an m/m/1/n queue with working breakdowns and vacations · 1b. deepa and 2k. kalidass 1department of...

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.

[email protected]

2Departments of Mathematics,

Faculty of Arts,

Science and Humanities,

Karpagam Academy of Higher Education,

Coimbatore, India

[email protected]

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

International Journal of Pure and Applied Mathematics Special Issue

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)


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


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


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


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


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


866

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


867

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


868

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


869

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


870

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


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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an m/m/1/n queue with working breakdowns and vacations · 1b. deepa and 2k. kalidass 1department of...

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