M/M/C QUEUE WITH MULTIPLE WORKING VACATIONS AND SINGLE WORKING VACATION UNDER ENCOURAGED ARRIVAL WITH IMPATIENT CUSTOMERS Текст научной статьи по специальности «Медицинские технологии»

M/M/C QUEUE WITH MULTIPLE WORKING VACATIONS AND SINGLE WORKING VACATION UNDER ENCOURAGED ARRIVAL WITH IMPATIENT CUSTOMERS

Prakati P, Julia Rose Mary K

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Departmentof Mathematics, Nirmala College f or Women, India [email protected] [email protected]

Abstract

This paper demonstrates an M/M/C queuing model with Multiple working vacations and also single working vacation under encouraged arrival with impatient customers. The queuing model with the servers adopting multiple working vacation policy and single working vacation are determined separately and it is observed that the servers during working vacation(s) will be serving the customers at a slower service rate when compared during regular busy period. In addition to the above conditions, if there is a rapid increase in the customers� arrival i.e, if encouraged arrival occurs and due to this sudden growth of the queue, there may be a impatience in the behaviour of the customer. With these considerations, an M/M/C Queuing model is analysed with two vacation policies separately by applying PGF method and thus the performance measures for an M/M/C Queue with Multiple Working Vacations and Single Working Vacation under Encouraged arrival with impatient customers are evaluated.

Keywords: Multiple Working Vacations(MWV), Single Working Vacation (SWV),Encouraged Arrival,Impatient behaviour,Performance Measures

1. Introduction

In our daily life, we meet up with the scenario of waiting in queues to get our work done, for example-tomakebankdeposit,mail apackage, obtainfoodincafeteria etc.Waiting inqueue is a matter of personal annoyance and it also costs the amount of time that we waste by waiting inqueue s.Itmay affect theefficienc ofthe service provided andisamajorfactorinboth the quality of life and also affecting the efficienc of a nation�s economy . Great inefficiencies also occur because of waiting.

Forexample, making machines wait toberepaired may result inless production, delay in telecommunication transmission due to saturated lines may cause data glitches etc. In fact, we have become accustomed to considerable amounts of waiting. Origin of Queuing theory in research was contributed byAgner KrarupErlang,whocreated modelstodescribethesystemof incoming calls at the Copenhagen Telephone Exchange Company .

AnM/M/s queuingsysteminwhich theservers undergoing vacation was analysedin[7]. In Queuing vacation policy, an overview of some general decomposition results were attained and the methodology used to obtain those results for two vacation models were analysed in [3]. Moreover, theliteratureonstatistical analysis ofqueuing systems were briefl discussed in[2].

Itcanbeobserved thatinnumerous industrial sector, theconceptofQueuing withservers� vacation is implemented. An M/G/1 Queue with vacation policy used in the scenarios like maintenance of production systems, wher e machines or equipment mainly degrade while being operated were evaluated and for such queuing model,an explicit expression for the distribution ofthe timeittakes untilthespecifie amountofwork hasbeenserved were derived in[1].

In General, Systems with vacations are usually modeled and analyzed by queuing theory. An appr oach for modeling and analyzing finite-sou ce multi-ser ver systems with single and multiple vacations of servers or all stations were presented using the Generalized Stochastic Petri nets modelin[11].During anyservice,the servers mayundergo breakdo wnsimultaneously bothin regular busy period andworking vacation period duetothefailure ofamaincontrolunit. This scenariowas discussed bymodeling andanalysing aMarkovian multiserver finitebuffer queue under synchronous working vacation policy in[5].

A multiser ver queuing system with customers ' impatience until the end of service under single and multiple vacation policies were examined in [6]. Situations like arrival of the customers following Poissondistributionbutthegeneraldistributionfollowed by theadministration render �ing service with various vacations were detailedly discussed in [10].

The concept of impatient behaviours like balking and reneging with the availability of hetero�geneous servers in an M/M/c queue was analysed in [16]. Moreover, the time-dependent system sizeprobabilities were derived explicitlyusinggenerating functionandalsothetime-dependent mean,variance, busyperioddistribution andsteady-state probabilities were alsoobtained. In addition to this, perfor mance of an M/M/c/K Queuing Models applied in Healthcar e Things for Medical Monitoring were evaluated in [14].

The impatient nature of the customer during any service may be expressed if there is a delay in the service and the delay may be due to lack of servers or slow service provided. Queues with slow servers andimpatient customers were considered andthethemeanqueuesizewere derived. Also, Several extreme cases were investigated and numerical results are presented in [12].

An M/M/1 queue with single and multiple working vacations with impatient customers were studied and Closed-for m solutions and various perfor mance measur es like, the mean queue lengthsandthemeanwaiting timeswere derived andthestochasticdecomposition properties were verifie forboth multipleandsingleworking vacation casesin[13].Likely, theimpatient behaviour ofthe customers withwithsingleandmultiple synchronous working vacations inan M/M/C queuewas analysedin [9].Performance nature ofaMarkovian QueuewithImpatient Customers andWorking Vacation were derived in[8].

It is obvious that in the case of any discounts or offers provided during any sale or if any sudden demand is created for a product or a service, then there will be a rapid growth in the arrival of the customers, which is termed as encouraged arrival. The concept of encouraged arrivalin anM/M/c/N queuingsystemswithreneging, retention andFeedbackcustomerswere discussed in[15]. Thestationarysystem sizeprobabilities were obtained recursively forthe above model, while the steady state behavior of the M/M/1/N queuing model with encouraged or discouraged arrivals and impatient customers are obtained in [4].

With the aid of the above discussed concepts, an M/M/C Queuing model during encouraged arrival undergoing single working vacation and multiple working vacations with impatient behaviour of the customers are analysed separately .

Inthis paper, between thetwo vacation policiesanalysed,multiple working vacation iscon�sidered firs inwhich ifaserver returns toanemptyqueue,thenhegoesforanothervacation immediately ,thus working vacation occurs multiple times. Whereas, in the later vacation policy, the server takes only a single vacation each time. Thus for an M/M/C Queue during encouraged arrivalwith impatient behaviour undergoing multiple working vacation isderived withexplicit formulations followed by the same queuing model with single working vacation.

2. Methods

An M/ M/ c queuing model with encouraged arrival following multiple working vacations with impatient custome rs is consider ed. Customers arriving to be served follow Poisson process and the arrival rate is denoted by the parameter .w. If there is a sudden increase in the arrival of the customers,i.e., encouraged arrival occuring in the system follows poisson process with the encouraged arrival rate .w(1 + .).

Since the consider ed model denotes �c� servers, there may be maximum of �c� servers available, to serve the customers according to FCFS rule. When a customer arrives and find all the servers in the system are busy, then he needs to wait until he gets served and thus the waiting line or the queue begins.

The time taken for each server to complete the work during regular busy period follows exponential distribution and denoted with the service rate �w. Thus the traffi intensity or the

.w(1+.)

stability of the system during regular busy period is consider ed as . = c�w < 1

After completion of a service, if there is no customer in the system,then all the �c� servers will take vacation promptly and the duration of working vacation for each servers is exponentially distributed with parameter .' .Asalltheservers inthe system undergo vacation, even ifasingle customer arrives,then anyone of the server will return from his vacation and start serving the arrived customer.Thus theconcept ofworking even during vacation forthe arrivalof customers is termed as working vacation period, and the service rate following exponential process during working vacation period is �wv anditisobserved thattheservice rateduring working vacation is slower than the regular busy period i.e., �wv < �w

Itisobvious thatifthe servers return from theirvacation andwhenthesystem isnonempty, the service rate of the servers changes from �wvto�v indicating that the regular busy period begins. Suppose, if the servers fin no customer waiting in the queue after returning from their vacation, they immediately leave for another vacation. In such cases,if a customer waits in the queue for a longer time, as all the �c� servers are in working vacation period,he may become impatient inwaiting andtheimpatient behaviour ofthe customer atthetimeTisisexponentially distributed with parameter .w which is consider ed to be independent of the customers in that moment.

The customer waiting in the queue may exit the queue and never returns if its service has not beencompleted before thetimeTexpires. Theinter arrivaltimes, service times,vacation duration times and impatient time are all taken to be mutually independent. To construct this system, we defin atwo dimensional continuous timediscrete stateMarkov chainas {(M(t), N(t)),t . 0}with state space s = {(0,0) .{(n, j)}, n . 1, j = 0,1}

Where M(t)denotes thetotalnumberofcustomers inthe system attimetand N(t)denotes the state of the system at time t with

N(t)={1when theservers are innon-vacation period attimet}and

N(t)={0when theservers are inworking vacation period attimet}.

2.1. Steady State Equations and its Solutions for Multiple Working Vacations Model:

The steady state transition probabilities are define by Pnj = P{M(t)= n, N(t)= j}, n . 0, j = 0,1 Now, the set of balance equations as

.w(1 + .)P00 =(�wv + .w)P1,0 + �wP1,1, (1)

[.w(1 + .)+ .' + n(�wv + .w)]Pn,0 = .w(1 + .)Pn.1,0 +(n + 1)((�wv + .w)Pn+1,0, ifn . 1, (2) (.w(1 + .)+ �w)P1,1 = .' P1,0 + 2�wP2,1, (3)

(.w(1 + .)+ n�w)Pn,1 = .w(1 + .)Pn.1,1 +(n + 1)�wPn+1,1 + .' Pn,0, if 2 . n . c . 1, (4)

(.w(1 + .)+ c�w)Pn,1 = .w(1 + .)Pn.1,1 + c�wPn+1,1 + .' Pn,0 ifn . c. (5)

By letting the probability generating functions as

P0(z)=

..

z

nPn,0,

n=0

..

z

nPn,1.

P1(z)=

n=1

'

with P0(1)+ P1(1)= 1 and P0(z)= .

n=1 nzn.1Pn,0. Now, By Multiplying Eq(2) with zn and adding over �n� and rearranging the terms, the differential equation is attained as :

(�wv + .w)(1 . z)P0 ' (z)=[.w(1 + .)(1 . z)+ .' ]P0(z) . (.' P0,0 + �wP1,1). (6)

Likely multiplying Eq(4) and Eq(5) by zn and adding over �n�, the following equation is obtained,

c

.

(1 . z)(.w(1 + .)z . c�w)P1(z)= .' zP0(z) . (.'

P0,0 + �wP1,1)z + �w(1 . z)

.

(n . c)z

nPn,1. (7)

n=1

Let us consider ,

A = .' P0,0 + �wP1,1. (8)

Then, for z .= 1,

' .w(1 + .) .' A

P0(z) . [+ ]P0(z)= . . (�wv + .w)(�wv + .w)(1 . z)(�wv + .w)(1 . z)

(9)

Eq(9)is anordinarylinear differential equationwithconstantcoefficients To solve theequa�tion, an integrating factor can be consider ed as

R . ' . '

.w(1+.) .w(1+.)z

. [+ ]dz .

I.F = e (�wv+.w)(�wv+.w)(1.z)= e (�wv+.w)(1 . z)(�wv+.w)

The General solution to Eq(9) is given by:

.w(1+.)z . ' .w(1+.)z . '

d ..A .

[e (�wv+.w)(1 . z)(�wv+.w)]P0(z)=[ ]e (�wv+.w)(1 . z)(�wv+.w) . (10)

dz (�wv + .w)(1 . z)

Now, integrating from 0to z,following equation isattained,

. ' Z . '

.w(1+.)zz .w(1+.)z

. A ..1

P0(z)=[e (�wv+.w)(1 . z)(�wv+.w)[Po(0) . e (�wv+.w)(1 . x)(�wv+.w) dx]. (11)(�wv + .w) 0

then,

.w(1+.) Z 1 .w(1+.)z . '.. '

A ..1

P0(1)= e (�wv+.w)[Po(0) . e (�wv+.w)(1 . x)(�wv+.w) dx] lim(1 . z)(�wv+.w) . (12) (�wv + .w) 0 z>1

..

. '

.

nPn,0 . 1and lim(1. z)

z>1

Since 0 . P0(1)=

(�wv+.w)= .,and thus the existing term is as follows

z

n=0

A

P0,0 = P0(0)= L (13)(�wv + .w)

Z 1 . .w(1+.)z . '.1

Where L = e (�wv+.w)(1 . x)(�wv+.w) dx. (14)0

Defin Z(.w(1 + .), .' )= ..w(1 + .)..' e..w(1+.)(..(.' , ..w(1 + .)) + .(.' )) (15)

wher e .(z) is the . function which is represented as Z .

.(z)= e.ttz.1dt (16)0

Z .

and .(y, z)= e.tty.1dt. (17) z

some calculations give

.w(1 + .) .'

L = Z( , ). (18)(�wv + .w)(�wv + .w)

By Eq(8) and Eq (13), it is obser ved that

.'

P0,0 + �wP1,1 L�w

P0,0 = L = P1,1. (19)

(�wv + .w) �wv + .w . .' L

Now, using the value of A from Eq(13) in Eq(11), P0(z) is obtined as

.w(1+.)z

Z . '

(�wv+.w) z .w(1+.)z

e 1 ..1

P0(z)= .' [1 . e (�wv+.w)(1 . x)(�wv+.w) dx]P0,0. (20)

L

(1 . z)(�wv+.w) 0

By applying L�Hospital�s rule to Eq(20), we get

(�wv + .w)

P0(1)= P0,0 (21)

.' L and now substituting the value ofP0,0 from Eq(19), the following relation is obtained

.' P0(1)= .' P0,0 + �wP1,1. (22) From Eq(7), P1(z) is attained as, [.'P0(z) . A]z �w

P1(z)= . F(z), (23)

(.w(1 + .)z . c�w)(1 . z)(.w(1 + .)z . c�w)

wher e,

c F(z)= . (n . c)znPn,1. (24) n=1

It is clear from Eq(20) that P0(z) is a function of P0,0 and the ratio betw een the time of the servers onworking vacation andthesystem isempty .Similarly from Eq(23), P1(z) is a function of P0(z), A and F(z). Hence, if P0,0 and Pj,1(j=1,2,...c) are obtained, P0(z) and P1(z) can be deter mined completely .

2.2. Perfor mance Measur es

By using L�Hospital�s rule in Eq(23), we get

[.''

P0(1) . A]+ .'P0(1) �w

P1(1)= + F(1), (25)

c�w . .w(1 + .) c�w . .w(1 + .)

wher e

c F(1)= . (c . n)Pn,1. (26) n=1

Using Eq(22) and Eq(8) in Eq(25),w e get,

.' �w

P1(1)= E(L0)+ F(1). (27)

c�w . .w(1 + .) c�w . .w(1 + .)

Now, byapplying L�hospital�s ruletoEq(6),we have '..w(1 + .)P0(1)+ .'P0 ' (1) ..w(1 + .)P0(1) . E(L0)

E(L0)= lim P0(z)= = which impliesz>1 .(�wv + .w)(�wv + .w) (28)

.' + �wv + .w

P0(1)= E(L0). (29)

.w(1 + .) As P0(0)+ P0(1)= 1, from Eq(27) and Eq(29),the expected number of customers during working vacation period is obtained as

.w(1+.)

.w(1 + .)(1 . .)

c

E(L0)= . F(1). (30)

.' + �wv(1 . .)+ .w(1 . .) .' + �wv(1 . .)+ .w(1 . .)

On substituting Eq(30) in Eq(29), the probability that the system in working vacation period is as

+�wv+.w

(1 . .)(.' + �wv + .w) .'

c

P(J = 0)= P0(1)= . F(1) (31)

.' + �wv(1 . .)+ .w(1 . .) .' + �wv(1 . .)+ .w(1 . .)

and the probability that the system is in busy period is found as

.' +�wv+.w

(.'.)

c

P(J = 1)= P1(1)= 1. P0(1)= + F(1).

.' + �wv(1 . .)+ .w(1 . .) .' + �wv(1 . .)+ .w(1 . .)

(32) E(L1) can be obtained by differentiating Eq(23) and using L�Hospital�s rule,

i.e., E(L1)= lim P1 ' (z)

z>1 ..w(1 + .)[z(.A + .'P0(z)) .A + .'P0(z)+ z.'P0 ' (z)]

= lim{ +

z>1 (1 . z)(.w(1 + .)z . c�w)2 (1 . z)(.w(1 + .)z . c�w) z(.A + .'P0(z) [(c�w . .w(1 + .)z)F' (z)+ .w(1 + .)F(z)]

++ �w } (33)

(1 . z)2(.w(1 + .)z . c�w)(c�w . .w(1 + .)z)2

.' (c�w . .w(1 + .)E(L0(L0 . 1)) + 2c�w.'E(L0) F' (1) .F(1)

= ++ (34)

2(c�w . .w(1 + .)z)2 c(1 . .)(c(1 . .)2 wher e

dF(z)

F ' (1)= at z=1

dz

c

=

(c . j)Pj,1 (35)

j=1

Now, the value of P0 '' (1) is obtained on differentiating Eq(6) twice on both sides as

'' ' ''

(.' + .w)(1 . z)P0 (z)+ 2.w(1 + .)P0(z)=[.w(1 + .)(1 . z)+ .' + 2(�wv + .w]P0 (z) (36)

wher e ''' d3P0(z)

P0 (z)=

dz3

'' 2.w(1 + .) '

By letting z=1 in Eq(36),w e get P0 (1)= P0(1) (37).' + 2(�wv + .w)

.

2.w(1 + .)EL0

or it can also be denoted as, E(L0(L0 . 1)) = . (38).' + 2(�wv + .w)

Now, substituting, Eq(38) into Eq(34),the Mean number of customers, when the system in regular busy period is obtained as

..' 11 1 .

E[L1]=[ +]E[L0]+ F ' (1)+ F(1)

(1 . .) .' + 2(�wv + .w) .w(1 + .)(1 . .) c(1 . .) c(1 . .)2 (39) Hence, E[L]= E[L0]+ E[L1]

.w(1+.)

..' 11 .w(1 + .)(1 . .) . F(1)

c

= 1 +[+][ ]

(1 . .) .' + 2(�wv + .w) .w(1 + .)(1 . .) .' + �wv(1 . .)+ .w(1 . .)

+ 1 F ' (1)+ . F(1) (40)

c(1 . . c(1 . .)2

k

Substituting Eq(31) in Eq(21) results in P(0,0)= .' P0(1)

(�wv+.w) (.' +�wv+.w)

.'k (1 . .)((.' + �wv + .w)

c

=[ . ]F(1). (41)

(�wv + .w) .' + �wv(1 . .)+ .w(1 . .) .' + �wv(1 . .)+ .w(1 . .)

Suppose, the state of the system is (n,1),then the service rates of the servers are n�w for n . c and c�w for n > c respectiv ely. Inthis manner, theexpected numberofcustomers served perunitoftime isgiven by

c

n�wPn,1 +

..

c�wPn,1 = �w[cP1(1) . F(1)] (42)

Ns

=

.

n=1 n=c+1

and the proportion of customers served per unit of time is given by

Ns 1

Ps = =[cP1(1) . F(1)] (43)

.w(1 + .) c. wher e P1(1)isgivenbyEq(32). If the state of the system is (n,1), n. 1, the rate of customer abandonment of a customer due to impatience is n.w.Thusthemeanrateofthe customerabandonment duetoimpatience isgiven by

Ra =

..

n.wPn,0 = .wE[L0]. (44)

n=1

Thus an M/M/c Queuing model with Multiple Working Vacations under encouraged arrival with impatient behaviour is evaluated.

2.3. Single Working Vacation Model:

A Single working Vacation policy define that the server(s) in the queuing system takes vacation immediately , when he find no customers waiting in the queue. At the end of the working vacation, if the server find the system non empty, then he starts his regular busy period by shifting his service rate from �wv to �w. If not,the server will remain idle in the system itself than going for vacation and waits until the customer arrives for the new busy period. To construct thissystem, we defin aMarkov chainas {(M(t), N(t)),t . 0} with state space as in Multiple Working Vacations for Single Working Vacation also. s = {(n, j)}, n . 0, j = 0,1}

Where M(t)denotes thetotalnumberofcustomers inthe system attimetand N(t)denotes

the state of the system at time t with N(t)={1when theservers are innon-vacation period attimet}and N(t)={0when theservers are inworking vacation period attimet}.

2.4. Steady State Equations and its Solutions for Single Working Vacation Model:

Now, the set of balance equations as

(.w(1 + .)+ .' )P00 =(�wv + .w)P1,0 + �wP1,1, (45)

[.w(1 + .)+ .' + n(�wv + .w)]Pn,0 = .w(1 + .)Pn.1,0 +(n + 1)((�wv + .w)Pn+1,0, ifn . 1, (46) (.w(1 + .))P0,1 = .' P0,0, (47)

(.w(1 + .)+ n�w)Pn,1 = .w(1 + .)Pn.1,1 +(n + 1)�wPn+1,1 + .' Pn,0, if 1 . n . c . 1, (48)

(.w(1 + .)+ c�w)Pn,1 = .w(1 + .)Pn.1,1 + c�wPn+1,1 + .' Pn,0 ifn . c. (49)

By letting the probability generating functions as

..

nPn,0,

R0(z)=

z

n=0

..

nPn,1.

R1(z)=

z

n=1

.with R0(1) + R1(1) = 1 and R0(z) = .n=1 nzn.1Pn,0.

Now, By Multiplying Eq(46) with zn and adding over �n� and rearranging the terms, the

differential equation is attained as :

'

'

(�wv + .w)(1 . z)R0(z)=[.w(1 + .)(1 . z)+ .' ]R0(z) . (�wP1,1). (50)

Likely multiplying Eq(48) and Eq(49) by zn and adding over �n�, the following equation is obtained,

c

(1. z)(.w(1+ .)z . c�w)R1(z)= .' zR0(z) . (.' P0,0 + �wP1,1)z + z2.'

P0,0 + �w(1. z)

.

(n . c)z

nPn,1.

Then, for z .= 1, n=1 (51)

' .w(1 + .) .' �wP1,1R0(z) . [ + ]R0(z) = . . (�wv + .w) (�wv + .w)(1 . z) (�wv + .w)(1 . z) (52)

Solving the differential equation, as in Multiple Working Vacations Model we get,

.w(1+.)z

Z . '

(�wv+.w) z .w(1+.)z

e 1 ..1

R0(z)= .' [1 . e (�wv+.w)(1 . x)(�wv+.w) dx]P0,0. (53)

L

(1 . z)(�wv+.w) 0

Thus a similar expression for R0(z) as in Multiple Working Vacations Model and here we arrive at, (�wP1,1)

R0(0)= P0,0 = L (54)(�wv + .w)

(�wv + .w)

R0(1)= P0,0 (55)

.'L

and from Eq(54) and Eq(55), the following relation is obtained .' R0(1)= �wP1,1. (56) From Eq(51),R1(z) is attained as, 2. '

[.'R0(z) . A]z + zP0,0 �w

R1(z)= . F(z), (57)

(.w(1 + .)z . c�w)(1 . z)(.w(1 + .)z . c�w)

wher e,

c

F(z)=

.

(n . c)z

nPn,1. (58)

n=1

It is clear from Eq(53) that R0(z) is a function of P0,0 and the ratio betw een the time of the servers on working vacation and the system is empty . Similarly from Eq(57), R1(z) is a function of R0(z), A and F(z). Hence, if P0,0 and Pj,1(j=1,2,...c) are obtained, P0(z) and P1(z) can be deter mined completely .

2.5. Perfor mance Measur es By using L�Hospital�s rule in Eq(57), we get

[.'E(L)0]+ B �w

R1(1)= + F(1) (59)

c�w . .w(1 + .) c�w . .w(1 + .)

wher e

c

B = .' (2 . c)P0,0 and F(1)=

.

(n . c)Pn,1.

(60)

n=1 Using Eq(22) and Eq(8) in Eq(25), the following equation is obtained,

.' �w

P1(1)= E(L0)+ F(1). (61)

c�w . .w(1 + .) c�w . .w(1 + .)

Now, applying L�hospital�s ruletoEq(6),we have '..w(1 + .)P0(1)+ .'P0 ' (1) ..w(1 + .)P0(1) . E(L0)

E(L0)= lim P0(z)= = which impliesz>1 .(�wv + .w) �wv + .w (62)

.' + �wv + .w

P0(1)= E(L0). (63)

.w(1 + .)

As P0(0)+ P0(1)= 1, from Eq(27) and Eq(29), is the expected number of customers during working vacation period is obtained as

.w(1+.)

.w(1 + .)(1 . .) ...' (2 . c)P0,0

c

E(L0)= .. F(1).

.' + �wv(1 . .)+ .w(1 . .) .' + �wv(1 . .)+ .w(1 . .) .' + �wv(1 . .)+ .w(1 . .)

(64)

On substituting Eq(30) in Eq(29), the probability that the system in working vacation period is as

(1..)(.' +�wv+.w) ..' (.' +�wv+.w)(2.c)P0,0

P(J = 0)= R0(1)= .'.

+�wv(1..)+.w(1..) .w(1+.)[.' +�wv(1..)+.w(1..)]

.'

+�wv+.w c

. F(1)

.' + �wv(1 . .)+ .w(1 . .)

wher e

..' (.' + �wv + .w)(2 . c)P0,0

X = , then

.w(1 + .)[.' + �wv(1 . .)+ .w(1 . .)

.' +�wv+.w

(1 . .)(.' + �wv + .w)

c

P(J = 0)= R0(1)= . X . F(1) (65)

.' + �wv(1 . .)+ .w(1 . .) .' + �wv(1 . .)+ .w(1 . .)

and the probability that the system is in busy period is as follows

(.'.) ..' (.' +�wv+.w)(2.c)P0,0

P(J = 1)= R1(1)= 1 . R0(1)= +

.' +�wv(1..)+.w(1..) .w(1+.)[.' +�wv(1..)+.w(1..)]

.'

+�wv+.w c

+ F(1).

.' + �wv(1 . .)+ .w(1 . .)

since we know that,

..' (.' + �wv + .w)(2 . c)P0,0

X = ,

.w(1 + .)[.' + �wv(1 . .)+ .w(1 . .)

we get

+�wv+.w

(.'.) .'

c

P(J = 1)= R1(1)= 1. R0(1)= + X + F(1).

.' + �wv(1 . .)+ .w(1 . .) .' + �wv(1 . .)+ .w(1 . .)

(66) Now, E(L1) can be obtained by differentiating Eq(58) and using L�Hospital�s rule,

E(L1)= lim R1 ' (z)

z>1

..w(1 + .)[z(.A + .'R0(z)) + z2.'P0,0] .A + .'R0(z)) + 2z.'P0 ' (z)+ z.'R0 ' (z)

= lim{ +

z>1 (1 . z)(.w(1 + .)z . c�w)2 (1 . z)(.w(1 + .)z . c�w)

z(.A + .'R0(z)+ z2.'P0,0 [(c�w . .w(1 + .)z)F' (z)+ .w(1 + .)F(z)]

++ �w } (67)

(1 . z)2(.w(1 + .)z . c�w)(c�w . .w(1 + .)z)2

.' (c�w . .w(1 + .)E(L0(L0 . 1)) + 2c�w.'E(L0)+ 2.' [(2(c�w . .w(1 + .) . c.w(1 + .)]P0,0

=

2(c�w . .w(1 + .)z)2

F' (1) .F(1)

++ (68)

c(1 . .)(c(1 . .)2

wher e

dF(z)

F ' (1)= atz = 1

dz

c

=

(c . j)Pj,1 (69)

j=1

Now, the value of R0 '' (1) is obtained on differentiating Eq(50) twice on both sides and proceed�ing similarly as in Multiple Working Vacations, We get

2.w(1 + .)

E(L0(L0 . 1)) = E(L0). (70).' + 2(�wv + .w)

Now, substituting, Eq(70) into Eq(69),the Mean number of customers, when the system in regular busy period is obtained as

..' 11 11

E[L1]= {[ +]E[L0]+[ . ]P0,0}

(1 . .) .' + 2(�wv + .w) .w(1 + .)(1 . .) .w(1 + .) �w(1 . .)

+ 1 F1 + . F(1). (71)

c(1 . .) c(1 . .)2 E[L]= E[L0]+ E[L1]

.w(1+.)

..' 11 .w(1 + .)(1 . .) . .B . F(1)

c

= {1 +[+][ ]}

(1 . .) .' + 2(�wv + .w) .w(1 + .)(1 . .) .' + �wv(1 . .)+ .w(1 . .)

+Y + 1 F ' (1)+ . F(1) (72)

c(1 . .) c(1 . .)2 ..' 11

wher eY =[ . ]P0,0

(1 . .) .w(1 + .) �w(1 . .)

Substituting Eq(65) in Eq(55) results in P(0,0)= .'kR01

(�wv+.w)

(.w(1+.).' +�wv+.w)

.'k .w(1 + .)(1 . .)((.' + �wv + .w) .

c

=[ ]. (73)

k.'2.(2.c)(.' +�wv+.w)

(�wv + .w) .' + �wv(1 . .)+ .w(1 . .)+

(�wv+.w)

Suppose, the state of the system is (n,1),then the service rates of the servers are n�w for n . c and c�w for n > c respectiv ely.

Thus, the expected number of customers served per unit of time is given by

c

n�wPn,1 +

..

c�wPn,1 = �w[cP1(1) . F(1)] (74)

Ns

=

.

n=1 n=c+1

and the proportion of customers served per unit of time is given by

Ns 1

Ps = =[cP1(1) . F(1)] (75)

.w(1 + .) c.

wher eP1(1) is given by Eq(66). If the state of the system is (n,1), n . 1, the rate of customer abandonment of a customer due to impatience is n.w.Thusthemeanrateofthe customerabandonment duetoimpatience isgiven by

Ra =

..

n.wPn,0 = .wE[L0]. (76)

n=1

Hence, an M/M/c Queuing model with Multiple Working Vacations under encouraged arriv al with impatient behaviour is evaluated.

3. Results

Inthis paper,an M/M/C QueuingmodelunderMultipleworking vacations andsingleworking vacation withimpatientbehaviour ofthe customerduringencouragedarrival are analysed. Itis obser ved that for the system of steady state equations, perfor mance measur es like Mean Queue length (E[L]), Probability that the system is in working vacation period (P[J=0]), Probability that the system is in regular busy period (P[J=1]) are evaluated for the two different vacation policies separately .

4. Discussion

Oncomparing theperformancemeasuresbetween thetwo vacation policies,from Eq(65)and Eq(31),itisobserved thatthedifference between theprobability ofthe system (P[J=0])insingle working vacation and that during multiple working vacations, we notice that by reducing the term "X" from the probability of the system in multiple working vacations, we attain the probability ofthesystem insingle working vacation .Likely, from Eq(66)and Eq(32),itisclearthatthe probability of the system in regular busy period during single working vacation is obtained by adding the term "X" to the probability of the system in regular busy period during multiple working vacation. Moreover, whilecomparing themeanqueuelengthduringthetwo different vacation policies, we observe thatfrom Eq(72)and Eq(40),E(L)insingle working vacation isthe addition of the term "Y" and the term .Bto the existing mean queue length of multiple working vacations.

5. Conclusion

As the M/M/c Queuing model with Multiple and single working vacation with impatient behaviour of the customers during encouraged arrival is analysed, apart from deriving the explicit formulations, some of the characteristic measur es are also discussed. It can be concluded that, with the impact of the terms "X","Y" and"B" in multiple working vacations an M/M/C Queuing model with impatient behaviour of the customer during encouraged arrival can be shifted toSingle working vacation.However,for anefficient functioning of the queue a single working vacation canbesuggested. Infuture work, numericalexamplesmay beevaluated to evident the obtained result.

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