an m/m/1/n queue with working breakdowns and vacations · 1b. deepa and 2k. kalidass 1department of...
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An M/M/1/N Queue with Working Breakdowns
and Vacations 1B. Deepa and
2K. Kalidass
1Department of Mathematics,
Faculty of Engineering,
Karpagam Academy of Higher Education,
Coimbatore, India.
2Departments of Mathematics,
Faculty of Arts,
Science and Humanities,
Karpagam Academy of Higher Education,
Coimbatore, India
Abstract In this paper, we consider an M/M/1/N queue with working breakdown
and two types of server vacations. We obtain the steady state probabilities
by using computable matrix technique. Finally, sensitivity analysis of the
model is performed.
Key Words:M/M/1/N queue, working breakdowns, single vacation
scheme.
Mathematics Subject Classification: 68M20, 90B22 and 60K25.
International Journal of Pure and Applied MathematicsVolume 119 No. 10 2018, 859-873ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu
859
1. Introduction
Queueing models with unreliable servers have been studied by various authors
in the past since the perfectly reliable server is virtually nonexistent. Studies on
the above topic are called queueing systems with server breakdowns. There is
an extensive literature on queueing systems with server breakdowns and for an
exhaustive review on this topic we refer the readers to Krishnamoorhty et al
(2012) [6].
In most of the previous research, it is generally assumed that the server
completely stops service during the lengthy and unpredictable breakdown
period. Kalidass and Kasturi (2012) [2] studied a new class of breakdowns
policies in which a customer is served at a slower rate when the system is
defective. Such kind of breakdowns is called working breakdowns. Li (2013)
[7] discussed equilibrium customer strategies in M/M/1 queue with working
breakdowns. Kim and Lee (2014) [5] studied M/G/1 queueing system with
disaster and working breakdowns. Recently, Yang and Wu (2014) [14],
numerically derived transient state probabilities for an M/M/1/N queue with
working breakdowns and server vacations. Recently Kalidass and Pavithra
(2016) [3] studied M/M/1/N queue with working breakdowns and Bernoulli
feedbacks.
In some situations, an idle server will start some other uninterruptible task
which is referred to as a 'vacation period'. For a comprehensive and complete
review on vacation queueing systems, we refer the readers to Doshi (1986) [1],
Ke et al(2010) [4] and Shweta Upadhyaya [11].
In this paper, we consider an M/M/1/N queue with working breakdowns and
vacations. From the practical point of view this type of model can be used to
study many real life situations. This motivates us to study the present model.
Exploiting the technique given by [Yue,D., Zhang,Y. and Yue, W. (2006) [15],
Yue, D. and Sun,Y. (2008) [13] and Zeng Hui and Guan Wei.(2011) [16]], we
obtain the steady state probabilities of our model.
The paper is organized as follows. The immediate section provides
mathematical description of our model. In section 3, the steady state analysis of
the system studied. Obtained by the above analysis, we present some
performance measures and numerical examples of our system in section 4 and 5
respectively.
2. The Model
In this model, we consider a single server queue system with arrival rate λ. The
service times during a normal period are exponentially distributed independent
and identically distributed random variables with mean service time
1. The
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860
server proceeds on a vacation whenever the system is empty. The duration of
the server vacation is assumed to be exponentially distributed with a parameter
. The server may get partial breakdowns while providing the service. After the
partial breakdowns server provides service in a slow manner. The partial
breakdown service times are exponentially distributed independent and
identically distributed random variables with mean service times .11
v
After every (breakdown) busy period, the server goes for a vacation. The
vacation time follows an exponential distribution with v . The repair times
while the server is in partial breakdown and vacation follows an exponential
distribution with parameter βv )( .
All inter arrival times, inter (normal or breakdown) service times, and inter
vacation times are independent of each other.
Let )(tC be the server state at time t. Then
vacation.and normalin isserver the,4
vacation,andbreakdown in working isserver the,3
breakdown, in working isserver the,2
,state normalin isserver the,1
)(tC
Let N(t) be the number of customers in the system at time t. Then
0,)(,)( ttNtC is a continuous time Markov chain.
3. Steady State Analysis
The steady state probabilities equations governing the model are obtained as
follows:
40302010 PPPP v (1)
International Journal of Pure and Applied Mathematics Special Issue
861
4131211011)( PPPPP v (2)
1413122111)( NvNNNN PPPPP (3)
NvNNNN PPPPP 432111)( (4)
4020)( PP v (5)
41221121)( PPPP vvv (6)
221421112)( NNvNvNNv PPPPP (7)
12412)( NNvNNv PPPP (8)
1130)( PP (9)
3031)( PP (10)
2313)( NN PP (11)
133 NN PP (12)
2140)( PP vvv (13)
4041)( PPvv (14)
2414)( NNvv PP (15)
144)( NNvv PP (16)
Let 440330220110 PPPPPPPPP where
NN PPPPP 11112111
NN PPPPP 21222112
NN PPPPP 31332313
NN PPPPP 41442414 be the steady state probability vector.
Then equations (1) to (16) can be expressed as
PQ=0 (17)
where
8887868584838281
78767472
6867666564636261
58565452
4847464544434241
38363432
2827262524232221
18161412
)(0
0)(0
00)(
000
AAAAAAAA
AAAA
AAAAAAAA
AAAA
AAAAAAAA
AAAA
AAAAAAAA
AAAA
Q
vvvv
4Nx4
N
AAAA
178563412 00
International Journal of Pure and Applied Mathematics Special Issue
862
767472585452383632181614 AAAAAAAAAAAA N
1
000
8785838167656361454341272321 AAAAAAAAAAAAAA
10
0
N
22A
NN
)(00000
)(0000
000)(0
0000)(
00000)(
24A
NN
00
00
00
25A
10
N
86686448462826 AAAAAAA
NN
000
000
000
42A
NN
00
00
00
NNvv
vv
vv
vv
v
A
)(00000
)(0000
000)(0
0000)(
00000)(
44
47A
10
N
v
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863
62A
NN
00
00
00
NN
A
000000
)(00000
000)(00
0000)(0
00000)(
66
82A
NNv
v
v
00
00
00
84A
NNv
v
v
00
00
00
NNvv
vv
vv
vv
vv
A
)(000000
)(00000
000)(00
0000)(0
00000)(
88
From equation (17), we have
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864
250
240)(
230
220)(
210
200)(
)19(0
)18(0
8847840
40472
6635630
30251
444234201
4020
4322211210
40302010
APAP
PAP
APAP
PAP
PAPAPP
PP
PPPAPAP
PPPP
vv
v
v
v
v
Solving Equations (19) to ( 25), we get
)(
)(
))((
)(
)(
47
1
0
1
1121040
25
1
1121030
47
1
0
1
1121020
1
887847
1
0
1
112104
1
665625
1
112103
1
0
1
112102
1
112101
vv
vv
v
vv
AAPP
AAPP
AAPP
AAAAPP
AAAAPP
APP
APP
From the normalizing condition, we have
Where TN
e
1
111
International Journal of Pure and Applied Mathematics Special Issue
865
)(1440330220110 AePPePPePPePP
eAAAA
eAAAA
eA
e
A
P
vvvv
vv
v
)()(
)()())((1
1
1
887847
1
047
1
0
1
665625251
047
1
0
1
112
10
After 10P is determined, the steady-state probabilities )4,3,2,1( iPi , and
403020 P,, PP can be computed.
4. Performance Measures
According to the distribution of the steady state, various system performance
measures can be developed.
1. Expected number of customers in the system =
k
i
N
k
ikkPNE1 0
)( .
2. Expected waiting time in the system =
.)(
NEWE
3. The proportion of time ,the serve being normal
N
k
kPQ0
11 .
4. The proportion of time, the server being in working breakdown
N
k
kPQ0
22 .
5. The proportion of time, the server being in normal and vacation
N
k
kPQ0
33 .
6. The proportion of time, the server being working breakdown and
vacation
N
k
kPQ0
44 .
5. Numerical Examples
In this section numerical experiments for various performance indices are
provided. For the computation purpose the system capacity N=3 is fixed.
Fig.1 displays the correlation between P10 and the arrival rate λ at the case of
.2,1,3,1,1,2 vvv
International Journal of Pure and Applied Mathematics Special Issue
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Fig. 1: P10 vs λ
From the figure 1 it is observed that the probability to the server being ideal in
the normal state decreases with an increasing value of for different values of µ
= 2,4,6.
Fig. 2: P10 vs λ
The figure 2 illustrates the relation of P10 and the arrival rate . In figure 2, it
can be noticed that P10 decreases with an increasing value of λ when µv =
1.75,1,1.5.
1 2 3 4 5 6 7 8 9 100
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
P10
*-- = 2
+-- = 4
o-- = 6
1 2 3 4 5 6 7 8 9 100
0.05
0.1
0.15
0.2
0.25
P10
*--v = 0.75
+--v = 1
o--v= 1.5
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Fig. 3: P40 vs α
Figure 3 exhibits the correlation of P40 by varying failure rate α for breakdown
service µv = 0.5, 1,1.5. It is seen that P40 increases with the increase of failure
rate α.
Fig. 4: P40 vs γv
The figure 4 displays the relation between P40 and working breakdown vacation
rate γv for µv = 0.5, 1,1.5. It can be observed that P40 decreases along with the
increase of γv.
1 2 3 4 5 6 7 8 9 100
0.005
0.01
0.015
0.02
P40
*--v = 0.5
+--v = 1
o--v = 1.5
1 2 3 4 5 6 7 8 9 100
0.002
0.004
0.006
0.008
0.01
0.012
0.014
v
P40
*--v = 0.5
+--v = 1
o--v = 1.5
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Fig. 5: P10 vs α
Fig. 6: P10 vs γ
Fig. 7: P10 vs β
1 2 3 4 5 6 7 8 9 100.01
0.02
0.03
0.04
0.05
0.06
P10
*-- = 2
+-- = 3
o-- = 4
1 2 3 4 5 6 7 8 9 100
0.02
0.04
0.06
0.08
0.1
P10
*-- = 2
+-- = 3
o-- = 4
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
P10
*-- = 1
+-- = 2
o-- = 3
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Fig. 8: P10 vs μ
Figures 5, 6, 7 and 8 give the correlation between P10 and α, γ, β and µ
respectively.
Fig. 9: E(N) vs λ
Figure 9 demonstrates the relation between mean system size E(N) and arrival
rate λ for µ=2,3,4. It is seen that E(N) increases with the increase of arrival rate
λ.
Fig. 10: E(N) vs λ
2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
P10
*-- = 1
+-- = 2
o-- = 3
1 2 3 4 5 6 7 8 9 100.5
1
1.5
2
2.5
3
E(N
)
*-- = 2
+-- = 3
o-- = 4
1 2 3 4 5 6 7 8 9 100.5
1
1.5
2
2.5
3
E(N
)
*--v = 0.5
+--v = 1
o--v = 1.5
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Fig. 11: E(N) vs λ
Fig. 12 :E(N) vs λ
Figures 10,11,12 display the correlation between E[N] and µv, γv and α..
6. Conclusion
In this paper, we have considered a finite capacity queueing system with
working breakdowns and vacations. It could be interest to develop this research
further by including the concept of retrial customers, in the model.
In this paper, we considered M/M/1/N queue with single vacation and
working breakdowns.
We have derived the explicit expression for P10 by using computable
matrix method.
References
[1] Doshi B.T., Queueing systems with vacations-a survey, Queueing Systems 1 (1986), 29-66.
[2] Kalidass K., Kasturi R., A queue with working breakdowns, Computers & Industrial Engineering 63 (2012), 779-783.
[3] Kalidass K., Pavithra K., An M/M/1/N queue with working breakdowns and Bernoulli feedbacks, International Journal of Applied Mathematics 6(2) (2016).
1 2 3 4 5 6 7 8 9 100.5
1
1.5
2
2.5
3
E(N
)
*--v = 1
+--v = 2
o--v = 3
1 2 3 4 5 6 7 8 9 100.5
1
1.5
2
2.5
3
E(N
)
*-- = 1
+-- = 2
o-- = 3
International Journal of Pure and Applied Mathematics Special Issue
871
[4] Ke J.C., Wu C.H., Zhang Z.G., Recent developments in vacation queueing models: a short survey, International Journal of Operational Research 7 (2010), 3-8.
[5] Kim B.K., Lee D.H., The M/G/1 queue with disasters and working breakdowns, Applied Mathematical Modelling 38 (2014), 1788-1798.
[6] Krishnamoorthy A., Pramod P.K., Chakravarthy S.R., Queues with interruptions: a Survey, Top (2012).
[7] Li L., Wang J., Zhang F., Equilibrium customer strategies in Markovian queues with partial breakdowns, Computers & Industrial Engineering 66 (2013), 751–757.
[8] Medhi J., Stochastic Processes, 2nd ed. Wiley Eastern Ltd (1994).
[9] Medhi J., Stochastic Models in queueing theory, Second Edition, Academic press (2002).
[10] Takagi H., Queueing Analysis: a Foundation of Performance Evaluation, Vacation and Priority Systems, North-Holland, Amsterdam (1991).
[11] Shweta Upadhyaya, Queueing systems with vacation: an overview, International Journal of Mathematics in Operational Research 9 (2) (2016), 167-213.
[12] Tian N., Zhang Z.G., Vacation Queueing Models-Theory and Applications, Springer, New York (2006).
[13] Yue D., Sun Y., The Waiting Time of the M/M/1/N Queuing System with Balking Reneging and Multiple Vacations, Chinese Journal of Engineering Mathematics 5 (2008), 943-946.
[14] Yang D.Y., Wu Y.Y., Transient behaviour analysis of a finite capacity queue with working breakdowns and server vacations, Proceedings of the International Multi Conference of Engineers and Computer Scientists (2014).
[15] Yue D., Zhang Y., Yue W., Optimal Performance Analysis of an M/M/1/N Queue System with Balking, Reneging and Server Vacation, International Journal of Pure and Applied Mathematics 28 (2006), 101-115.
[16] Zeng Hui, Guan Wei., The two-phases-service M/M/1/N queuing system with the server breakdown and multiple vacations, ICICA 2011, LNCS 7030 (2011), 200-207.
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