma2, Bin Deng2
1School of Management, Chongqing Jiaotong University, Chongqing, China; 2School of Management and Economics, University of
Electronic Science and Technology of China, Chengdu, China.
E-mail: liumingwu2007@yahoo.cn
Received June 4th, 2010; revised August 9th, 2010; accepted September 27th, 2010.
ABSTRACT
This paper develops a new method for calculating the system size distribution on two different M/G/1 queueing system
under N-policy with general startup/closedown. Firstly, the stochastic decomposition property is used to derive the p.g.f.
of the system size distribution. By the Leibniz formula of derivation, we investigate the additional system size distribu-
tion, and then, we get the recursion expression of system sizes distribution. Finally, several examples are given for il-
lustrating the application of the recursion expression and sensitivity analysis is also performed.
Keywords: Queue-Length, N-Policy, Stochastic Decomposition, Recursion Expression, Leibniz Formula
1. Introduction
This paper presents a new approach for study M/G/1
queueing system under N-policy with general startup/
closedown times. N-policy is an effective mechanism for
cutting down operating cost, which has been widely ap-
plied in modeling queue-like production system. As soon
as each service is completed, the server will be turned off
for a random closedown time and a random startup time
is needed before commencing his service, which reflects
more actual situation.
Queueing system under N-policy first considered by
Yadin and Naor [1] has been extensively expanded by
many authors. Kella [2] discussed N-policy M/G/1
queueing system with server vacations. Bothakur et al. [3]
extended Baker’s model [4] with exponential startup
time to the general startup time. Lee et al. [5-7] investi-
gated the batch arrival N-policy M/G/1 queueing system
with a single vacation and multiple vacations. The p.g.f.
(probability generating function) of the system size dis-
tribution was derived, which shown the famous stochas-
tic decomposition property proposed by Fuhrmann and
Cooper [8] still came into existence. The concept of clo-
sedown time was introduced by Takagi [9]. Ke [10] de-
veloped an M/G/1 queueing system under varieties and
extensions of NT queueing systems with breakdowns,
startup and closedown.
It is difficult to get the analytic steady-state system
size distribution based on the p.g.f. under the general
distribution of service time. Wang and his co-authors
[11-14] derived the analytic steady-state solutions of the
N-policy M/M/1 queueing system, the N-policy M/Hk/1,
the N-policy M/Ek/1, and the M/H2/1 respectively. Re-
cently, a maximum entropy approach was used to ana-
lyze the steady-state characteristics of M/G/1 queueing
system [15,16]. Based on this method, Wang et al. [17-
20] developed the approximate probability distribution of
the system size for the N-policy M/G/1 queue with vari-
ous cases. Also, the maximum entropy approach could
be succeeded applied to batch arrival queueing system
[21-23]. Tang [24] developed a total probability decom-
position method for deriving the recursion expression of
system size distribution in equilibrium. This method
could be used to discuss batch arrival continuous-time or
discrete-time queueing system [25,26], while the analysis
procedure was very complicated.
This paper will present a new method for analyzing
the N-policy queueing system. Our method can derive
the analytical expressions of system size distribution
which is different from the maximum entropy approach.
It is to be noted that the analytic system size distribution
under N-policy is very difficult to be derived by Tang’s
method. Our method is much simpler than Tang’s in
analyzing the N-policy queueing system.
The arrangement of this paper is as follows: First, we
This work was supported by the New Century Excellent Talents in Min-
istry of Education Support Program under Grant NCET-05-0811.
The System Size Distribution for M/G/1 Queueing System under N-Policy with Startup/Closedown
Copyright © 2010 SciRes. iB
364
develop a new approach for derive the recursion expres-
sion of system size distribution in equilibrium. Second,
some special cases are presented in this paper for our
new method. Finally numerical examples are given for
illustrating the accurate calculation of the system size
distribution and then sensitivity analysis is investigated.
2. Model Description and Assumption
In this paper, we discuss two different kinds M/G/1
queueing system under N-policy with startup/closedown.
The models are described in detailed as follows:
Assumption of the model 1.
1) Customers arrive according to a Poisson process
with rate
. Arriving customers form a single waiting
line and are served according to the order of his arrivals.
The server can process only one customer at a time.
2) The service time provided by a single server is an
independent and identically distributed random vari-
able ()Gwith a general distribution function

Gt .
3) Whenever the system becomes empty, the sever
shuts down with a random closedown time()D. When
the number of arrivals in the queue reaches a predeter-
mined threshold N, the server immediately reactivates
and performs a startup time with random length ()U
before starting service. When the server completes his
startup, he starts serving the waiting customers until the
system becomes empty.
4) If a customer arrives during a closedown time, the
server is immediately started without operating N-policy
and without a startup time.
Assumption of the model 2. The first three assump-
tions are the same as those in model 1. However the
fourth assumption now is that even if a customer arrives
during a closedown time, the server still be shut down.
As soon as the closedown is over, the server immediately
reactivates. Before starting service, the server still need a
random startup time()U, however without satisfying the
conditions of N-policy. In many literatures [9,10], they
suppose that the server will resume shutting down and
offers service when a customer arrives during a close-
down time. However, in some real-world, the server
(machine) restarts until the closedown is finished for
protecting the machine when a customer arrives during
the closedown time.
3. Preliminary Formula for the System Size
Distribution for M/G/1 Queueing System
Before discussing our models, let us recall some results
in the ordinary M/G/1 queueing system. For the sake of
convenience, we define several items. We call a time
interval when the server is working a busy period. A time
interval when the server is unavailable (such as startup/
closedown, or idle) an idle period. Let

00,1,
j
pj

be the probability distribution of the system size in equi-
librium. From Tang [24], we have
0
01p
(1)
0(1) ,1
jj
pj

 (2)
where
 

1
0
11
() 1!
j
t
j
t
eGt dt
gj



 
1
0
10
11,1.
!
i
jk
t
jk
ki
t
edGtj
gi








 
0
t
g
edGt
, and if0j,
1
0
j
k
.
We denote
by the probability distribution of system
size during the busy period at stationary. Note that
EG

is traffic intensity and it should to be sup-
posed to be less than unity, which ensures the system
reaching at equilibrium. The p.g.f. of the queue size of
the M/G/1 queueing system is given by



0
11 1
1
zg z
zgzz

 
  (3)
4. Analysis of Model 1
4.1. The p.g.f. of the System Size
In this section we derive the p.g.f. of the system size dis-
tribution following the similar arguments as in [10].
Model 1 is as an extension of the M/G/1 queueing sys-
tem given in Section 3. At the beginning of the busy pe-
riod for Model 1 can be described by
Case 1: no customers arrive when the server is shut-
ting down, which occurs with probability
D
. For
this system, the server will start up until the number of
customer arrival reach N during the idle period. The p.g.f.
of the system size at the beginning of the busy is given
by
N
zU z
, where
Us
is LST of U.
Case 2: Some customers arrive when the server is
shutting down, which occurs with probability
1D
.
In this case, the service is started immediately without a
startup time and without N-policy. There is only one
customer at the beginning of the busy period.
By the stochastic decomposition results [8], the p.g.f
of the number of customers found at the beginning of the
busy period is give by


11
N
zD zUzDz
 
 
 (4)
where
Uz
denotes the p.g.f. of the number of
customers arrive during the startup time. Following the
result of Medhi and Templeton [27], the stochastic de-
The System Size Distribution for M/G/1 Queueing System under N-Policy with Startup/Closedown
Copyright © 2010 SciRes. iB
365
composition of the p.g.f. of number of customers in the
N policy M/G/1 queueing system with startup/closedown
still comes into existence. We obtain
 
 
1
10
'
1
1
11
z
zz z



(5)
where
 


'
111DNEU D


 is the ex-
pected number of customers that arrive during the idle
period plus startup time.
4.2. Additional System Size Distribution
From Equation (5) we see that the stationary system size
is sum of two random variables, one of which is the sys-
tem size of the general M/G/1queue. The other is the
additional system size distribution due to N-policy and
startup/closedown. The p.g.f. of additional system size
distribution is give by
 
 
1
'
1
1
11
a
z
zz


(6)
Following the definition of p.g.f., the probability of
additional system size distribution can be written by can
be written by
 

0
1|
!
n
a
az
n
dz
pn ndz
 , 0,1,n (7)
Notice that, when iN, we have
1
0
1, 0
(()1))
|1(),1
0,2, ,1
i
z
i
i
dz Di
dz iN


 

, (8)
And when iN, we have


1
0
1|
i
z
i
dz
dz





0
0
|
kN ik
i
z
kik
k
dzd Uz
i
Dkdz dz









0
!!
iN
tt
Die dUt
iN


(9)
Substitute (8) and (9) into (7), we get the additional
system size distribution
 



1
01
a
pDNEU D



(10)
 




,
1
a
D
pn DNEUD



1, ,1nN  (11)

  




0
0
1!
,
1
m
nN t
m
a
t
DedUt
m
pn DNEU D
 







,1,nNN
  (12)
4.3. System Size Distribution
In this section we will derive the stationary-state system
size distribution. The chain rule of differentiate is used as
the main tool for calculation. Obviously, the probability
of the number of customers in the queue which equals to
zero is given by
 



01 0
1
|.
1
z
pz
DNEU D


 
(13)
When 1,,1nN
 , by the Leibniz formula, from
(5), the probability of the number of customers is n in
the queue is also given by




0
0
'
0
1
11 |
!
1
knk
n
nz
knk
k
dzd z
k
pn
ndz dz


 


 


0
0
'
0
1
1|
!
1
nk
n
k
z
nk
k
dz
p
nk dz

(14)
where, 0
k
p is given by Formulas (1) and (2). We de-
note ()z
by 1() 1
1
z
z
, using the rule of chain of
derivation again, conditioning on ,nk N thus we
have
0
|
nk
z
nk
dz
dz
 

1
0
1
0
1!
1|.
1
i
nk nki
z
inki
i
dz nki
i
nk dz z


 





!1 1!nk nkDnk

!nk D
  (15)
Substitute (15) into (14) and combing (2), we get the
system size distribution
 




1
1
1
.
1
n
nk
k
n
DD
pDNEU D
 
 



 



1, ,1nN
 (16)
When nN, if ,nk N
that is knN , we
have
The System Size Distribution for M/G/1 Queueing System under N-Policy with Startup/Closedown
Copyright © 2010 SciRes. iB
366


 

0
|!1,
nk
z
nk
dz nk DUk
dz
 , (17)
From the (15), (17) and combing (2), we obtain the
system size distribution.
 




*
11
110, 1,
1
nN n
kk
kknN
n
DU DUkD
pDNEUD

 
 



 

 



, nN (18)
where
  
0
0
,!
m
nk Nt
m
t
Uke dUt
m

,
 
0
0
0, !
m
nN
t
m
t
UedUt
m
.
The recursion expressions of system size distribution are
given by (13), (16) and (18), which can be used for cal-
culating the probability of system size in equilibrium.
Remark 4.1 If the startup time equals to zero, which
implies
0EU and
 
0, 1,, 1UUk

, model
1 can be reduced to the M/G/1 queueing system with
delayed N-policy. In this case, these results coincide with
those of Tang’s system [28]. The approach discussed in
this paper is much easier than the total probability de-
composition method for the system size distribution used
in [28].
5. Analysis of Model 2
5.1. The p.g.f. of the System Size
In this section, we consider another N-policy for M/G/1
queueing system with startup/closedown. In this system,
the only difference from the model 1 is that if some cus-
tomers arrive during the closedown period, the server
will still be shut down in order to protecting the server.
After the closedown, the server immediately commences
a startup with a random time. The server will offer ser-
vice without operating N-policy until the startup is fin-
ished.
For this system, from the similar arguments as for mod-
el 1, the p.g.f. for the number of customers found in the
system at the beginning of the busy period is given by
2()( )()
N
zD zUz


 
(1( ))()()DD zUz
 
 
 , (19)
where

Dz
denotes the p.g.f. of the number of
customers arrive during the closedown time. And the
expected number of customers at the beginning of busy
period is as follows
 

'
21DNEU





1DEDEU
 (20)
From (19) and (20), we have the distribution for the
system size at an arbitrary time
 
 
2
20
'
2
1
11
z
zz z


. (21)
5.2. System Size Distribution
In this section, we derive the system size distribution.
Similarly, from (19), (20), we have the probability that
the system is empty
 
 

0'
2
111
.
1
DDU
p



(22)
As in Section 4, when 1,,1,nN  the probability
of the system size equals tonis given by
0
0
'
0
1
1(())
|
()!
(1)
nk
n
k
nz
nk
k
pdz
pnk dz

ٛ
, (23)
where, we denote
z by

21
1
z
z

. From (19), we
get
2
0
1|
i
z
i
dz
dz


 



0
11,0
,
1,,1,,1
i
l
DDU i
l
DiliN
i







(24)
where
 
00
,lil
tt
ile tdDte tdUt






Thus, we get




0
|11 !
nk
z
nk
dz DDU nk
dz








10
!1,
!
nk i
il
nk l
Dil
i
i


 



. (25)
Take (25) into (23), we get



 
'
1
111
1
n
pDDU



The System Size Distribution for M/G/1 Queueing System under N-Policy with Startup/Closedown
Copyright © 2010 SciRes. iB
367



10
1
1,
!
ni
il
l
Dil
i
i


 






10
1
1,
!
nk i
il
l
Dil
i
i


 





 
1
11
n
k
k
DDU





,
1, ,1nN . (26)
When nN, we analyze the following two cases.
1) If ,nk N that is knN , we have


 

0
|1 ,!
nk
z
nk
dz UkDn k
dz

 



 
00
!
1,
!
nk i
il
nk l
Dil
i
i


 


 , (27)
2) If ,nk N which implies knN , we get



 
0
00
!
|11,
!
nk nk i
z
nk
il
dz nk l
Dil
i
i
dz


 



(28)
Substitute (27) and (28) into (23), we have
0
0
'
0
1
1(())
|
()!
(1)
nk
nN
k
nz
nk
k
pdz
pnk dz

 


0
0
'
1
1
1|
!
1
nk
n
k
z
nk
knN
dz
p
nk dz
 

  


'
00
1
11
10, 1,
!
1
ni
il
l
UD Dil
i
i
 



 






1
1,
nN
k
k
Uk D







00
1
1,
!
nki
il
l
Dil
i
i


 






100
1
11 ,
!
nnki
k
knNi l
l
Dil
i
i
 
 








,
nN (29)
We get the recursion expressions of the system size
distribution for the model 2, which are shown by (22),
(26) and (29).
Remark 5.1 Suppose that we let
01PU, which
indicates
0EU and
1U
. In this case, our
model 1 can be simplified to M/G/1 queueing system
with general closedown time. It is different from Tang’s
system [28] because the server will be shut down, even if
some customer arrives during the closedown time. We
get the system size distribution as follows:
 

 


0
111
1
DD
pDND ED





 (30)
 


1
1
n
pDND ED









10
1
111 ,
!
ni
il
l
DDD il
i
i
 
 



 






1
11
n
k
k
DD






10
1
1,
!
nk i
il
l
Dil
i
i


 


 , 1, ,1nN (31)
 


1
1
n
pDND ED






00
1
1,
!
ni
il
l
Dil
i
i


 






100
1
1,
!
nN nki
k
kil
l
Dil
i
i




 






100
1
11 ,
!
nnki
k
knNi l
l
Dil
i
i
 
 

 



,nN (32)
where
  
0
0, .
,,.
l
t
il
il etdDtil

Remark 5.2 If we let the closedown time equal to zero,
which implies
0ED
and

1D
. In this case,
our model 1 and 2 can describe the N-policy M/G/1
queueing system with general startup time. The system
size distribution can be simplified to the following ex-
pressions:

0
1
pNEU
(33)

1
11
n
nk
k
pNEU


, 1, ,1nN 
(34)
 

11
110,1 ,
nN n
kk
kknN
n
UUk
pNEU


 




nN. (35)
The System Size Distribution for M/G/1 Queueing System under N-Policy with Startup/Closedown
Copyright © 2010 SciRes. iB
368
6. Numerical Example
To illustrate the implication of calculating the system
size distribution, we consider the model of an M/G/1
queueing system under N-policy with general startup/
closedown, which based on the model 1 discussed in
Section 4. Here we assume that the service time follows
the negative exponential with mean
1EG
. The
startup time is 3-stage Erlang distribution with mean
3EU
and the closedown time is 4-stage Erlang
distribution with mean
4ED
. The numerical re-
sults are illustrated in Table 1.
Observing the Table 1, it is clear that as
increases,
1) the probability of the system being empty increases
when other system parameters are constant; 2) when
1, ,nN , the probability of system size being n in-
creases; 3) when 1,2,nNN the probability of
system size being n decreases. The expected system size
(EL) also decreases as
increases. Deserve to be men-
tioned is that the mean system size is not enough for the
system design and control. As we can see the expected
system size is no more than 3 while the sum of probability
of system size exceeding the mean system size presented
in the last is about 30%, which can not be neglected.
7. Conclusions
In this paper, we derive the recursion expressions of sys-
tem size distribution for two different N-policy M/G/1
queueing systems with general startup/closedown. We
first utilize the derivation of the chain rule combing with
the famous stochastic decomposition results for devel-
oping the probability distributions of system size in the
system. We present several examples to illustrate the
implement of calculating the system size distribution
from these recursion expressions, and investigate the
effects for different system parameters. We get the sys-
tem size distribution characteristic, which is important in
practice. The method in this paper can be used in N- pol-
icy batch arrival queueing system.
Table 1. The distribution of the system size.
(1,2, 2,5N

 )
2 3 4 5 6 7 8 9
P0 0.2396 0.2516 0.2580 0.262 0.2647 0.2667 0.2682 0.2694
P1 0.1672 0.1755 0.1799 0.1827 0.1846 0.186 0.1871 0.1879
P2 0.1309 0.1374 0.1409 0.1431 0.1446 0.1457 0.1465 0.1472
P3 0.1128 0.1184 0.1214 0.1233 0.1246 0.1255 0.1262 0.1268
P4 0.1037 0.1089 0.1117 0.1134 0.1146 0.1154 0.1161 0.1166
P5 0.0932 0.0978 0.1003 0.1019 0.1029 0.1037 0.1043 0.1048
P6 0.0405 0.0386 0.0372 0.036 0.035 0.0342 0.0336 0.033
P7 0.0304 0.0259 0.0232 0.0214 0.0201 0.0192 0.0185 0.0179
P8 0.0206 0.0156 0.0131 0.0116 0.0107 0.01 0.0095 0.0092
P9 0.0131 0.0089 0.0070 0.0061 0.0055 0.0051 0.0048 0.0046
P10 0.0081 0.0049 0.0037 0.0031 0.0028 0.0026 0.0024 0.0023
P11 0.005 0.0026 0.0019 0.0016 0.0014 0.0013 0.0012 0.0012
P12 0.0032 0.0015 0.0010 0.0008 0.0007 0.0007 0.0006 0.0006
P13 0.0021 0.0009 0.0005 0.0004 0.0004 0.0003 0.0003 0.0003
P14 0.0016 0.0005 0.0003 0.0002 0.0002 0.0002 0.0002 0.0002
P15 0.0013 0.0004 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001
P16 0.0011 0.0003 0.0001 0.0001 0.0001 0.0001 0 0
P17 0.001 0.0003 0.0001 0.0001 0 0 0 0
P18 0.001 0.0002 0.0001 0 0 0 0 0
P19 0.001 0.0002 0.0001 0 0 0 0 0
EL 2.7203 2.4915 2.4203 2.3885 2.3704 2.3594 2.3501 2.3438
Sum 0.3279 0.3077 0.3006 0.2698 0.2945 0.2929 0.2916 0.2908
The System Size Distribution for M/G/1 Queueing System under N-Policy with Startup/Closedown
Copyright © 2010 SciRes. iB
369
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