on the m /g/1 retrial queue with regular and optional re ...xajzkjdx.cn/gallery/491-mar2020.pdf ·...

On the MX/G/1 retrial queue with regular and optional re-service under

modified Bernoulli server vacation and delaying repair

C. Revathi

Department of Mathematics

D.K.M College for women(Autonomous), Vellore – 632 001,Tamil Nadu, INDIA

e-mail: [email protected]

L. Francis Raj

PG & Research Department of Mathematics

Voorhees College, Vellore – 632 001,Tamil Nadu, INDIA

e-mail: [email protected]

M.C. Saravanarajan

School of Advanced Sciences

VIT University, Vellore – 632 014,Tamil Nadu, INDIA

ABSTRACT: We study the analysis of a single server batch arrival retrial queue with optional re-service under modified

Bernoulli vacation which consists of a breakdown and delay time. The main assumption in this paper is that the repair process

does not start immediately after a breakdown and there is a delay time waiting for repairs to start. We also assume that the

customers that the customers who find the server busy are queued in the orbit in accordance with an FCFS discipline. Once a

service is completed the customer may leave the system or some of them have the option to demand re-service for the same

service taken. At the completion epoch of each service, if there are no customers in the system, the server will wait for the next

customer to arrive with probability 1-a and chooses vacation with probability a. The steady state have been found by using

supplementary variable technique and compared with known results.

KEY WORDS:Batch arrival, Retrial queue, optional re-service,modified Bernoulli vacation, breakdowns

I.INTRODUCTION

In recent years, the retrial queueing system has been studied extensively, due to its applicability in

telephone switching system, telecommunication networks and computer networks. Extensive surveys of retrial

queues can be seen in Artalejo(1999) and Gomez – Corral(2008). A review of the literature on retrial queues and

their applications can be found in Falin(1997) and Yang and Templeton(1987). Most of the recent studies have been

devoted to batch arrival vacation models under different vacation policies. A batch arrival queue with single

vacation policy is discussed by Choudhury (2002).

We can find limited studies on queuing system with re-service, researcherslike Rajadurai et al.(2014) and

Radha et al.(2017). Re-service is also an important factor in the theory of queuing system and have many

applications in the real world. There may be situations where re-service is desired, for instance while visiting a

doctor, the patient may be recommended for some investigations, after which he may need to see the doctor again

while in some situations offering public service a customer may find his service unsatisfactory and consequently

demand re-service. In fact, authors like Madan et al. (2004), Jeyakumar and Arumuganathan (2011) studied queues

with re-service.

Choudhury and Deka (2008) have analyzed the M/G/1retrial queue with server subjected to repairs and

breakdowns. A number of papers (Krishnakumar and Arivudainambi (2002), Krishnakumar and PavaiMadheswari

(2003) have recently appeared in the queueing literature in which the concept of general retrial time has been

considered along with the Bernoulli vacation schedule. However in the retrial queueing systems the server may not

be aware of the status of the customers in the system as in the case of Keilson and Servi’smodel (1986). Hence it is

Journal of Xi'an University of Architecture & Technology

Volume XII, Issue III, 2020

Issn No : 1006-7930

Page No: 5247

mailto:[email protected]

mailto:[email protected]

necessary to modify the Bernoulli schedule introduced by them in the retrial context. This special kind of Bernoulli

vacationis called as modified Bernoulli vacation schedule considered by Madan (2004), PavaiMadheswari,

Krishnakumar and Suganthi (2017). Wang and Li (2008) have studied a similar model with breakdown and repair.

Here we have discussed the case of optional re-service. In this case a customer has the choice of selecting

any one of the two types of regular and optional re - service, subsequently has the option to repeat the service taken

by him or leaves from the system. We assume customers arriving in batches and the system is providing in regular

and optional re – service, the customer can choose re –service for the same service taken [17]. We can find many

real life applications of this model. Such situations are mostly observed in telephone consultation of medical service

systems and in the area of computer processing systems, banks, post offices etc. where the customer has option of

choosing re- service [14].

The rest of this paper is organized as follows; the description of the mathematical model is given in section

2. In section 3, we introduce the definitions and notations of this chapter. In section 4, all the steady state equations

governing the mathematical system of our model formulated, while in section 5, to obtain steady state results in

terms of the probability generating functions for the number of customers in the queue, the average number of

customers & the average waiting time in the queue. Some special cases are given in section 6 to show the relations

between this work and previous works done by other researchers. Conclusion of this work is presented in section 7.

II.THE MATHEMATICAL MODEL

The detailed description of the model is given as follows:

1. We consider an MX/G/1queueing system, according to an arrival of a job follows a compound Poisson

process with Let Xi denotes the number of customers belonging to the ith arrival batch where Xi,i = 1, 2,

3,….. P[Xi = n] = iC , n = 1, 2, 3,….. andX (z) denotes the probability generating function of X.

2. If there is no waiting space and therefore if an arriving customer finds the server being busy to serve his

request immediately, all these customers leave the service area and join a pool of blocked customers called

orbit. The customers in the orbit, later on try to get their service. Successive Inter retrial times of any

customer has an arbitrary probability distribution A(x) with the corresponding density function )(xa and

Laplace-Stieljie’s transform (LST) A*(x). The conditional completion rates for retrial time is

)(1

)()(

xA

xdAdxxa

3. A single server provides regular and optional re – service to each job. The service discipline is FCFS(First

come First served). As soon as regular service is completed may opt for optional re – service for the sane

service taken without joining the orbit with probability r or may leaves the system with probability

rr 1 . It is assumed that the service time follows general random variable B1 and B2 with distribution

function B1(t) for normal service and B2(t) for optional re-service and LST B1*(x), B2*(x). The conditional

completion rates for service is )(1

)()(,

)(1

)()(

2

22

1

11

xB

xdBdxx

xB

xdBdxx

4. After completion of service to each customer, the server may go for a vacation of random length V with

probability a )10( a or may wait in the service facility to serve the next customer with probability 1 –

a. It is assumed that when there are no customers waiting in the orbit, the server waits in the system for a

new customer with probability 1 – a. At the end of vacation, if there are no customers waiting in the orbit,

the server waits for the one to arrive. The vacation time of the server V has distribution function V(t) and

LST V*(x). The conditional completion rates for vacation is )(1

)()(

xV

xdVdxx

5. The system may breakdown at random, and the service channel will fail for a short interval of time. The

server’s life times are generated by exogenous Poisson Process with rates 1 and 2 for regular and

optional re – servicerespectively.



Issn No : 1006-7930

Page No: 5248

Vacation

terminationServer goes

For vacation

Repair termination Repair

termination

Server idle

Batch arrival Departure

1 - r

ServerbusyRetrial 1 2

Figure 1: The model

6. Once the system breakdown, their repairs do not start immediately and there is a waiting period of the

server. We define the waiting time as delay time. The delay time follows a general distribution with density

function W1 (t) for normal service, W2 (t) for optional re – service and LST W1*(y), W2*(y) respectively.

The repair time involves F1 (t) for normal service, F2 (t) for optional re – service and LST F1*(y), F2*(y)

respectively. The conditional completion rates for delaying repair on normal service, delaying repair on

optional re – service, repair on normal service, repair on optional re – service vacation respectively are

)(1

)()(,

)(1

)()(

2

22

1

11

yW

ydWdyy

yW

ydWdyy

&

)(1

)()(,

)(1

)()(

2

22

1

11

yF

ydFdyy

yF

ydFdyy

III. DEFINITIONS AND NOTATIONS

All stochastic processes involved in the system are assumed to be independent of each other.

Now we introduce some more notations which will be used for the mathematical formulation of this model.

In addition let )(),(),(),(),(),(),(),( 002

01

02

01

02

01

0 tVtFtFtWtWtBtBtA be the elapsed retrial time, regular

service time, optional re – service time, delaying repair time on regular service, delaying repair time on

optional re – service time, repair time on regular service, repair time on optional re – service and vacation

time respectively at time t

In the steady state, we assume that

1)(,0)0(,1)(,0)0(,1)(,0)0(,1)(,0)0( 2211 VVBBBBAA are continuous at x = 0

and ,1)(,0)0(,1)(,0)0( 1111 FFWW 1)(,0)0(,1)(,0)0( 2222 FFWW are

continuous at y = 0

The state of system at time t can be described by the Markov process{C(t), X(t), t 0 } where C(t) denotes

the server state 0, 1, 2, 3, 4, 5, 6, 7 according to the server is idle, busy, re-service, delaying repair on

normal service, delaying repair on optional re - service, repair on normal service, repair on optional re -

service vacation respectively. We introduce the random variable

r

Service area

S

Regular Service

Service

Re - Service

Vacation

Delaying repair Delaying repair

Repair Repair

Orbit



Issn No : 1006-7930

Page No: 5249

tat timen on vacatio isserver theif7,

tat time service -re optionalon repair isserver theif6,

tat time service normalon repair isserver theif5,

tat time service -re optionalon repair delaying isserver theif4,

tat time service normalon repair delaying isserver theif3,

tat time service-reon isserver theif,2

tat timebusy isserver theif1,

tat time idle isserver theif,0

)(tC

Ergodicity Condition:We analyze the embedded Markov chain at departure completion epochs

Let }/{ Nntn be the sequence of epochs at which either a service period completion occurs or a vacation time

ends or a repair period ends. The sequence of random vectors )}(),({ nnn tXtCZ form a Markov chain,

which is embedded Markov chain for our retrial queueing system.

Theorem:The embedded Markov chain }/{ Nnzn is ergodic if and only if 1 where

)]())]()((1)[())]()((1)[()[())(1)(( 22221111 VaEFEWEBrEFEWEBEXEAXE

Steady state distribution: In this section, we first develop the stationary distribution for the system under

consideration. For the process {X(t), t 0 }. We define the probability

}0)(,0)({)(0 tXtCPtP

1},)(,)(,0)({),( 0 ndxxtAxntXtCPdxtxPn

})(,)(,1)({),( 0,1 1 dxxtBxntXtCPdxtxQ n fort 0 , x 0 , n 0

})(,)(,2)({),( 02,2 dxxtBxntXtCPdxtxQ n fort 0 , x 0 , n 0

})(/)(,)(,3)({),,( 01

01,1 xtBdyytWyntXtCPdytyxD n

})(/)(,)(,4)({),,( 02

02,2 xtBdyytWyntXtCPdytyxD n

})(/)(,)(,5)({),,( 01

01,1 xtBdyytFyntXtCPdytyxR n fort 0 , (x, y) 0 , n 0

})(/)(,)(,6)({),,( 02

02,2 xtBdyytFyntXtCPdytyxR n fort 0 , (x, y) 0 , n 0

0},)(,)(,7)({),( 0 ndxxtVxntXtCPdxtxn

The following probabilities are used in sequent sections

)(0 tP is the probability that the system is empty at time t. ),( txPn is the probability that at time t there are exactly n

customers in the orbit with the elapsed retrial time of the rest customer undergoing retrial is x.

),(,1 txQ n is the probability that at time t there are exactly n customers in the orbit with the elapsed service time of

the test customer undergoing service is x.



Issn No : 1006-7930

Page No: 5250

),(,2 txQ n is the probability that at time t there are exactly n customers in the orbit with the elapsed re-service time of

the test customer undergoing service is x.

),,(,1 tyxD n is the probability that at time t there are exactly n customers in the orbit with the elapsed service time of

the test customer undergoing service is x and the elapsed delaying repair of server is y.

),,(,2 tyxD n is the probability that at time t there are exactly n customers in the orbit with the elapsed re - service

time of the test customer undergoing service is x and the elapsed delaying repair of server is y.

),,(,1 tyxR n is the probability that at time t there are exactly n customers in the orbit with the elapsed service time of

the test customer undergoing service is x and the elapsed repair time of server is y.

),,(,2 tyxR n is the probability that at time t there are exactly n customers in the orbit with the elapsed re-service time

of the test customer undergoing service is x and the elapsed repair time of server is y.

),( txn is the probability that at time t there are exactly n customers in the orbit with the elapsed vacation time is x.

We assume that the stability condition is fulfilled in the sequence and so that the limiting probabilities

)(lim 00 tPPn

fort 0 , ),(lim)( txPxP nn

n

for t 0 , x 0 , n 1

),(lim)( ,1,1 txQxQ nn

n

fort 0 , x 0 ,n 0 , ),(lim)( ,2,2 txQxQ nn

n

for t 0 , x 0 , n 0

),(lim),( ,1,1 txDyxD nn

n

fort 0 , ),(lim),( ,2,2 txDyxD nn

n

for t 0

),(lim),( ,1,1 txRyxR nn

n

fort 0 , ),(lim),( ,2,2 txRyxR nn

n

for t 0 , ),(lim)( txx nn

n

for t 0 exist

IV. STEADY STATE EQUATIONS

By the method of supplementary variable technique, we obtain the system of equations that govern the dynamics of

the system behavior under steady state as

0

0

0

20,2

0

10,10 )()()()()()()1( dxxvxVdxxxQdxxxQraP (4.1)

0)())(()(

xPxadx

xdPn

n ,n 1 (4.2)

0

0,110,1110,1

),()()()]([)(

dyyxRyxQxdx

xdQ , n = 0 (4.3)

0

,11,1

1

,111,1

),()()()()]([)(

dyyxRyxQCxQxdx

xdQnkn

n

k

knn

, n 1 (4.4)

0

0,220,2220,2

),()()()]([)(

dyyxRyxQxdx

xdQ , n = 0 (4.5)



Issn No : 1006-7930

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0

,22,2

1

,222,2

),()()()()]([)(

dyyxRyxQCxQxdx

xdQnkn

n

k

knn

, n 1 (4.6)

0),())((),(

0,110,1

yxDydy

yxdD , n = 0 (4.7)

),(),())((),(

,1

1

,11,1

yxDCyxDydy

yxdDkn

n

k

knn

, n 1 (4.8)

0),())((),(

0,220,2

yxDydy

yxdD , n = 0 (4.9)

),(),())((),(

,2

1

,22,2

yxDCyxDydy

yxdDkn

n

k

knn

, n 1 (4.10)

0),())((),(

0,110,1

yxRydy

yxdR , n = 0 (4.11)

),(),())((),(

,1

1

,11,1

yxRCyxRydy

yxdRkn

n

k

knn

, n 1 (4.12)

0),())((),(

0,220,2

yxRydy

yxdR , n = 0 (4.13)

),(),())((),(

,2

1

,22,2

yxRCyxRydy

yxdRkn

n

k

knn

, n 1 (4.14)

0)())(()(

00

xx

dx

xd , n = 0 (4.15)

)()())(()(

1

xCxxdx

xdkn

n

k

knn

, n 1 (4.16)

The steady state boundary conditions are

1,)()()()()1()()()0(

0

2,2

0

1,1

0

ndxxxQdxxxQradxxxVP nnnn

(4.17)

PCdxxPCdxxaxPQ nkn

n

k

knn 1

0

)1(

10

1,1 )()()()0(

, n 1 (4.18)

0

1,1,2 )()()0( dxxxQrQ nn ,n 0

(4.19)



Issn No : 1006-7930

Page No: 5252

)()0,( ,11,1 xQxD nn ,n 0 (4.20)

)()0,( ,22,2 xQxD nn ,n 0 (4.21)

0

1,1,1 )(),()0,( dyyyxDxR nn ,n 0 (4.22)

0

2,2,2 )(),()0,( dyyyxDxR nn ,n 0 (4.23)

0

20,2

0

10,1 )()()()()0( dxxxQdxxxQrn ,n 0 (4.24)

0

2,2

0

1,1 )()()()()0( dxxxQadxxxQra nnn ,n 1 (4.25)

The normalizing condition is

1

),(),(),(

),()()()(

)(

0

0 0

,2

0 0

,1

0 0

,2

0 0

,1

00

,2

0

,1

1 0

n

nnn

nnnn

n

no

dxdyyxRdxdyyxRdxdyyxD

dxdyyxDdxxdxxQdxxQ

dxxPP

(4.26)

V. QUEUE SIZE DISTRIBUTION

We define the generating functions

1

)(),(

n

nn zxPzxP ;

1

)0(),0(

n

nn zPzP

0

,11 )(),(

n

nn zxQzxQ

0

,11 )0(),0(

n

nn zQzQ

0

,22 )(),(

n

nn zxQzxQ

0

,22 )0(),0(

n

nn zQzQ

0

,11 ),(),,(

n

nn zyxDzyxD

0

,11 )0,(),0,(

n

nn zxDzxD

0

,22 ),(),,(

n

nn zyxDzyxD

0

,22 )0,(),0,(

n

nn zxDzxD

0

,11 ),(),,(

n

nn zyxRzyxR

0

,11 )0,(),0,(

n

nn zxRzxR

0

,22 ),(),,(

n

nn zyxRzyxR

0

,22 )0,(),0,(

n

nn zxRzxR

0

)(),(

n

nn zxzx ;

0

)0(),0(

n

nn zz

Now multiply the steady state equation and steady state boundary conditions (4. 2) to (4.25) by nz and summing

overn,



Issn No : 1006-7930

Page No: 5253

0),())((),(

zxPxadx

zxdP

(5.1)

0

111111 ),,()(),()]()([

),(dyzyxRyzxQxzX

dx

zxdQ

(5..2)

0

222222 ),,()(),()]()([

),(dyzyxRyzxQxzX

dx

zxdQ

(5.3)

0),,()]()([),,(

111 zyxDyzX

dy

zyxdD

(5.4)

0),,()]()([),,(

222 zyxDyzX

dy

zyxdD

(5.5)

0),,()]()([),,(

111 zyxRyzX

dy

zyxdR

(5.6)

0),,()]()([),,(

222 zyxRyzX

dy

zyxdR

(5.7)

0),()]()([),(

zxxzXdx

zxd

(5.8)

The steady state boundary conditions at x = 0 and y = 0 are

0

0

22

0

11

0

)(),()(),()1()(),(),0( PdxxzxQdxxzxQradxxzxzP

(5.9)

0

00

1 ),()(

)(),(1

),0( PdxzxPz

zXdxxazxP

zzQ

(5.10)

0

112 )(),(),0( dxxzxQrzQ

(5.11)

),(),0,( 111 zxQzxD ,n 0 (5.12)

),(),0,( 222 zxQzxD ,n 0 (5.13)

0

111 )(),,(),0,( dyyzyxDzxR

(5.14)

0

222 )(),,(),0,( dyyzyxDzxR

(5.15)



Issn No : 1006-7930

Page No: 5254

0

22

0

11 )(),()(),(),0( dxxzxQadxxzxQraz ,n 0 (5.16)

Solving the above PDE (5.1) to (5.16), it follows that

xexAzPzxP )](1)[,0(),( (5.17)

xzAexBzQzxQ

)(111

1)](1)[,0(),(

(5.18)

xzAexBzQzxQ

)(222

2)](1)[,0(),(

(5.19)

yzheyWzxDzyxD )(111 )](1)[,0,(),,(

(5.20)

yzheyWzxDzyxD )(222 )](1)[,0,(),,(

(5.21)

yzheyFzxRzyxR )(111 )](1)[,0,(),,(

(5.22)

yzheyFzxRzyxR )(222 )](1)[,0,(),,(

(5.23)

xzhexVzzx )()](1)[,0(),( (5.24)

Where ))](())((1[)()( 1111 zhFzhWzhzA ,

))](())((1[)()( 2122 zhFzhWzhzA and ))(1()( zXzh

Inserting (5.17) in (5.10), we obtain

01

)()]())(1()([

),0(),0( P

z

zXAzXzX

z

zPzQ

(5.25)


))((),0(),0( 1112 zABzrQzQ (5.26)


xzAexBzQzxD

)(1111

1))(1)(,0(),0,(

(5.27)


xzAexBzQzxD

)(2222

2))(1)(,0(),0,(

(5.28)


))((),0,(),0,( 111 zhWzxDzxR (5.29)


))((),0,(),0,( 222 zhWzxDzxR (5.30)



Issn No : 1006-7930

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Inserting (5.24) in (5.16), we get

))((),0())((),0(),0( 222111 zABzaQzABzQraz (5.31)

Inserting (5.18), (5.19), (5.24) in (5.9), we get

)(

)(),0(

zDr

zNrzP (5.32)

zrazVrazABrazarVzABzXPzNr ))1())((())(())1())((())(()()( 2211

))1())((())(())1())((())(()())(1()()( 2211 razVrazABrazarVzABAzXzXzzDr


)()]()1)([(),0( 01 zDrAzXPzQ (5.33)


)())(()]()1)(([),0( 1102 zDrzABAzXrPzQ (5.34)


)())(1)(()1)((),0,()(

11011 zDrexBAzXPzxD

xzA (5.35)


)())(1))((()()1)((),0,()(

2112022 zDrexBzABAzXPrzxD

xzA (5.36)


)())(())(1)(()1)((),0,( 1)(

11011 zDrzhWexBAzXPzxR

xzA (5.37)


)())(())(1))((()()1)((),0,( 2)(

2112022 zDrzhWexBzABAzXPzxR

xzA (5.38)

Inserting (5.33), (5.34) in (5.31), we obtain

)())](())(())((()()1)(([),0( 2211110 zDrzABzAarBzABraAzXPz (5.39)

Using (5.32) - (5.39) in (5.17) - (5.24) then we get the limiting probability generating functions

),(),,,(),,,(),,,(),,,(),,(),,(),,( 212121 zxzyxRzyxRzyxDzyxDzxQzxQzxP



Issn No : 1006-7930

Page No: 5256

xexAzDr

zNrPzxP

))(1(

)(

)(),( (5.40)

zrazhVrazABrazharVzABzXzNr ))1())((())(())1())((())(()()( 2211

))1())((())(())1())((())(()())(1()()( 2211 razhVrazABrazharVzABAzXzXzzDr

)())(1)](()1)([(),()(

1011 zDrexBAzXPzxQ

xzA (5.41)

)())(1))((()]()1)([(),(

)(21102

2 zDrexBzABAzXPrzxQxzA

(5.42)

)())(1())(1)(()1)((),,()(

1)(

11011 zDrexBeyWAzXPzyxD

xzAyzh (5.43)

)())(1())(1))((()()1)((),,()(

2)(

2112022 zDrexBeyWzABAzXrPzyxD

xzAyzh (5.44)

)())(1))((())(1)(()1)((),,()(

11)(

11011 zDrexBzhWeyFAzXPzyxR

xzAyzh (5.45)

)())(())(1))((())(1)(()1)((),,( 11)(

22)(

22022 zDrzABexBzhWeyFAzXrPzyxR

xzAyzh (5.46)

)(]))(1))((())((()()1)(([),( )(11220 zDrexVzABzAarBraAzXPzx xzh

(5.47)

The marginal orbit size distributions due to system state of the server, thelimiting probability generating functions,

),(),,,(),,,(),,,(),,,(),,(),,(),,( 212121 zxzyxRzyxRzyxDzyxDzxQzxQzxP

We define the partial probability generating functions as

00

22

0

22

0

11

0

11

0

22

0

22

0

11

0

11

0

22

0

11

0

),()(),()(,),,(),(

,),()(,),,(),(),()(,),,(),(

,),()(,),,(),(,),()(,),()(,),()(

dxzxzdxzxRzRdxzyxRzxR

dxzxRzRdxzyxRzxRdxzxDzDdxzyxDzxD

dxzxDzDdxzyxDzxDdxzxQzQdxzxQzQdxzxPzP

Note that )(),(),(),(),(),(),(),( 212121 zzRzRzDzDzQzQzP

is the probability function of orbit size when the server

is idle, busy, re-service, repair on normal service, repair on re-service, delaying repair on normal service, delaying

repair on re-service, on vacation respectively.

)(

)()(

zDr

zNrzP

(5.48)

zrazhVrazABrazharVzABzXAPzNr ))1())((())(())1())((())(()())(1()( 2211

))1())((())(())1())((())(()())(1()()( 2211 razhVrazABrazharVzABAzXzXzzDr

)()()()(

1))(()(

1

1101 zDrAzh

zA

zABPzQ

(5.49)



Issn No : 1006-7930

Page No: 5257

)()())(()()(

1))(()( 11

2

2202 zDrAzABzh

zA

zABrPzQ

(5.50)

)(1))(()(

))((1)()( 1

1

1111 zDrzhW

zA

zABPAzD

(5.51)

)())((1))(()(

))((1)()( 112

2

2222 zDrzABzhW

zA

zABPrAzD

(5.52)

)())((1))(()(

))((1)()( 11

1

1111 zDrzhWzhF

zA

zABPAzR

(5.53)

)())(())((1))(()(

))((1)()( 2112

2

2222 zDrzhWzABzhF

zA

zABPrAzR

(5.54)

)())(())((()1))(()(()( 11220 zDrzABzAarBrazhVAPzV (5.55)

SinceP0 is the probability that the server is idle when the server is idle when no customer in the orbit and it

can be determined using the normalizing condition

1)1()1()1()1()1()1()1()1( 212121 VRRQQPP . Thus by setting z = 1 in (5.48) - (5.55) and

applying L-Hospital’s rule whenever necessary and we get

)(

)()()(1)()()(1)()()( 22221111

A

VaEFEWEBrEFEWEBEAXEP (5.56)

We define the probability generating functions of the number of customer in the system is

)()()()()()()()()( 212121 zRzRzDzDzQzQzzzPPzK

)))(()(1))(())((())())(1()((

)))(((1))(()()(

2211

2211

zArBrazhaVzABAzXzXz

zArBrzzABAPzK

(5.57)

We define the probability generating functions of the number of customers in the orbit is

)()()()()()()()()( 212121 zRzRzDzDzQzQzzPPzH

)))(()(1))(())((())()1((

1)()(

2211 zArBrazaVzABAzzz

zAPzH

(5.58)

VI.PERFORMANCE MEASURES

We derive the system performance of our model.

The mean number of customers in the system Lsunder steady state condition is obtained by differentiating (5.57)

with respect to z and evaluating at z = 1

)(lim1

zKLz

s



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)()))()(1)((

)))()((1)(()()))()(1)(()()(2

)()))()((1)((

)))()((1)((

)()))()(1)(()))()(1)((

)(2

)))()(1)((2))())()((1)((2

)))()((1)(()))()((1)((2

)()))()((1)(()))()((1)((

)(

2222

11112222

2222

2111

22222111

22221111

22221111

22222

22

2111

21

2

VaEFEWEBrE

FEWEBEVaEFEWEBrEXEA

AFEWEBrE

FEWEBE

VaEFEWEBrEFEWEBE

XE

FEWEBarEVEFEWEBaE

FEGEBEFEWEBrE

VaEFEWEBrEFEWEBE

XE

Ls

The mean number of customers in the system Lqunder steady state condition is obtained by differentiating (5.58)

with respect to z and evaluating at z = 1

)(lim1

zHLz

q

)()()))()((1)(()))()((1)(()(2

1)()))()(1)((2)()))()((1)((2

)))()(1)(()))()((1)((2

)()))()((1)(()))()((1)((

)(

)()()((1)(())()((1)(()())(1(

22221111

22221111

22221111

22222

22

2111

21

2

22221111

AVaEFEWEBrEFEWEBEXE

VEFEWEBarEVEFEWEBaE

FEWEBEFEWEBrE

VaEFEWEBrEFEWEBE

XE

VaEFEWEBrEFEWEBEXEA

Lq

The average time a customer spends in the system Ws and in the orbit Wq under steady state condition due to Little’s

formula, we obtain Ws = Ls / λE(X) and Wq = Lq / λE(X)

We obtain other performance measures

Let U be the steady state probability that the server is busy (server utilization), I be the steady state probability that

the server is idle during the retrial time or on vacation

)())()((1)())()((1)()()(

))()((1)())()((1)()()()()(

22221111

2222111121

VaEFEWEBrEFEWEBEXEA

FEWEBrEFEWEBEXEAPzQzQU

)())()((1)())()((1)()()(

1))()((1)())()((1)()()()1()1(

22221111

22221111

VaEFEWEBrEFEWEBEXEA

FEWEBrEFEWEBEXEAPVPPI

VI.1SPECIAL CASES

In this section, special cases of our model can be deduced.

Case(i):Here we assume that there is no re – service, then r = 0.Our queue size distribution reduces to

)1))(())((()())(1()(

1))(()(

11

11

azhaVzABAzXzXz

zzABPzK

Where )(

)()()(1)()()( 1111

A

VaEFEWEBEAXEP



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Case(ii): No vacation, No breakdown, No retrial this model can be reduced to the following form

Let a = 1, 1)(,021 A

))(()))(())(((

)))(((1))(()(

21

21

zhVzhrBrzhBz

zhrBrzzhBPzK

)()()(1)(where 21 VEBrEBEXEP

Case(iii): Single Poisson arrival, No vacation, No breakdown and No optional re-service then our model can be

reduced to M/G/1 retrial queue. The following form and results agree with Gomez-Corral(1999)

)(

)()(where

))(()())(1()(

1))(()()( 1

1

1

A

BEAP

zhBAzXzXz

zzhBAPzK

VII.CONCLUSION

In this paper, we have discussed a single server queue with customers batches in a system of variable size, but are served

one by one. Further we assume that the service times, vacation times, delay times, repair times each have a general distribution.

We derived the probability generating functions of number of customers in the orbit for different states. Various performance

measures and special cases have analyzed. The results of this work finds applications in mailing system, software designs of

various computer communication systems and satellite communication etc. This work can also be extended further by

incorporating, the concepts of impatient customers, Balking / Reneging, Orbit search, Starting failure, Working Vacation policies

REFERENCES

[1] Artalejo.J.R and Gomez – corral, A retrial queueing system, A computational Approach, Springer Berlin, 2008

[2] Artalejo.J.R, A classified bibliography of research on retrial queues, progress in 1990-1999, 187-211, 7(2), 1999

[3] Artalejo.J.R, Accessible bibliography on retrial queues, Mathematical and Computer modeling, 1-6, 30(3), 1999

[4] Baruah.M, Madhan.K.C, Eldabi.T, Balking and re-service in a vacation queue with batch arrival and two types of heterogeneous service, Journal of Mathematics research, 114-214, 4, 2012

[5] Choudhury.G, A batch arrival queue with a vacation time under single vacation policy,Computers and operations Research,29,14, 2002

[6] Choudhury.G, Deka.K, An M/G/1 retrial queueing system with the two phases of service subject to the server breakdown and repair performance Evaluation, 714-724, 65, 2008

[7] Falin.G.I and Templeton.T.G.C., Retrial queues, Chapman and Hall, London,1997

[8] Falin.G.I, A survey on retrial queues, Queueing systems, 127-167, 7, 1990

[9] Gomez-Corral.A, Stochastic analysis of single server retrial queue with the general retrial times, Naval Research Logistics, 561-581, 1999

[10] Jeyakumar.S & Arumuganathan, A Non-Markovian Queue with Multiple Vacations and Control Policyon Request for Re-service,Quality Technology of Quantitative Management, 253-269, 8, 2011

[11] Keilson.J and Servi.L.D, Oscillating random walk models for G1/G/1 vacation system with Bernoulli schedules, Journal Applied of Probability, 790-802, 23, 1986

[12] Krishnakumar.B, Arivudainambi.D, The M/G/1 retrial queue with Bernoulli schedules and general retrial times, Computer and Mathematics with Application, 15-30, 43, 2002

[13] Kulkarni.V.G and Choi.B.D, Retrial queues with server subject to breakdowns and repairs, Queueing systems, 191-208, 7(2), 1990

[14] Karthikeyan T., Sekaran, K., Ranjith D., Vinoth Kumar V., & Balajee J M. (2019). Personalized Content Extraction and Text Classification Using Effective Web Scraping Techniques. International Journal of Web Portals, 11(2), 41–52. https://doi.org/10.4018/ijwp.2019070103.

[15] Madan.K.C and Baklizi.A., An M/G/1 queue with additional 2nd stage service and optional re-service, Information and Management science, 2002

[16] PavaiMadheswari.S, Krishnakumar.B and Suganthi.P, Analysis of M/G/1 retrial queues with second optional service and customer balking under two types of Bernoulli vacation schedules, Operations research, 2017

[17] Praveen Sundar, P. V., Ranjith, D., Karthikeyan, T., Vinoth Kumar, V., & Jeyakumar, B. (2020). Low power area efficient adaptive FIR filter for hearing aids using distributed arithmetic architecture. International Journal of Speech Technology. https://doi.org/10.1007/s10772-020-09686-y.

[18] Rajadurai.P, Saravanarajan.M.C and Chandrasekaran.V.M, Analysis of an MX/(G1,G2)/1 retrial queueing system with balking, optional re-service under modified vacation policy & service interruption, Ain Shams Engineering Journal, 2014

[19] Wang.J, An M/G/1 queue with second optional service and server breakdowns, Computer and Mathematics with Applications, 13-23, 2004

[20] Yang.T and Templeton.C.J.G, A survey on retrial queues, Queueing systems, 201-233, 2, 1987.



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