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24 CHAPTER 2 ANALYSIS OF A BATCH ARRIVAL GENERAL BULK SERVICE QUEUEING SYSTEM WITH MULTIPLE VACATIONS, SETUP TIME AND SERVER’S CHOICE OF ADMITTING RE-SERVICE 2.1 INTRODUCTION In many practical situations one can observe that the leaving batch of customers may request for re-service. Avi - Itzhak and Naor (1963) have analyzed an M/G/1 queueing model with repair of service station on request by a leaving customer. Rosenberg and Uri Yechiali (1993) analyzed a bulk arrival general service single server queue with single and multiple vacations under LIFO service regime. Huan et al (1995) discussed a computational analysis of M(n)/G/1/N queues with setup time. The interdeparture time distribution for each class in the /1 i /G i M X queue with setup times and repeated server vacations was studied by Frans (1999). Choudhury (2000) analyzed single server Poisson bulk arrival general service queue with a setup period and a vacation period. Madan and Baklizi (2002) considered an M/G/1 queueing model, in which the server performs first essential service to all arriving customers. As soon as the first service is over, they may leave the system with the probability (1- ș) and second optional service is provided with probability ș . Arumuganathan and Ramaswami (2003) analyzed a non-Markovian bulk

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CHAPTER 2

ANALYSIS OF A BATCH ARRIVAL GENERAL BULK

SERVICE QUEUEING SYSTEM WITH MULTIPLE

VACATIONS, SETUP TIME AND SERVER’S CHOICE

OF ADMITTING RE-SERVICE

2.1 INTRODUCTION

In many practical situations one can observe that the leaving batch

of customers may request for re-service. Avi - Itzhak and Naor (1963) have

analyzed an M/G/1 queueing model with repair of service station on request

by a leaving customer. Rosenberg and Uri Yechiali (1993) analyzed a bulk

arrival general service single server queue with single and multiple vacations

under LIFO service regime. Huan et al (1995) discussed a computational

analysis of M(n)/G/1/N queues with setup time. The interdeparture time

distribution for each class in the /1i

/Gi

MX queue with setup times and

repeated server vacations was studied by Frans (1999). Choudhury (2000)

analyzed single server Poisson bulk arrival general service queue with a setup

period and a vacation period.

Madan and Baklizi (2002) considered an M/G/1 queueing model, in

which the server performs first essential service to all arriving customers. As

soon as the first service is over, they may leave the system with the

probability (1- ) and second optional service is provided with probability .

Arumuganathan and Ramaswami (2003) analyzed a non-Markovian bulk

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arrival general bulk service queueing systems with instantaneous Bernoulli

feedback and multiple vacations. Madan et al (2004) analyzed a single server

bulk arrival queue, in which the leaving batch of customers might opt for

re-service. They obtained various performance measures too. Hur et al (2005)

studied single server bulk arrival queueing system with vacations and server

setup. Arumuganathan and Judeth Malliga (2006) analyzed a bulk queue with

re-service of service station and set up time. Al-khedhairi and Lotfi Tadj

(2007) discussed a bulk service queue with a choice of service and re-service

under Bernoulli schedule. Lotfi Tadj and Ke (2008) studied a hysteretic bulk

quorum queue with a choice of service and optional re-service. Madhu Jain

et al (2010) discussed an optimal repairable MX/G/1 queue with multi -

optional services and Bernoulli vacation.

In the literature, queueing models with re-service, the server

accepts all requests for re-service, irrespective of other constraints like

number of customers in the queue, the cost of power and so on. But in reality,

one can observe that, the server may reject the request for re-service with

some probability. In many systems, before the commencement of service, the

server will do some preparatory work such as warming up the machine or

booting the computer, etc; in queueing terminology, such term is referred to

as setup time. Server vacation models are useful for the system in which the

server wants to utilize the idle time for different purposes. Application of

vacation models can be found in production systems, designing local area

networks and data communications systems. Addressing this, the proposed

model /1ba,/GXM queueing system with multiple vacations, setup time

and server’s choice of admitting re-service is developed.

In this chapter, a bulk queueing system with server’s choice of

admitting re-service, multiple vacations and setup time is considered. At a

service completion, the leaving batch may request for re-service with

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SERVICE

RE-SERVICE

MULTIPLE

VACATIONS

aQaQ

aQaQ

aQ

aQ

aQ

1

1

aQ

SETUP TIME

Bulk

arrival

aQ

probability and it is not mandatory to accept it; the server admits this

request with a probability . After the re-service or service completion

without request for re-service, if the queue length is less than a, the server

leaves for a secondary job (vacation) of random length. After this vacation, if

the queue length is still less than ‘a’, the server leaves for another vacation

and so on, until he finally finds at least ‘a’ customers waiting for service. At a

vacation completion epoch, if the server finds at least ‘a’ customers waiting

for service, he requires a setup time to start the service. After a setup time or

on service completion or on re-service completion, if the server finds at least

‘a’ customers waiting for service say , he serves a batch of min ( ,b )

customers, where b a . Analytical treatment of this model is obtained by the

supplementary variable technique. The model under study is schematically

represented in Figure 2.1.

Figure 2.1 Schematic Representation of the Queueing Model

(Q – Queue Length)

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The motivation of the model comes from a real life situation

observed in the Environmental Sensor Networks (ESN). ESN system can

potentially provide a new data for environmental science (eg. climate models)

as well as vital hazard warnings (eg. flood alerts) etc. This is particularly

important in remote or dangerous environments where many fundamental

processes have rarely been studied due to their inaccessibility. A sensor

network is designed to transmit the data from an array of sensors to a data

repository on a server. Monitoring the behavior of ice caps and glaciers is an

important part of our understanding of the Earth’s climate. The environmental

sensor nodes gather data such as glaciers, movement of stones and sediment

under the ice, temperature, pressure, vibration etc. autonomously and pass the

data to the cluster heads. The cluster heads pass the gathered messages

(customers) to the base station. After obtaining the required information, the

base station operator generates various reports and transmits (service) the

report to the application nodes. After getting the reports, the application node

may request some more reports with the same data (re-service). The request

may be accepted or rejected by the base station based on the importance of the

current report generation at that instant. If the number of messages received

by the base station to produce the report is inadequate, then the base station

will do some other associated work such as antivirus running, backup process,

etc., At the completion epoch of the associated work (vacations), if the

number of messages is inadequate, then, the base station will repeat the

associated works until the required number of information reaches to process

it. When the operator returns from the associated work and finds the required

number of messages available in the queue, then the operator begins some

preparatory work, such as checking the application nodes, type of reports

required, etc, for which some amount of time is required, called as setup time.

The above situation can be modeled as /1ba,/GXM queueing system with

multiple vacations, setup time and server’s choice of admitting re-service.

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For the proposed model, the probability generating function (PGF)

of the steady state queue size distribution at an arbitrary time epoch is

obtained using supplementary variable technique. Particular cases and some

special cases are discussed. Various performance measures are derived. A

cost model for the queueing system is developed. Numerical solution for

particular values of parameters is presented.

2.2 MATHEMATICAL MODEL

Let X be the group size random variable of the arrival, be the

Poisson arrival rate. gk

be the probability that ‘k’ customers arrive in a batch

and X (z) be its probability generating function (PGF). Let ‘ ’ be the

probability that a leaving batch request for re-service and ‘ ’ be the

probability that the server accepts a re-service. Let S(x) (s(x)) {~S( ) }[ S

0(x)]

be the cumulative distribution function (probability density function)

{Laplace-Stieltjes transform}[remaining service time] of service. Let V(x)

(v(x)) {~V( ) }[ V

0(x)] be the cumulative distribution function (probability

density function) {Laplace-Stieltjes transform}[ remaining vacation time] of

vacation. Let U(x) (u(x)) { U( )}[ U0(x)] be the cumulative distribution

function (probability density function) {Laplace-Stieltjes transform}

[remaining set up time] of set up. Let R(x) (r(x)) {~R( )}[ R

0(x)] be the

cumulative distribution function (probability density function) {Laplace-

Stieltjes transform} [remaining re-service time] of re-service. Nq(t) denotes

the number of customers waiting for service at time t, Ns(t) denotes the

number of customers under service at time t. The different states of the server

at time ‘t’ are defined as follows:

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0, if theserver is busy with service

1, if theserver is busy with re serviceC(t)

2, if theserver ison vacation

3, if theserver ison setup work

thZ(t) j if theserver ison j vacation startingfrom theidleperiod

To obtain the system equations, the following state probabilities are

define

0P (x, t)dt = Pr N (t) = i, N (t) = j, x S (t) x + dt,C(t) = 0 , a i b, j 0s qi, j

0R (x, t)dt Pr N (t) n, x R (t) x dt,C(t) 1 , n 0n q

0Q (x, t)dt = Pr N (t) = n, x V (t) x + dt,C(t) = 2, Z(t) = j , j 1, n 0qj,n

and

0U (x, t)dt Pr N (t) n, x U (t) x dt,C(t) 3 , n an q

Now, the following system equations are obtained for the queueing

system, using supplementary variable technique:

bP (x t, t t) = P (x, t) 1 t + (1- ) P (0, t)s(x) ti,0 i,0 m,im=a

b+ (1- ) P (0, t)s(x) t R (0, t)s(x) t U (0, t)s(x) t

m,i i im=a

a i b

jP (x - t, t + t) = P (x, t) 1 t + P (x, t) t, a i b - 1, j 1

i, j i, j i, j-k kk=1

bP (x t, t t) = P (x, t) 1 t + (1- ) P (0, t) s(x) tb, j b,j m,b+ jm=a

jb+ (1 - ) P (0, t) s(x) t + P (x, t) g t

m,b+ j b, j-k km=a k=1

+R (0, t) s(x) t + U (0, t) s(x) t; j 1b+ j b+ j

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bQ (x - t, t + t) = Q (x, t) 1 t + (1 - ) P (0, t) v(x) t

1,0 1,0 m,0m=a

b(1 - ) P (0, t) v(x) t R (0, t) v(x ) t

m,0 0m=a

bQ (x - t, t + t) = Q (x, t) 1 t + (1 - ) P (0, t) v(x) tm,n1,n 1,n m=a

b n+ (1 - ) P (0, t) v(x) t + Q (x, t) tm,n 1,n-k km=a k=1

+R (0, t) v(x) t, 1 n a -1n

nQ (x - t, t + t) = Q (x, t) 1 t + Q (x, t) t, n a

1,n 1,n 1,n-k kk=1

Q (x - t, t + t) = Q (x, t) 1 t + Q (0, t) v(x) t, j 2j,0 j,0 j-1,0

Q (x - t, t + t) = Q (x, t) 1 t + Q (0, t) v(x) tj,n j,n j-1,n

n+ Q (x, t) t, 1 n a 1, j 2

j,n-k kk=1

nQ (x - t, t + t) = Q (x, t) 1 t Q (x, t) t, n a, j 2

j,n j,n j,n-k kk=1

bR (x - t, t + t) = R (x, t) 1 t + P (0, t) r(x) t

0 0 m, 0m = a

b nR (x - t, t + t) = R (x, t) 1- t + P (0, t) r(x) t + R (x, t) tn n m,n n-k km=a k=1

n 1

n-aU (x - t, t + t) = U (x, t) 1- t + U (x, t) t + Q (0, t) u(x) t, n an n n-k kk=1 =1 ,nl l

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2.3 STEADY STATE QUEUE SIZE DISTRIBUTION

From the above equations, the steady state queue size equations are

obtained as follows:

b bd- P (x) = - P (x) + (1- ) P (0) s(x) + (1- ) P (0) s(x)

i,0 i,0 m,i m,im=a m=adx

R (0)s(x) U (0)s(x); a i bi i

(2.1)

jd- P (x) = - P (x) + P (x)g , a i b - 1, j 1

i, j i, j i, j-k kdx k=1 (2.2)

b bd- P (x) = - P (x) + (1- ) P (0) s(x) + (1- ) P (0) s(x)

b, j b, j m,b+ j m,b+ jm=a m=adx

j+ P (x) g + R (0) s(x) + U (0) s(x); j 1

b, j-k k b+ j b+ jk=1

(2.3)

bd- Q (x) = - Q (x) + (1- ) P (0) v(x)

1,0 1,0 m,0m=adx

b+ (1 - ) P (0) v(x) + R (0) v(x)

m,0 0m=a

(2.4)

b bd- Q (x) = - Q (x) + (1- ) P (0) v(x) + (1- ) P (0) v(x)m,n m,n1,n 1,n m=a m=adx

n+ Q (x)g + R (0) v(x), 1 n a -1n1,n-k kk=1

(2.5)

nd- Q (x) = - Q (x) + Q (x)g , n a

1,n 1,n 1,n -k kdx k =1 (2.6)

d- Q (x) = - Q (x) + Q (0) v(x) j 2

j,0 j,0 j-1,0dx (2.7)

nd- Q (x) = - Q (x)+Q (0)v(x)+ Q (x)g , 1 n a-1, j 2

j,n j,n j-1,n j,n-k kdx k=1 (2.8)

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nd- Q (x) = - Q (x) + Q (x)g , n a, j 2

j,n j,n j,n-k kdx k=1 (2.9)

bd- R (x) = - R (x) + P (0) r(x)

0 0 m,0m=adx (2.10)

b nd- R (x) = - R (x) + P (0)r(x) + R (x) g , n 1n n m,n n-k km=adx k=1

(2.11)

n-ad- U (x) = - U (x) + U (x) g + Q (0) u(x), n an n n-k kdx k=1 =1 ,nl l

(2.12)

The Laplace –Stieltjes transforms (LST) of P (x)j,n

, Q (x)j,n

, U (x)n and

R (x)n are defined as

x - xP ( ) = e P (x)dx, Q ( ) = e Q (x)dxj,n j,n j,n j,n

0 0,

xU ( ) = e U (x)dxn n0

and xR ( ) = e R (x)dxn n0

Taking LST on both sides of the Equations (2.1) through (2.12), we have

b

am

b

am)(S

~)0(

i,mP)1()(S

~)0(

i,mP)1()(

0,iP~

)0(0,i

P)(0,i

P~

R (0)S( ) U (0)S( ), a i bi i

(2.13)

j

1k1j,1bia,

kg)(

kj,iP~

)(j,i

P~

)0(j,i

P)(j,i

P~

(2.14)

b

am)(S

~)0(

jb,mP)1()(

j,bP~

)0(j,b

P)(j,b

P~

b

am)(S

~)0(

jb,mP)1(

j

1k kg)(

kj,bP~

R (0)S( ) U (0)S( )b j b j

j 1 (2.15)

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bQ ( ) Q (0) Q ( ) (1 ) P (0)V( )

1,0 1,0 1,0 m,0m a

b(1 ) P (0)V( ) R (0) V( )

m,0 0m a

(2.16)

b bQ ( ) Q (0) Q ( ) (1 ) P (0)V( ) (1 ) P (0)V( )m,n m,n1,n 1,n 1,n m a m a

n- Q ( )g - R (0)V( ), 1 n a -1n1,n-k kk=1

(2.17)

nQ ( ) Q (0) Q ( ) Q ( ) g , n a

1,n 1,n 1,n 1,n k kk 1 (2.18)

Q ( ) Q (0) Q ( ) Q (0) V( ) j 2j,0 j,0 j,0 j 1,0

(2.19)

nQ ( ) Q (0) Q ( ) Q ( ) g Q (0) V( ),

j,n j,n j,n j,n k k j 1,nk 1

1 n a 1; j 2

(2.20)

nQ ( ) Q (0) Q ( ) Q ( )g , n a; j 2

j,n j,n j,n j,n k kk 1 (2.21)

bR ( ) R (0) R ( ) P (0)R( )

0 0 0 m,0m a (2.22)

b nR ( ) - R (0) = R ( ) - P (0)R( ) - R ( g ; n 1n n n m,n n-k km=a k=1

(2.23)

n-aU ( ) - U (0) = U ( ) - U ( g - Q (0)U( ); n an n n n-k k ,nk=1 =1 ll

(2.24)

2.3.1 Probability Generating Function

Lee (1991) developed a new technique to find the steady state

probability generating function (PGF) of the number of customers in the

queue at an arbitrary time epoch. To apply the technique, first the following

probability generating functions are defined.

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jP (z, ) = P ( )zi i,jj=0

andj

P (z,0) = P (0)z a i bi i,jj=0

nQ (z, ) = Q ( )zj j,nn=0

and nQ (z,0) = Q (0)z j 1j j,nn=0

nR(z, ) = R ( )znn=0

and nR(z,0) = R (0)znn=0

nU(z, ) = U ( )znn=a and nU(z,0) = U (0)znn=a

(2.25)

whereb

P (z, )ii=a

is the PGF of joint transform of the number of customers in

the queue during busy period. At = 0 , one can get the PGF of the number

of customers in the queue during busy period at an arbitrary time. Similarly,

the other joint transforms Q (z, )j

, R(z, ) and U(z, ) are defined.

The probability generating function P(z) of the number of

customers in the queue at an arbitrary time of the proposed model can be

obtained using the following equation.

b-1P(z) = P (z,0) + P (z,0) + Q (z,0) + R(z,0) + U(z,0)

bi ji=a j=1 (2.26)

In order to find P (z,0)i

, P (z,0)b

,Q (z,0)j

, R(z,0) and U(z,0) , the

following sequence of operations are done.

Multiplying (2.16) by z0, (2.17) by z

n (1 n a-1), (2.18) by z

n

(n a), summing up from n = 0 to and using (2.25), we get

a 1 b n n( X(z))Q (z, ) Q (z,0) V( ) (1 ) P (0)z R (0)zm,n n1 1 m an 0

(2.27)

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Multiplying (2.19) by z0, (2.20) by z

n (1 n a-1), (2.21) by z

n

(n a) , summing up from n = 0 to and using (2.25), we get

a-1 n( X(z))Q (z, ) = Q (z,0) - V( ) Q (0)z , j 2

j j j-1,nn=0 (2.28)

Multiplying (2.22) by z0, (2.23) by z

n ( n 1), summing up from

n = 0 to and using (2.25), we get

b( X(z))R(z, ) R(z,0) R( ) P (z,0)mm a

(2.29)

Multiplying (2.24) by zn ( n a) , summing up from n = a to and

using (2.25), we get

a 1 n( X(z))U(z, ) U(z,0) U( ) Q (z,0) Q (0)z

nn 01 l l,l (2.30)

Multiplying (2.13) by z0, (2.14) by z

j ( j 1) , summing up from

j = 0 to and using (2.25), we get

b( X(z))P (z, ) = P (z,0) -S( ){(1- ) P (0)

i i m,im=a

b+ (1- ) P (0) +R (0) U (0)}

m,i i im=a, a i b-1 (2.31)

Multiplying (2.13) by z0 with i = b, (2.15) by z

j ( j 1) , summing

up from j = 0 to and using (2.25), we get

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b bX(z))z P (z, ) = z P (z,0)b b

b b-1 j(1- ) (P (z,0) - P (0)z )m m,jm=a j=0

b b-1 j-S( ) + (1- ) (P (z,0) - P (0)z )m m,jm=a j=0

b-1 b-1n n+(R(z,0) - R (0)z ) +(U(z,0) - U (0)z )n nn=an=0

(2.32)

By substituting = – X(z) in the Equations (2.27) through

(2.32), we get

a-1 b n nQ (z,0) = V( X(z)) (1- ) P (0)z + R (0)zm,n n1 m=an=0

(2.33)

a-1 nQ (z,0) = V( X(z)) Q (0)z j 2

j j-1,nn=0 (2.34)

bR(z,0) = R( X(z)) P (z,0)mm=a

(2.35)

a-1 nU(z,0) = U( X(z)) Q (z,0) - Q (0)z

,nn=0=1 l ll (2.36)

b bP (z,0) = S( X(z)) (1- ) P (0) + (1- ) P (0) + R (0) +U (0)i m,i m,i i im=a m=a

a i b-1 (2.37)

and

b b-1 j(1- ) P (z,0) - P (0)zm m,jm=a j=0

b b-1 jbz P (z,0) = S( X(z)) (1- ) P (z,0) - P (0)zmb m,jm=a j=0

b-1 b-1n n+ R(z,0) - R (0)z + U(z,0) - U (0)zn nn=an=0

(2.38)

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Letb

p = P (0), q = Q (0), u = U (0), r = R (0)i m,i i i i i i i

m = a =1l,

l

and

k 1 p q ri i i i

, c 1 p u ri i i i

From the Equation (2.38), we get

S( X(z)) f(z)P (z,0) =b b

z - (1- )S( X(z)) - S( X(z))R( X(z)) (2.39)

where f(z)=b-1 m(1- ) + R( X(z)) S( X(z)) - z cmm=a

a-1 m+ U( X(z)) V( X(z))k - q - (1- )p + r zm m m m

m=0

From the Equations (2.27) and (2.33), we get

a-11 nQ (z, ) = V( X(z)) - V( ) (1- )p + r zn n1 X(z) n=0 (2.40)

From the Equations (2.28) and (2.34), we get

a-11 nQ (z, ) = V( X(z)) - V( ) Q (0)z , j 2j j-1,nX(z) n=0

(2.41)

From the Equations (2.29) and (2.35), we get

b-11R(z, ) = R( X(z)) - R( P (z,0)mm=aX(z)

(2.42)

From the Equations (2.30) and (2.36), we get

a-11 nU(z, ) = U( X(z)) - U( ) Q (z,0) - q znX(z) n=0=1 ll (2.43)

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From the Equations (2.31) and (2.37), we get

1P (z, ) = S( X(z)) -S( ) c , a i b 1i iX(z)

(2.44)

From the Equations (2.32) and (2.39), we get

S( X(z)) -S( ) f(z)P (z, ) =b b

X(z) z - (1- )S( X(z)) - S( X(z))R( X(z))(2.45)

Substituting P (z,0),i

P (z,0)b

, Q (z,0)j

, R(z,0) and U(z,0) from the

Equations (2.40) – (2.45) in the Equation (2.26), the probability generating

function of the queue size P(z) at an arbitrary time epoch is obtained as

b 1S( X(z)) 1 c

f (z) S( X(z)) 1ii aP(z)X(z) h(z)( X(z))

a 1 b 1iV( X(z)) 1 k z R( X(z)) 1 S( X(z))ci mi 0 m a

X(z) ( X(z))

R( X(z)) 1 S( X(z))f (z)

h(z)( X(z))

a 1 a 1n n[U( X(z)) 1] V( X(z)) k z q zn nn 0 n 0

( X(z))

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On further simplification, we get

m m

m m

b

(S( X(z))-1) b-1 b m(z -z )cmR( X(z))-1 (S( X(z)) m=a

1-S( X(z)) a-1m+ (1- )p +r z

R( X(z))-1 S( X(z)) m=0

a-1b m+ (z -1) U( X(z)) V( X(z))k -q z

m=0

z -S( X(-

P(z)=

z)) a-1 m(k -q )zm mR( X(z))-1 S( X(z)) m=0

h(z)(- X(z)) (2.46)

where bh(z) z (1 )S( X(z)) S( X(z))R( X(z))

The probability generating function P(z) has to satisfy the condition

P(1) = 1. In order to satisfy the condition, applying L’Hospital’s rule in and

evaluating P(z)1z

lim and equating the expression to 1, we have to satisfy

b 1 2 E(X) E(S) + E(R) (b m)cm

m a

a 1 a 1(X,S, R) (1 )p r (X,S, R) q km m m m

m 0 m 0

a 12b E(X)E(V)k 2 E(X) b - E(X) E(S) + E(R)m

m 0

where

2 2 2 2(X,S, R) 2 mE(S)E(X) E (X)E(S ) E(S)E(X )

2 2 2 2 2 22 mE(R)E(X) 2 E (X)E(R) E(R)E(X ) E(R )E (X)

and

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2(X,S,R) 2m b E(X)E(S) E(X)E(R) b(b 1) E(X )E(S)

2 2 2 2 2 2 2 2E (X)E(S ) E (X)E(R ) 2 E (X)E(S)E(R)

2E(X )E(R) m(m 1) (m b)(m b 1) 2b E(X)E(G)

Since p , q , r and ui i i i

are the probabilities of ‘i’ customers being

in the queue at service completion epoch, vacation completion epoch, re-

service completion epoch and set up time completion epoch, respectively, it

follows that the left hand side of the above expression must be positive. Thus

P(1) = 1 is satisfied if and only if b - E(X) E(S) + E(R) 0 .

Define ‘ ’ asE(X) E(S) + E(R)

b. Thus < 1is the condition

to be satisfied for the existence of steady state for the model under

consideration.

2.3.2 Computational Aspects of Unknown Probabilities

Equation (2.46) gives the probability generating function P (z) of

the number of customers in the queue at an arbitrary time epoch, which

involves ‘b + 2a’ unknown probabilities namely, p ,p ,p ,.......p ,0 1 2 a 1

r ,r , r ,.......r ,0 1 2 a 1

q ,q ,q ,.......q ,0 1 2 a 1

c ,c ,.....,ca a 1 b 1. Using Lemma (2.1)

and theorems (2.1) and (2.2), q and ri i

; i = 0 to a-1 are expressed in terms of

pi; i = 0 to a-1.

Now, the Equation (2.46) contains only ‘b’

unknowns p ,p ,p ,.......p , c ,c ,.....,ca0 1 2 a 1 a 1 b 1. By Rouche’s theorem, the

expression bz (1 )S( X(z)) S( X(z))R( X(z)) has b-1

zeros inside and one on the unit circle 1z . Since P (z) is analytic within

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and on the unit circle, the numerator of (2.46) must vanish at these points,

which gives ‘b’ equations and ‘b’ unknowns. These equations can be solved

by suitable numerical techniques.

Lemma 2.1

Let i be the probability that ‘i’ customers arrive during a vacation.

The probability generating function of i is given by

iz V X(z)i

i=0

Proof:

Conditioning on the actual vacation length, number of arrivals and

the group size, we get

m (m)ti e t g

i d V(t)i m=0 m !0

where(m)

gi

is the m – fold convolution of gi with itself (i.e., total of m

arrivals make ‘i’ customers)

Multiplying the above equation by z i

and taking the summation

from i = 0 to , we get

mti - t (m) i

z = e g z dV(t)i im!i=0 0 m=0 i=m

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m mt) [X(z)]t

= e d V(t)m !0 m=0

= V X(z) (2.47)

Hence theLemma

Theorem: 2.1

The constants rn involved in P(z) are expressed in terms of pn as,

nr = pn i n - ii=0

, n=0,1,2,…a-1, whereiis the probability that ‘i’

customers arrive during a re-service period.

Proof:

From the Equation (2.35), we have

bR(z,0) P (z,0)R( X(z))mm a

b jnR (0)z = R( X(z)) P (0)zn m,jm=an=0 j=0

jnr z = R( X(z)) p zn jn=0 j=0

which, using Lemma 2.1, gives

jn nr z z p zn n jn 0 n 0 j 0

Equating the coefficients of zn

, n=0,1,2,…a-1, we get

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nr = pn i n-ii=0

; n=0,1,2,…a-1 (2.48)

Hence the theorem

Theorem: 2.2

The constants qn involved in P(z) are expressed in terms of pn as,

n

0i ip

inbnq ; n = 0,1,2,3,……….a-1, where

(1 )0 0 0b

0 10

and

n n(1 ) bn n i i i n ii 0 i 1bn

10

; n = 1,2,3,…..…a-1 and i is

the probability that ‘i’ customers arrive during a vacation period.

Proof:

From the Equations (2.33) and (2.34), we have

1j)0,z(

jQ

a 1 a 1n nV( X(z)) (1 )p r z q zn n n

n 0 n 0

a 1n nq z V ( X(z)) (1 )p r q zn n n nn 0 n 0

which, using Lemma 2.1, gives

a 1n n nq z z (1 )p r q zn n n n n

n 0 n 0 n 0

a 1 n a 1n n(1 )p r q z (1 )p r q z

n i i i i n i i i in an 0 i 0 i 0

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Equating the coefficients of zn on both sides of the above equation

for n = 0,1,2,3,…a-1, we getn

q (1 )p r qn n i i i ii 0

Substituting ri and solving it for qn, we get

n n n n 1(1 ) p p q

n i i n k i i k n i ii 0 i 0k 0 i 0qn1

0

Coefficient of pn in qn is(1 )

0 0 0 b01

0

Coefficient of pn-1 in qn is [ 1 + 1 Coefficient of pn-1 in qn-1]/1- 0

(1 ) b1 1 0 0 1 1 0

b11

0

Coefficient of pn-2 in qn is [ 2 + 1 Coefficient of pn-2 in qn-1

+ 2 Coefficient of pn-2 in qn- 2]/1- 0

(1 ) ( b b )2 2 0 1 1 0 2 1 1 2 0

b21

0

Proceeding like this, we get

Coefficient of p0 in qn is

n n(1 ) bn n i i i n ii 0 i 1 bn

10

Therefore,n

0i ip

inbnq ; n = 0,1,2,3,……….a-1 (2.49)

Hence the theorem

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2.4 PERFORMANCE MEASURES

In this section, some useful performance measures of the proposed

model like, expected number of customers in the queue E(Q), expected

length of idle period E(I), expected length of busy period E(B) are derived

which are useful to find the total average cost of the system. Also, probability

that the server is on setup work P(U), probability that the server is on vacation

P(V) and probability that the server is busy P(B) are derived.

2.4.1 Expected Queue Length

The expected queue length E (Q) (i.e. mean number of customers

waiting in the queue) at an arbitrary time epoch, is obtained by differentiating

P(z) at z = 1and is given by

)Q(E)z(P1z

lim

b-1 a-1f X,S, R c + f X,S, R,U, V km m1 2m=a1 m=0

E(Q) =2 2 2 a-124 E (X)T

f (X,S, R) - f (X,S,R, U) (1- )p + r )1 m m3 4m=0

(2.50)

where S1 = E(X)E(S) , 2 2 2S2 = X (1)E(S) + E (X)E(S ) ,

2 2S3 = E (X)E(S)E(R) ,S4 = X (1)E(S) , 2 2S5 = X (1)E(X)E(S ) ,

3 3 3S6 = E (X)E(S ) , S7 = X (1)E(S) , 2 2S8 = X (1)E(S ) + X (1)E(R ) ,

2S9 = E(S) + E(R) X (1) , R1 = E(X)E(R) ,

2 2 2R2 = X (1)E(R) + E (X)E(R ) , 2 2R3 = E (X)E(X)E(S)E(R)

R4 = X (1)E(R) , 2 2R5 = X (1)E(X)E(R ) , 3 3 3R6 = E (X)E(R ) ,

R7 = X (1)E(R) , 2 2R8 = E(R )E(S) + E(R)E(S ) ,

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V1 = E(X)E(V) , 2 2 2V2 = X (1)E(V) + E (X)E(V ) ,

U1 = E(X)E(U) , 2 2 2U2 = X (1)E(U) + E (X)E(U ) , 2 2U3 = E (X)E(U)E(V) ,

T1 = b -S1- R1 , T2 = S1+ R1 , T3 = S2 + R2 , T6 = S6 + R6 ,

T7 = S6 + S7 + 3S5 , 3 3T8 = 3R5 + 6R3 + R4 + R6 + 3 E (X)R8 + 6mS3 ,

f1 = 6 E(X) b(b -1) -S2 - R2 - 2 S3+ 6 X (1)T1 , f2 = 2m T2 + T3+ 2 S3 ,

2 3 3f3 = 4 b(b -1)(b - 2)E(X) -12 E(X) 3R3 + E (X)R8 - 4 E(X)T6 -8 X (1)T2

2 3 2+4 X (1)b -6 S9 + 6 b(b -1)X (1) -18 E (X)S8 ,

2 2E1 = (b - b + m - m )T2 + (b - m)(T3+ 2 S3) , E2 = 2(b - m) f1 T2 ,

2E3 = 3(m - m)T2 + 3mT3+ T7 + T8 ,

E4 = 3m(m -1)T1+ 3m(b(b -1) - S2 - ( R2 + 2S3)) ,

3 3E5 = b(b -1)(b - 2) + S6 -S4 -3S5 + (R6 -3 E (X)R8- R4 -6R3-3R5) ,

E6 = (m + b)(m + b -1)(m + b - 2) - m(m -1)(m - 2) ,

E7 = 2mT1+ b(b -1) -S2 - (R2 + 2S3) ,

E8 = (m + b)(m + b -1) - m(m -1) , 2E9 = 2(2mb + b - b)V1+ +3b(V2 + 2U3)

f (X,S, R) = 12 E(X)T1E1-E21

; f (X,S, R,V, U) = 4 E(X)T1E9-f1(2b)V12

f (X,S,R) = f1f2 + 3f3T2-4 E(X)T1E33

and

2f (X,S, R, U) = 4 E(X)T1 E4 + E5 - E6 - U1 3(2mb + b - b) - 3b U24

-f1 E7 - 2b U1- E8 - 3f3 T1- b

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2.4.2 Expected Length of Idle Period

Let I be the idle period random variable, then the expected length of

the idle period is given by,

E(I) = E(I1) + E(U),

where I1 is the random variable denoting ‘ idle period due to multiple vacation

process’ and E(U) is the expected length of setup time.

To find E(I1), another random variable U1 is defined as,

0, if the server findsat least ‘a’ customers after the first vacationU =

1 1, if the server finds less than‘a’ customers after the first vacation

Now, the expected length of idle period due to multiple vacations

E(I1) is given by

E(I1) = E(I1 / U1 = 0)P(U1 = 0) + E(I1 / U1= 1)P(U1 = 1)

= E(V)P(U1 = 0) + (E(V) + E(I1))P(U1 = 1)

where E(V) is the expected vacation time.

Solving for E(I1), we have

E(V)E I

1P U 0

1

(2.51)

To find P (U1 = 0), we do some algebra using the Equation (2.33),

then

a 1n nQ (z,0) Q (0)z V( X(z)) (1 )p r zn n1 1,nn 0 n 0

a 1n nz (1 )p r zn n n

n 0 n 0

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Equating the coefficients of zn (n=0,1,2,…..a-1) on both sides, we

get

nQ (0) (1 )p r

1,n i n i n ii 0

Therefore,

a 1P(U 0) 1 Q (0)

1 1,nn 0

a 1 n1 (1 )p r

i n i n in 0 i 0

(2.52)

Using (2.51) and (2.52), the expected length of idle period E(I) is obtained as

E(V)E(I) = + E(U)

a-1 n1- (1- )p + r

i n-i n-in=0 i=0

(2.53)

where E(V) is the expected vacation time

2.4.3 Expected Length of Busy Period

Let B be the busy period random variable. Let ‘T’ be the residence

time that the server is rendering service or under re-service. Therefore, T=S

with probability (1 ) and T=S+R with probability .

Another random variable J is defined as,

0, if the server finds less than‘a’ customers after a residence timeJ=

1, if the server findsat least ‘a’ customers after a residence time

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Now, the expected length of busy period E(B) is given by

E(B) = E(B / J = 0)P(J = 0) + E(B / J = 1)P(J = 1)

Solving for E(B), we get

E(B)E(S)

E(R)P(J 0)

Thus, the expected length of busy period is obtained as

E(S)E(B) = + E(R)

a-1(1- )p + rn n

n=0

(2.54)

where E(R) is the expected re-service time and E(S) is the expected service

time.

2.4.4 Probability that the Server is on Setup Work

Let U be the setup period random variable and P (U) be the

probability that the server is on setup work.

From the Equation (2.43), we have

a-11 nU(z,0) = U( X(z)) -1 V( X(z)) (k - q )zn nX(z) n=0

Now, the probability that the server is on setup work is given by

P (U) = lim U(z,0)z 1

=

a-1 nU( X(z)) -1 V( X(z)) (k - q )zn nn=0

limz 1 X(z)

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P (U) =a-1

E(U) (1- )p + rn nn=0

(2.55)

where E(U) is the expected setup period.

2.4.5 Probability that the Server is on Vacation

Let V be the random variable for multiple vacations and P (V) be

the probability that the server is on multiple vacations at time t.

From the Equations (2.40) and (2.41), we have

a-1 a-1V( X(z)) -1 n nQ (z,0) = (1- )p + r z + Q (0)zn n j-1,nj X(z) n=0 n=0j=2j = 1

a-1 a-1V( X(z)) -1 n n= (1- )p + r z + q zn n nX(z) n=0 n=0

a-1V( X(z)) -1 n= k znX(z) n=0

, where k 1 p q rn n n n

Now, the probability that the server is on vacation is given by

P(V) = lim Q (z,0)jz 1 j = 1

=a-1V( X(z)) -1 nlim k zn

z 1 X(z) n=0

P(V) =a-1

E(V) knn=0

(2.56)

where E(V) is the expected vacation period.

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2.4.6 Probability that the Server is Busy

Let B be the busy period random variable and P (B) be the

probability that the server is busy at time t.

From the Equations (2.42), (2.44) and (2.45), we have

P(B) =b

lim P (z,0) R(z,0)iz 1 i=a

=b-1

lim P (z,0) + P (z,0) + R(z,0)i bz 1 i=a

Now, the probability that the server is busy is given by

P(B) =b-1 b-1f (1) E(R)

E(S) c + + f (1) + (T1)ci iT1 T1i=a i=a

(2.57)

where

m m

b-1 a-1f (1) = E(X)E(S) + E(R)E(X) - m c + (1- )p + r E(X)E(U)m

m=a m=0

and T1 = b - E(X) E(S) + E(R)

2.5 PARTICULAR CASES

In this section, some of the existing models are deduced as a

particular case of the proposed model.

Case (i): If no request for re-service (i.e. 0 ) and if there is no setup time

i.e., U( ( )) 1X z , then the Equation (2.46) reduces to

1a

0m

mzm

k)1bz(]1))z(X(V~

[

1b

amm

k)m

zb

z(]1))z(X(S~

[

))z(X)(z(h

1)z(P

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where ))z(X(S~bz)z(h and mqmpmk , which exactly coincides

with the result XM / G(a,b) /1 and multiple vacations without setup time and

N-policy of Krishna Reddy et al (1998).

Case (ii): Considering single service (i.e. a = b = 1), no request for re-service

(i.e. 0 ) and if there is no setup time (i.e. U( ( )) 1X z ), then (2.46)

becomes

V( X(z)) -1 z -1 k0P(z) =

z -S( X(z)) X(z) - , where k = p + q

0 0 0

which coincides with the result XM / G /1 queueing system and multiple

vacations without N-Policy of Lee et al (1994).

2.5.1 Special Cases

The model so developed is general in nature as the service time,

set-up time and vacation time are arbitrary. But for practical purposes,

service time, set-up time and vacation time with particular distribution is

required. In this section, some special cases of the proposed model by

specifying vacation time random variable as exponential distribution, setup

time random variable as exponential distribution and bulk service time

random variable as hyper exponential distribution are discussed.

Case (i): /1ba,/GXM queueing system with setup time, server’s choice of

admitting re-service and exponential vacation time

If the vacation time is assumed as exponential with probability

density functionx

v(x) e , where is the parameter, then,

V( X(z))(1 X(z))

. Substituting this expression for V( X(z)) in

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(2.46) and after some algebra, the PGF of the queue size distribution of this

special case of the queueing model is obtained as,

(S( X(z)) 1) b 1 b m(z z )cmm aR( X(z)) 1 (S( X(z))

1 S( X(z)) a 1 m(1 )p r zm m

R( X(z)) 1 S( X(z)) m 0

a 1 b m(z 1) U( X(z)) k q zm m(1 X(z))m 0

bz

P(z)

a 1S( X(z)) m(k q )zm mm 0R( X(z)) 1 S( X(z))

h(z)( X(z))

where bh(z) z (1 )S( X(z)) S( X(z))R( X(z))

Case (ii): /1ba,/GXM queueing system with multiple vacations, server’s

choice of admitting re-service and exponential setup time

In case of exponential setup time random variable with probability

density function uxu(x) u e , where ‘u’ is the parameter, then,

uU( X(z))

u (1 X(z)). Substituting this expression for U( X(z)) in

(2.46) and after some algebra, the PGF of the queue size distribution of this

special case of the queueing model is obtained as,

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(S( X(z)) 1) b 1 b m(z z )cmm aR( X(z)) 1 (S( X(z))

1 S( X(z)) a 1 m(1 )p r zm m

R( X(z)) 1 S( X(z)) m 0

a 1 ub m(z 1) V( X(z))k q zm m

u (1 X(z))m 0

bz S( X(z

P(z)

a 1)) m(k q )zm m

m 0R( X(z)) 1 S( X(z))

h(z)( X(z))

and bh(z) z (1 )S( X(z)) S( X(z))R( X(z))

Case (iii): /1ba,/GXM queueing system with hyper exponential bulk

service time, setup time and server’s choice of admitting re-service

If the service time is assumed as hyper- exponential service time

with the probability density functionyx wxs(x) cye (1 c)we , where y and

w are the parameters, then,yc w(1 c)

S( X(z))y (1 X(z)) w (1 X(z))

.

Substituting this expression for S( X(z)) in (2.46) and after some algebra,

the PGF of the queue size distribution of this special case of the queueing

model is obtained as,

b 1 b m(z z )cmm a

P(z)

yc w(1 c)1

y (1 X(z)) w (1 X(z))

yc w(1 c)R( X(z)) 1

y (1 X(z)) w (1 X(z))

yc w(1 c)1

y (1 X(z)) w (1 X(z)) a 1 mz

m 0

a 1 b m(z 1) U( X(z)) V( X(z))k q zm m

m 0

(1 )p rm myc w(1 c)

R( X(z)) 1y (1 X(z)) w (1 X(z))

yc w(1 c)bz

y (1 X(z)) w (1 X(z)) a 1 m(k q )zm m

m 0

h(z)( X(z))

yc w(1 c)R( X(z)) 1

y (1 X(z)) w (1 X(z))

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where

yc w(1- c)bh(z) = z - (1- ) +

y + (1- X(z)) w + (1- X(z))

yc w(1- c)- + R( X(z))

y + (1- X(z)) w + (1- X(z))

2.6 COST MODEL

Cost analysis is the most important phenomenon in any practical

situation at every stage. Cost involves startup cost, operating cost, holding

cost, setup cost, re-service cost and reward cost. It is quite natural that the

management of the system desires to minimize the total average cost and to

optimize the cost. Addressing this, in this section, the cost model for the

proposed queueing system is developed and the total average cost is obtained

with the following assumptions:

Cs : Startup cost per cycle

Ch

: Holding cost per customer per unit time

Co : Operating cost per unit time

Cr : Reward cost per cycle due to vacation

Crs : Re-service cost per unit time

Cu : Setup cost per cycle

Since the length of the cycle is the sum of the idle period and busy

period, from the Equations (2.53) and (2.54), the expected length of cycle,

cE(T ) is obtained as

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E (Tc) = E (length of the Idle Period) +E (length of the Busy Period)

E (Tc) =E(V)

+ E(U)a 1 n

1 (1 )p ri n i n in 0 i 0

+E(S)

E(R)a 1

(1 )p rn nn 0

Now, the total average cost (TAC) per unit time is obtained as

Total Average Cost = Start-up cost per cycle + holding cost of number of

customers in the queue per unit time + Operating cost

per unit time * + re-service cost per unit time +

Setup cost per cycle – reward due to vacation per

cycle.

TAC =E(V) 1

C - C + C E(U) + C E(R) + C E(Q) + C us r rs ohP(U = 0) E(T )c

(2.58)

It is difficult to have a direct analytical result for the optimal value

a* (minimum batch size in M

X/G(a,b)/1 queueing system) to minimize the

total average cost. The simple direct search method to find optimal policy for

a threshold value a* to minimize the total average cost, is defined.

Step 1: Fix the value of maximum batch size ‘b’

Step 2: Select the value of ‘a’ which will satisfy the following relation

*TAC a TAC(a), 1 a b

Step 3: The value a* is optimum, since it gives minimum total average cost.

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Using the above procedure, the optimal value of ‘a’ can be

obtained, which minimizes the total average cost function. Some numerical

example to illustrate the above procedure is presented in the next section.

2.7 NUMERICAL ILLUSTRATION

In this section, the consistency of the theoretical results obtained in

the sections 2.3 – 2.4 are justified numerically with the following assumptions

and notations:

Service time distribution is 2 – Erlang with parameter

Batch size distribution of the arrival is geometric with mean 2

Probability of requesting re-service

Probability of admitting re-service

Re-service time is exponential with parameter

Vacation time is exponential with parameter

Setup time is exponential with parameter u

Minimum service capacity a

Maximum service capacity b

2.7.1 Effects of Various Parameters on the Performance Measures

The effects of various parameters such as arrival rates, expected

queue length, expected idle period, expected busy period, probability that the

server is on vacation, probability that the server is busy, probability of request

for re-service, probability of server’s choice of admitting customers for re-

service and threshold value ‘a’ are analyzed numerically and presented in

Tables 2.1 – 2.3 and represented in Figure 2.2. All numerical results are

obtained using Mat Lab software.

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The effects of different probabilities of server’s choice of admitting

re-service ‘ ’ for a fixed ‘a’ and ‘b’ on various performance measures are

obtained numerically. These results are tabulated in Table 2.1. From the

table, the following observations are made:

If the probability of server’s choice of admitting re-service

increases, then

the probability that the server is on vacation decreases

the probability that the server is busy increases

the expected queue length increases

the expected idle period decreases

the expected busy period increases

The effects of different arrival rates for a fixed ‘a’ and ‘b’ are

compared with various performance measures and are presented in Table 2.2.

From the table, it is clear that, if the arrival rate increases, then

the probability that the server is on vacation decreases

the probability that the server is busy increases

the expected queue length increases

the expected idle period decreases

the expected busy period increases

In Table 2.3 and Figure 2.2, for various threshold values ‘a’, the

expected queue length, the expected idle period and the expected busy period

are compared with different probabilities of server’s choice of admitting re-

service. From the table and the figure, the following points are observed:

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When the probability of admitting re-service increases, the

mean queue size and the mean busy period increase

When the threshold value increases, the mean queue size and

mean busy period increase whereas the mean idle period

decreases

2.7.2 Optimal Cost

In this section, a numerical example is analyzed to illustrate how

the management of an Environmental Sensor Networks (ESN) can effectively

use the results obtained in the sections 2.3, 2.4 and 2.6 to make the decision

regarding the threshold value to minimize the total average cost.

It is assumed that, the maximum capacity of the ESN is 10 units

(i.e. b = 10 messages). If the ESN operator starts the process even for a single

message (i.e. a = 1) without waiting for further arrival, clearly, the operating

cost will increase. On the other hand, if they start the process until all 10

messages arrive, the holding cost may increase; hence, there must be some

value between 1 and 10 that will optimize the cost. An optimal policy

regarding the threshold value ‘a’ which will minimize the total average cost is

wished to be obtained.

The total average costs are obtained numerically with the following

assumptions:

Startup cost : ` 3.00

Operating cost per unit time : ` 5.00

Holding cost per customer : ` 0.25

Re-service cost per unit time : ` 2.00

Reward cost per unit time due to vacation : ` 1.00

Setup cost per unit time : ` 1.00

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The effects of threshold value ‘a’ on the total average cost with b = 10

are reported in Tables 2.3 and 2.4 and represented in Figures 2.3 and 2.4.

From the Table 2.3 and the Figure 2.3, it is clear that, for an

environmental sensor networks center with the capacity of 10 messages at a

time, the management has to fix the threshold value a = 5 to minimize the

total average cost for the probability of requesting re-service 0.6

(i.e. 0.6 ) and the probability of server’s choice of admitting re-service 0.2

(i.e. 0.2 ). Similarly, the management has to fix the threshold value

a = 4 to minimize the total average cost for the probability of requesting

re-service 0.6 (i.e. 0.6 ) and the probability of server’s choice of admitting

re-service 0.4 (i.e. 0.4 ).

Similarly, the management has to fix the threshold value ‘a’ to

minimize the total average cost for various probabilities of requesting re-

service and different probabilities of server’s choice of admitting re-service.

These values are presented in Table 2.4 and represented in Figure 2.4.

2.8 CONCLUSION

In this chapter, a “ /1ba,/GXM queueing system with server’s

choice of admitting re-service, multiple vacations and setup time” is analyzed.

The probability generating function for the queue size at an arbitrary time

epoch is derived. Various performance measures are also obtained. Some

particular cases and special cases are also discussed. The theoretical

development of the model is justified with numerical results. The results so

obtained in this chapter can be used for managerial decision to optimize the

overall cost and search for the best operating policy in a waiting line system.

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Table 2.1 Probability of Server’s Choice of Admitting Customers (Vs)

Performance Measures

(For 2.5 , a =3, b = 4, 2.0 , 0.6 , 6 , 5 , u = 4)

P(V) P(B) E(Q) E(I) E(B)

0.0 0.3062 0.5995 4.2853 0.5319 1.3258

0.2 0.2864 0.6269 4.7467 0.5228 1.4617

0.4 0.2651 0.6528 5.2338 0.5175 1.5626

0.6 0.2445 0.6796 5.8252 0.5108 1.7060

0.8 0.2239 0.7065 6.5247 0.5043 1.8749

1.0 0.2031 0.7334 7.3580 0.4982 2.0689

P (V) - Probability that the server is on multiple vacations; P (B) - Probability that the

server is busy; E(Q) – Expected queue length; E(I) – Expected idle period; E(B) – Expected

busy period

Table 2.2 Arrival Rate (Vs) Performance Measures

(For 2.0 , a =3, b = 4, 0.4 , 0.6 , 6 , 5 , u = 4)

P(V) P(B) E(Q) E(I) E(B)

1.5 0.5295 0.3659 2.6919 0.5625 1.2346

2.0 0.3916 0.5081 3.5863 0.5467 1.2866

2.5 0.2651 0.6528 5.2338 0.5175 1.5626

3.0 0.1488 0.7978 9.9946 0.4868 2.3810

3.5 0.0414 0.9418 44.3704 0.4597 7.5073

P (V) - Probability that the server is on multiple vacations; P (B) - Probability that the

server is busy; E(Q) – Expected queue length; E(I) – Expected idle period; E(B) – Expected

busy period

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Table 2.3 Threshold Value (Vs) Performance Measures and Total

Average Cost

(For 0.5 , 2.0 , b = 10, 3 , 8 , u = 7)

a0.2; 0.6 0.4; 0.6

E(Q) E(I) E(B) TAC E(Q) E(I) E(B) TAC

1 0.2829 0.3260 1.5222 1.9854 0.2931 0.3242 1.5950 1.9897

2 0.5855 0.3115 1.9065 1.7934 0.5924 0.3108 1.9673 1.8142

3 0.9931 0.3024 2.3051 1.6931 0.9969 0.3020 2.3607 1.7199

4 1.4492 0.2961 2.7196 1.6502 1.4524 0.2960 2.7694 1.6806

5 1.9356 0.2917 3.1419 1.6492 1.9409 0.2916 3.1836 1.6828

6 2.4456 0.2884 3.5650 1.6800 2.4524 0.2884 3.6025 1.7146

7 2.9823 0.2860 3.9757 1.7382 2.9899 0.2860 4.0125 1.7726

8 3.5578 0.2842 4.3552 1.8237 3.5682 0.2842 4.3873 1.8588

9 4.1997 0.2829 4.6750 1.9417 4.2139 0.2830 4.7014 1.9780

10 4.9706 0.2822 4.8792 2.1101 4.9867 0.2823 4.9078 2.1462

E(Q) – Expected queue length; E(I) – Expected idle period; E(B) – Expected busy period;

TAC – Total average cost

Table 2.4 Threshold Value (Vs) Total Average Cost for various and

(For 0.5 , 2.0 , b = 10, 3 , 8 , u = 7)

a Total Average Cost

0.7; 1.0 1.0; 1.0 1.0; 0.5 1.0; 0.8

1 2.0087 2.0221 2.0000 2.0128

2 1.8872 1.9336 1.8557 1.9016

3 1.8229 1.8871 1.7795 1.8441

4 1.7998 1.8738 1.7498 1.8248

5 1.8091 1.8877 1.7553 1.8360

6 1.8456 1.9272 1.7898 1.8731

7 1.9063 1.9903 1.8497 1.9343

8 1.9935 2.0790 1.9357 2.0216

9 2.1130 2.2000 2.0545 2.1419

10 2.2833 2.3719 2.2238 2.3128

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Figure 2.2 Threshold Value (Vs) Expected Queue Length

(For various probability of admitting re-service ‘ ’)

Figure 2.3 Threshold Value (Vs) Total Average Cost

(For various probability of admitting re-service ‘ ’)

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Figure 2.4 Threshold Value (Vs) Total Average Cost for = 0.7 , = 1.0

(For 0.5 , 2.0 , b = 10, 3 , 8 , u = 7)