Research ArticleA Discrete-Time Queue with Balking, Reneging, andWorking Vacations
Veena Goswami
School of Computer Application, KIIT University, Bhubaneswar 751024, India
Correspondence should be addressed to Veena Goswami; veena [email protected]
Received 21 May 2014; Accepted 9 October 2014; Published 29 October 2014
Academic Editor: Onesimo Hernandez-Lerma
Copyright Β© 2014 Veena Goswami. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper presents an analysis of balking and reneging in finite-buffer discrete-time single server queue with single and multipleworking vacations. An arriving customer may balk with a probability or renege after joining according to a geometric distribution.The server works with different service rates rather than completely stopping the service during a vacation period. The servicetimes during a busy period, vacation period, and vacation times are assumed to be geometrically distributed. We find the explicitexpressions for the stationary state probabilities. Various system performance measures and a cost model to determine the optimalservice rates are presented. Moreover, some queueing models presented in the literature are derived as special cases of our model.Finally, the influence of various parameters on the performance characteristics is shown numerically.
1. Introduction
Discrete-time queueing models with server vacations havebeen studied extensively in recent years by many researchersbecause these systems are better suited than their continuous-time counterparts to analyze and design digital transmittingsystems. Extensive analysis of a wide variety of discrete-timequeueing models with vacations has been reported in Takagi[1]. Queues with balking and reneging often arise in practice,when there is a tendency of customers to be impatient dueto a long queue. As a result, the customers either balk (i.e.,decide not to join the queue) or renege (i.e., leave after joiningthe queue without getting served). The significance of thissystem emerges in many real life situations such as telephoneservices, computer and communication systems, productionline systems, machine operating systems, inventory systems,and the hospital emergency rooms.
In the study of working vacation models, the serverremains active during the vacation period and serves thecustomers generally at a slower rate. At a vacation termi-nation epoch, if the queue is nonempty, a regular serviceperiod begins with normal service rate; otherwise the servertakes another vacation.This vacation policy is called multipleworking vacations (MWV). In the single working vacation
(SWV) policy, the server takes only one working vacationwhen the system becomes empty. When the server returnsfrom a SWV, it stays in the system waiting for customers toarrive instead of taking another working vacation.Otherwise,it alters the service rate back to the normal service rate asin the MWV policy. An extensive report of a wide variety ofworking vacation queueingmodels can be found in Servi andFinn [2], Wu and Takagi [3], Tian et al. [4], Li and Tian [5],and the references therein.
Impatience is the most important feature in queueingsystems, as individuals always feel concerned while waitingfor service. Performance analysis of queueing systems withbalking and reneging in real life congestion problems isbeneficial because one finds new managerial insights. Whilemaking decision for the service rate of servers neededin the service system to meet time-varying demand, thebalking and reneging probabilities can be used to estimatethe amount of lost revenues as reported in Liao [6]. Thestudy of an π/π/1/π queue with balking, reneging, andmultiple server vacations has been studied in Yue et al. [7].The optimal balking strategies in a Markovian queue witha single exponential vacation have been discussed in Tianand Yue [8]. Ma et al. [9] examined an equilibrium balkingbehavior in the Geo/Geo/1 queueing system with multiple
Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2014, Article ID 358529, 8 pageshttp://dx.doi.org/10.1155/2014/358529
2 International Journal of Stochastic Analysis
vacations. Zhang et al. [10] have investigated the equilibriumbalking strategies for observable and unobservable π/π/1queue with working vacations. Laxmi et al. [11] analyzedanπ/π/1/π queueing system with balking, reneging, andworking vacations. Laxmi et al. [12] studied an optimizationof balking and reneging queue with vacation interruptionunderπ-policy.
The study of the discrete-time queueing systems withimpatient customersβ behavior and vacations seems to be arecent endeavor. A discrete-time single server queue withbalking has been examined by Lozano and Moreno [13].Vijaya Laxmi et al. [14] have considered a Geo/Geo/1/πqueue with balking and multiple working vacations. Queue-ing models with balking, reneging, and server working vaca-tions accommodate the real world situations more closelyfrom an economic viewpoint. However, most of the researchworks in discrete-time queueing systems have not yet con-sidered balking and reneging with server working vacationssimultaneously. It appears to be more pragmatic that thebalking and reneging behaviors are studied with workingvacations in the discrete-time queuing models.
This paper focuses on a finite buffer balking and renegingin discrete-time single server queue with single and multipleworking vacations. The interarrival times of customers andservice times are assumed to be independent and geo-metrically distributed. The service times during a vacationperiod and vacation times are assumed to be geometricallydistributed. The customers are allowed to decide whether tojoin or balk or renege, that is, leave after joining the queuewithout getting served. We present a recursive method toobtain the stationary-state probabilities for late arrival systemwith delayed access. Various performance measures havebeen discussed. Numerical results in the form of tables andgraphs are presented to display the impact of the systemparameters on the performance measures. We use quadraticfit search method to minimize the total expected cost perunit time. The proposed cost model may be useful to systemengineers in finding the optimal service rate desired foroperating the system smoothly at optimum cost. This modelhas potential applications in many congestion situations ofcommunication systems, manufacturing systems, produc-tion/inventory systems, transportation, and many others.
This paper is structured as follows. In Section 2, wepresent the description of the queueing model. The analysisof the model for the stationary-state probabilities is given inSection 3. Performancemeasures and cost analysis are carriedout in Section 4. Some special cases which are matchedwith existing results in the literature are demonstrated inSection 5. Numerical results in the form of tables and graphsare presented in Section 6. Section 7 concludes the paper.
2. Model Description
Weconsider a finite buffer discrete-time balking and renegingsingle server queue with single and multiple working vaca-tions under the late arrival system with delayed access (LAS-DA). Let us assume that the time axis is slotted into intervalsof equal length with the length of a slot being unity, and it
is marked as 0, 1, 2, . . . , π‘, . . .. Here, a potential arrival takesplace in (π‘β, π‘) and a potential departure occurs in (π‘, π‘+).More specifics on discrete-time queue have been reported inHunter [15] and Gravey and Hebuterne [16].
We assume that the interarrival times π΄ of customers areindependent and geometrically distributed with probabilitymass function (p.m.f.) π
π= π(π΄ = π) = π
πβ1
π, π β₯ 1, 0 <π < 1, where for any real number π₯ β [0, 1], we denoteπ₯ = 1 β π₯. If on arrival customer finds the system busy, thearriving customer either decides to join the queue or balk.Let ππrepresent the probability, when the system size is π
with which the customer decides to join the queue in orderto be served or balk with probability 1 β π
π, when there are
π customers in the system upon arrival (π = 0, 1, . . . , π β 1),where the capacity of the system is finiteπ. Furthermore, weassume that 0 β€ π
π+1β€ ππ< 1, 1 β€ π β€ π β 1, π
0= 1, ππ= 0.
The customers are served on a first-come, first-served(FCFS) discipline. Once service commences it always pro-ceeds to completion. The service times of customers areindependent and geometrically distributed with probabilitymass function (p.m.f.): π(π = π) = π
πβ1
π, π β₯ 1, 0 <π < 1. The service times πV in a working vacation periodare independent and geometrically distributed with commonp.m.f.: π(πV = π) = π
πβ1
π, π β₯ 1, 0 < π < 1. Theserver commences multiple working vacations on the instantwhen the system becomes empty. If the system is emptywhen the server returns from a working vacation, it beginsanother working vacation. Otherwise, the server changes to aservice period and a regular busy period begins. Under thesingle working vacation, when the system becomes empty,the server takes only one working vacation. It remains inthe system waiting for customers to arrive rather than takinganother working vacation. Otherwise, it switches the servicerate back to the busy period as under the MWV policy.The vacation times π are independent and geometricallydistributed with common p.m.f.: π(π = π) = π
πβ1
π, π β₯ 1,0 < π < 1.
After joining the queue each customer will wait a certainlength of time π for service to begin. If it has not begunby then, he will get impatient and leave the queue withoutgetting service.This time π is independent and geometricallydistributed with common p.m.f.: π(π = π) = πΌπβ1πΌ, π β₯ 1,0 < πΌ < 1. Since the arrival and the departure of theimpatient customers without service are independent, theaverage reneging rate of the customer can be given by π(π) =(π β 1)πΌ, 1 β€ π β€ π.
3. Analysis of the Model
In this section, we present the analytic analysis of finite bufferdiscrete-time balking and reneging single server queue withsingle andmultiple working vacations. We analyze both mul-tiple working vacations and single working vacation modelsconcurrently. For that reason we introduce an indicatorfunction (πΏ) as follows: πΏ = 1 yields the results for the SWVmodel and πΏ = 0 gives the results for the MWVmodel.
International Journal of Stochastic Analysis 3
At steady-state, let us define ππ,0, 0 β€ π β€ π, to be the
probability that there are π customers in the system when theserver is in working vacation and π
π,1, 1βπΏ β€ π β€ π, to be the
probability that there are π customers in the system when theserver is in regular busy period.
Based on the one-step transition analysis, the steady-stateequations can be written as
π0,0= π (1 β ππΏ) π
0,0+ s1(π) π1,0
+ t2(π) π2,0+ s1(π) π1,1+ t2(π) π2,1,
(1)
ππ,0= πrπ(π) ππ,0+ πqπβ1(π) ππβ1,0
+ πsπ+1(π) ππ+1,0
+ πtπ+2(π) ππ+2,0, 1 β€ π β€ π β 2,
(2)
ππβ1,0
= πrπβ1(π) ππβ1,0
+ πqπβ2(π) ππβ2,0
+ πsπ(π) ππ,0,
(3)
ππ,0= πrπ(π) ππ,0+ πqπβ1(π) ππβ1,0
, (4)
π0,1= ππ0,1+ πππΏπ
0,0, (5)
π1,1= r1(π) π1,1+ s2(π) π2,1+ t3(π) π3,1
+ q0(π) πΏπ
0,1+ πr1(π) π1,0+ πq0(π) π0,0
+ πs2(π) π2,0+ πt3(π) π3,0,
(6)
ππ,1= rπ(π) ππ,1+ qπβ1(π) ππβ1,1
+ sπ+1(π) ππ+1,1
+ tπ+2(π) ππ+2,1
+ πrπ(π) ππ,0+ πqπβ1(π) ππβ1,0
+ πsπ+1(π) ππ+1,0
+ πtπ+2(π) ππ+2,0,
1 β€ π β€ π β 2,
(7)
ππβ1,1
= rπβ1(π) ππβ1,1
+ qπβ2(π) ππβ2,1
+ sπ(π) ππ,1+ πrπβ1(π) ππβ1,0
+ πqπβ2(π) ππβ2,0
+ πsπ(π) ππ,0,
(8)
ππ,1= rπ(π) ππ,1+ qπβ1(π) ππβ1,1
+ πrπ(π) ππ,0+ πqπβ1(π) ππβ1,0
,
(9)
where
qπ(π₯) = {
π : π = 0,
ππππ₯ (1 β (π β 1) πΌ) : π = 1, . . . , π β 1,
rπ(π₯) =
{{{{{{{
{{{{{{{
{
(1 β ππ1) π₯ + ππ
1π₯ : π = 1,
(1 β πππ) π₯ (1 β (π β 1) πΌ)
+ πππ(π₯ (1 β (π β 1) πΌ)
+ π₯ (π β 1) πΌ) : π = 2, . . . , π β 1,
π₯ (1 β (π β 1) πΌ) : π = π,
sπ(π₯)
=
{{{{{{{
{{{{{{{
{
(1 β ππ1) π₯ : π = 1,
(1 β πππ) (π₯ (1 β (π β 1) πΌ)
+ π₯ (π β 1) πΌ)
+ ππππ₯ (π β 1) πΌ : π = 2, . . . , π β 1,
π₯ (1 β (π β 1) πΌ) + π₯ (π β 1) πΌ : π = π,
tπ(π₯) = {
(1 β πππ) π₯ (π β 1) πΌ : π = 2, . . . , π β 1,
π₯ (π β 1) πΌ : π = π.
(10)
The steady-state probabilities ππ,0(0 β€ π β€ π) and π
π,1, (1 β
πΏ β€ π β€ π) are computed recursively by solving the system of(1) to (9). Solving (2) to (4) recursively, it can be simplified as
ππ,0= ππππ,0, 0 β€ π β€ π, (11)
where
ππ= 1, π
πβ1= (
1 β πrπ(π)
πqπβ1(π)
) ππ,
ππβ2
=
(1 β πrπβ1(π)) π
πβ1β πsπ(π) ππ
πqπβ2(π)
,
ππ=
(1 β πrπ+1(π)) π
π+1β πsπ+2(π) ππ+2β πtπ+3(π) ππ+3
πqπ(π)
,
π = π β 3, . . . , 1, 0.
(12)
Using (11) in (5) and (7)β(9) yields ππ,1, (1 β πΏ β€ π β€ π), in
terms of ππ,0
and ππ,1
as
ππ,1= ππππ,1+ ππππ,0, 0 β€ π β€ π, (13)
where ππ= 1, π
π= 0,
ππβ1
=1 β rπ(π)
qπβ1(π)
,
ππβ1
= βπ(rπ(π) ππ+ qπβ1(π) ππβ1
qπβ1(π)
) ,
ππβ2
=(1 β r
πβ1(π)) π
πβ1β sπ(π) ππ
qπβ2(π)
,
ππβ2
= ((1 β rπβ1(π)) π
πβ1
β π (qπβ2(π) ππβ2+ rπβ1(π) ππβ1+ sπ(π) ππ))
Γ (qπβ2(π))β1
ππβ1=(1 β r
π(π)) π
πβ sπ+1(π) ππ+1β tπ+2(π) ππ+2
qπβ1(π)
,
π = π β 2, . . . , 1,
4 International Journal of Stochastic Analysis
ππβ1=(1 β r
π(π)) π
πβ sπ+1(π) ππ+1β tπ+2(π) ππ+2
qπβ1(π)
β π ( (rπ(π) ππ+ qπβ1(π) ππβ1
+ sπ+1(π) ππ+1+ tπ+2(π) ππ+2)
Γ (qπβ1(π))β1
) , π = π β 2, . . . , 2,
π0= 0, π
0= (
πππΏπ0
π) .
(14)
Now, using (11) and (13) in (6), the probability ππ,1
can beexpressed in terms of π
π,0as
ππ,1= Ξ₯ππ,0, (15)
where
Ξ₯
= π(r1(π) π1+ q0(π) π0+ s2(π) π2+ t3(π) π3
(1 β r1(π)) π
1β s2(π) π2β t3(π) π3
)
+ (s2(π) π2+ t3(π) π3+ πΏq0(π) π0β (1 β r
1(π)) π
1
(1 β r1(π)) π
1β s2(π) π2β t3(π) π3
) .
(16)
Finally, the only unknown ππ,0
is obtained from the normal-ization conditionβπ
π=0ππ,0+ βπ
π=1βπΏππ,1= 1 and is given by
ππ,0= [
π
β
π=0
ππ+
π
β
π=1βπΏ
(ππ+ Ξ₯ππ)]
β1
. (17)
This completes the evaluation of steady-state probabilities.
4. Performance Measures
As the system length distributions at various epochs areknown, performance measures such as average number ofcustomers in the system (queue) πΏ
π (πΏπ) are given by
πΏπ =
π
β
π=1
πππ,0+
π
β
π=1
πππ,1,
πΏπ=
π
β
π=1
(π β 1) ππ,0+
π
β
π=1
(π β 1) ππ,1.
(18)
The busy probability of the server (ππ΅), probability that the
server is in working vacation (πwv), and probability that theserver is idle (πid) are given as
ππ΅=
π
β
π=1
ππ,1, πwv =
π
β
π=0
ππ,0, πid = πΏπ0,1. (19)
When there are π customers in the system upon arrival, theprobability that a customer balks from the system is 1βπ
π, and
the immediate balking rate is π(1 β ππ). The average balking
rate (B.R.) is given by
B.R. =π
β
π=1
π (1 β ππ) ππ,0+
π
β
π=1
π (1 β ππ) ππ,1. (20)
If there are π customers in the system and server is available,then there are (π β 1) waiting customers in the queue. Sinceany one of the (π β 1) customers in the queue may renege, theinstantaneous reneging rate is (π β 1)πΌ. The average renegingrate (R.R.) is given by
R.R. =π
β
π=1
(π β 1) πΌππ,0+
π
β
π=1
(π β 1) πΌππ,1. (21)
Finally, the average rate of customer loss (L.R.) is the sum ofthe average balking rate and the average reneging rate andyields
L.R. = B.R. + R.R. (22)
4.1. Cost Model. In this subsection, we develop an expectedcost function to determine an optimum regular service rate,πβ, and the optimum total expected cost, πΉ(πβ). Let us define
the following cost elements:
πΆ1β‘ service cost per unit time when the server is in
regular busy period,πΆ2β‘ service cost per unit time when the server is on
working vacation period,πΆ3β‘ cost per unit time when a customer joins the
queue and waits for service,πΆ4β‘ cost per unit time when a customer balks or
reneges.
Based on the definitions of each cost element listed above andits corresponding system performance measures, the totalexpected cost function per unit time is given by
πΉ (π) = πΆ1π + πΆ
2π + πΆ3πΏπ+ πΆ4L.R., (23)
where πΏπand L.R. are given in the previous section. The
first two costs are obtained by the server, the third one bythe customerβs waiting in the queue, and the fourth one bythe customer loss. The objective is to determine the optimalservice rate πβ to minimize the cost function πΉ. As theexpected cost function is highly complex, it is a difficulttask to develop analytic results for the optimum value of π.We use the quadratic fit search method to solve the aboveoptimization problem as given in Rardin [17]. Given a 3-pointpattern, the unique optimum π₯ of the quadratic functionagreeing with π(π₯) at (π₯
0, π₯1, π₯2) occurs at
π₯
=1
2
π (π₯0) (π₯2
1β π₯2
2) + π (π₯
1) (π₯2
2β π₯2
0) + π (π₯
2) (π₯2
0β π₯2
1)
π (π₯0) (π₯1β π₯2) + π (π₯
1) (π₯2β π₯0) + π (π₯
2) (π₯0β π₯1).
(24)
International Journal of Stochastic Analysis 5
5. Special Cases
In this section, some special cases which are available inthe literature are deduced from our model by taking specificvalues for the parameters π, π, π
π, and πΌ.
Case 1. ConsiderπΌ β 0.That is, the customers never renege.The model reduces to Geo/Geo/1/π queue with balking andmultiple working vacations (for πΏ = 0) and single workingvacation (for πΏ = 1). For πΏ = 0, our results match the resultsof Geo/Geo/1/π queue with balking and multiple workingvacations of [14].
Case 2. Consider π β 0. That is, there is no service duringvacation. The model reduces to Geo/Geo/1/π queue withbalking, reneging, and multiple vacations (for πΏ = 0) andsingle vacation (for πΏ = 1).
Case 3. πΌ β 0, ππ= 1, 0 β€ π β€ π β 1; that is, the customers
neither balk nor renege. The model reduces to Geo/Geo/1/πqueue with working vacations. The steady-state probabilitiesare
ππ,0= ππππ,0, 0 β€ π β€ π,
ππ,1= ππππ,1+ ππππ,0, 0 β€ π β€ π,
(25)
where q0(π₯) = π, q
π(π₯) = ππ₯, 1 β€ π β€ π β 1, r
π(π₯) = ππ₯ + ππ₯,
1 β€ π β€ πβ1, rπ(π) = π₯, s
π(π₯) = ππ₯, 1 β€ π β€ πβ1, s
π(π₯) = π₯,
and tπ(π₯) = 0 βπ.
Case 4. Consider π β 0, π β 1. That is, the server nevertakes any vacation. The model reduces to Geo/Geo/1/πqueue with balking and reneging. In this case the vacationprobabilities (π
π,0, 1 β€ π β€ π) do not exist. If we take π
0,0as
π0and π
π,1as ππfor 1 β€ π β€ π, we get the results of balking
and reneging Geo/Geo/1/π queue without vacations.
Case 5. Consider π β 0, π β 1, πΌ β 0. That is, the servernever takes any vacation.Themodel reduces to Geo/Geo/1/πqueue with balking and without vacation. To obtain relationssteady-state probabilities, let us define π
0,0= π0and π
π,1= ππ
for 1 β€ π β€ π. In this case, the vacation probabilities (ππ,0,
1 β€ π β€ π) do not exist. Taking q1(π) = π, q
π(π) = ππ
ππ, 1 β€
π β€ πβ 1, rπ(π) = (1 β ππ
π)π + ππ
ππ, 1 β€ π β€ πβ 1, r
π(π) = π,
sπ(π) = (1 β ππ
π)π, 1 β€ π β€ πβ 1, s
π(π) = π, and t
π(π) = 0 βπ,
we get steady-state probabilities after simplification as
ππ= ππππ, 0 β€ π β€ π,
ππ=
πππ
πππβ1π
πβ2
β
π=π
π (1 β πππ+1)
ππππ
,
ππ= [
[
1 +π
πππβ1π
πβ1
β
π=0
πβ2
β
π=π
π(1 β πππ+1)
ππππ
]
]
β1
.
(26)
Our results match analytically the results posed in [13].
Case 6. Consider π β 0, π β 1, πΌ β 0, and ππ= 1, 0 β€
π β€ πβ1. Themodel reduces to Geo/Geo/1/π queue without
balking, reneging, and vacation and our results match theresults available in the literature. It is to be noted that π
π,0,
1 β€ π β€ π, will not occur in this case.
6. Numerical Analysis
In this section, we present a variety of numerical results toanalyze the operating and economic performance measuresfor MWV and SWV policies. The balking function is takenas ππ= 1 β (π/π
2
), 1 β€ π β€ π β 1, π0= 1, π
π= 0, and
the various cost elements are πΆ1= 25, πΆ
2= 20, πΆ
3= 21,
and πΆ4= 18. We fix the capacity of the system as π =
30. Table 1 shows the minimum cost πΉ(πβ), the optimumvalue of πβ, and some performance measures for MWV andSWV policies. It demonstrates a numerical analysis of someperformance measures such as πΏ
π, πΏπ , L.R., π
π΅, πwv, and πΉ(π)
for different values of reneging rate (πΌ). The parameters aretaken as π = 0.2, π = 0.4, π = 0.2, π = 10, and π
π=
1 β πβπ. We observe that as πΌ increases (i) the optimum πβ
decreases; (ii) the optimum cost and the πwv decrease forboth the models; (iii) the other performance indices increasefor both the models; (iv) the single working vacation modelhas better performance measures than the multiple workingvacations model.
We demonstrate the minimum cost πΉ(πβ), the optimumvalue of πβ, and the variations in the system performancemeasures for MWV and SWV policies in Table 2. It displayscomparative numerical results of some performance mea-sures such as πΏ
π, πΏπ , L.R., π
π΅, πwv, and πΉ(π) for different
values of service rate during vacation (π). The parametersare taken as π = 0.2, π = 0.4, πΌ = 0.01, π = 10, andππ= 1βπ/π
2. It can be seen that as π increases, (i) the optimumπβ decreases; (ii) the optimum cost and the πwv increase for
both the models; (iii) the other performance indices decreasefor both the models; (iv) the single working vacation modelhas better performance measures than the multiple workingvacations model.
Figure 1 illustrates the effect of service rate during vaca-tion π on the average rate of customer loss (L.R.) for variousvacation parameters πwithMWV policy.The parameters aretaken as π = 0.2, π = 0.5, π = 10, and πΌ = 0.1. We observethat the average rate of customer loss in the system decreasesas service rate during vacation increases. When π > π, L.R.decreases with the increase of vacation rate (π), but whenπ < π, L.R. increaseswith the increase of vacation rate.Hence,a better performance result can be achieved by choosing thevalue of π less than π, which is coherent with the intuition thatthe more frequently customers balk, the more customers arein queue and the more possibility to lose customers.
Figure 2 shows the effect of traffic intensity (π) on theaverage queue length (πΏ
π) for different service rates during
vacation period π. The parameters are taken as π = 0.2, π =0.7, π = 10, π = 0.1, and πΌ = 0.05. We observe that theexpected queue length πΏ
πmonotonically increases as traffic
intensity (π) increases for both MWV and SWV models.When π is kept fixed, πΏ
πdecreases with the increase of π.
Furthermore, MWV policy outperforms the SWVmodel.
6 International Journal of Stochastic Analysis
Table 1: Effect of πΌ on performance characteristics for MWV and SWVmodels.
πΌ = 0.03 πΌ = 0.05 πΌ = 0.07 πΌ = 0.1MWV SWV MWV SWV MWV SWV MWV SWV
πβ 0.3648 0.3683 0.3381 0.3439 0.3133 0.3215 0.2807 0.2903πΏπ
0.2169 0.1879 0.2207 0.1933 0.2241 0.1976 0.2263 0.2027πΏπ
0.7214 0.6618 0.7386 0.6827 0.7553 0.7018 0.7760 0.7282ππ΅
0.3840 0.4248 0.4008 0.4417 0.4174 0.4579 0.4405 0.4812πwv 0.6160 0.2515 0.5992 0.2441 0.5826 0.2370 0.5595 0.2268L.R. 0.0358 0.0337 0.0411 0.0386 0.0465 0.0435 0.0545 0.0512πΉ(πβ
) 17.5619 17.0943 17.0414 16.6545 16.5628 16.2478 15.9083 15.6871
Table 2: Effect of π on performance characteristics for MWV and SWVmodels.
π = 0.05 π = 0.1 π = 0.2 π = 0.3MWV SWV MWV SWV MWV SWV MWV SWV
πβ 0.4506 0.4463 0.4475 0.4448 0.4402 0.4422 0.4341 0.4397πΏπ
0.3049 0.2376 0.2876 0.2293 0.2591 0.2152 0.2346 0.2046πΏπ
0.8532 0.7270 0.8230 0.7127 0.7706 0.6873 0.7228 0.6668ππ΅
0.4189 0.4332 0.4084 0.4291 0.3895 0.4214 0.3712 0.4149πwv 0.5811 0.2526 0.5916 0.2530 0.6105 0.2537 0.6288 0.2541L.R. 0.0048 0.0038 0.0045 0.0037 0.0041 0.0035 0.0038 0.0034πΉ(πβ
) 17.8250 16.4920 18.4318 17.3027 19.7305 18.9820 21.1296 20.7263
0.0020.3 0.4 0.5 0.6 0.7 0.8
0.006
0.01
0.014
0.018
0.022
π = 0.1
π = 0.2
π = 0.3
π
L.R.
Figure 1: Impact of π on L.R.
Figure 3 plots the impact of π on the state probabilitiesof the server for various values of vacation parameter π withSWV policy. It is observed that as π increases, the probabilitythat the server is busy (π
π΅) decreases, the probability that
the server is in working vacation (πwv) increases, and theprobability that the server is idle (πid) increases. It is alsoseen that as vacation parameter π increases both π
π΅and πid
0.14
0.5
1.5
2.5
0.34 0.54 0.74 0.94 1.140
1
2
3
Lq
π
π = 0.1, MWV
π = 0.3, MWVπ = 0.5, MWVπ = 0.5, SWV
π = 0.1, SWVπ = 0.3, SWV
Figure 2: Effect of π on πΏπ.
increase. But, with the increase of vacation parameter π, πwvdecreases.
The impact of π on the state probabilities of the server forvarious values of vacation parameter π for MWV model isshown in Figure 4. It is observed that the probability that theserver is busy (π
π΅) decreases, whereas the probability that the
server is in working vacation (πwv) increases as π increases.
International Journal of Stochastic Analysis 7
0.15
0.25
0.35
0.45
0.55
0.6
0.5
0.4
0.3
0.1
0.2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Stat
e pro
babi
lity
of th
e ser
ver
Service rate during vacation, π
PB
π = 0.4
π = 0.4π = 0.4
π = 0.25
π = 0.25
π = 0.25
Pwv
Pid
Figure 3: Impact of π on state probability of the server, SWV.
0.7
0.6
0.65
0.55
0.45
0.35
0.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.5
Stat
e pro
babi
lity
of th
e ser
ver
PB
Service rate during vacation, π
π = 0.4
π = 0.4
π = 0.6
π = 0.6
π = 0.8
π = 0.8
Pwv
Figure 4: Impact of π on state probability of the server, MWV.
As expected, ππ΅increases as vacation parameter π increases
but πwv decreases with the increase of vacation parameter π.The impact ofπ on the average rate of customer loss (L.R.)
with MWV policy for different balking functions (i) ππ= 1 β
(π/π2
), (ii) ππ= 1β(π/π), (iii) π
π= 1/(π+1), and (iv) π
π= 1βπ
βπ
is shown in Figure 5. It is evident from the figure that as πincreases the average rate of customer loss (L.R.) decreasesfor different balking functions. But the average waiting timein the system is lower for the balking function given in (i).
Figure 6 depicts the total expected cost (πΉ(π)) as afunction of regular service rate (π) when the server is workingwith different service rates during vacation period for SWV
0
0.005
0.01
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
L.R.
π
bi = 1 β (i/N2)bi = 1 β (i/N)
bi = 1/(i + 1)
bi = 1 β eβi
Figure 5: Impact of π on L.R. for MWV.
220.3 0.4 0.5 0.6 0.7 0.8 0.9
24
26
28
30
32
34
36
Tota
l exp
ecte
d co
st,F(π)
Regular service rate, π
π = 0.1
π = 0.2
π = 0.3
Figure 6: Effect of π on total expected cost for SWV.
model. The parameters are taken as π = 10, π = 0.1, π =0.2, π = 0.1, πΌ = 0.01, πΆ
1= 25, πΆ
2= 20, πΆ
3= 18, and
πΆ4= 15. One may observe that for fixed π, the total expected
cost decreases when service rate π β€ 0.4 and increases forπ > 0.4. However, when the service rate during vacation π isincreased the total expected cost reduces substantially. Thisinfers that increasing the service rate π beyond a certain levelcannot further reduce total expected cost. These computedperformance measures illustrate the performance effects ofassigning job ability during the vacation period. Also thebalking and reneging have a clear effect on the probabilitiesas we can see that there is a significant initial deviations in thestate probabilities of the server.
8 International Journal of Stochastic Analysis
7. Conclusion
In this paper, we have carried out an analysis of balking andreneging in finite-buffer discrete-time single server queuewith single andmultiple working vacations for the late arrivalsystem with delayed access from an economic viewpoint.The balking and reneging factors make our system morepractical. The customers are allowed to balk or leave afterjoining the queue without getting served due to renege. Wehave developed closed-form expressions of the stationarystate probabilities using recursive method. Various perfor-mance measures are presented. Numerical results in theform of tables and graphs are discussed to display the effectof the system parameters on the performance measures.The cost model proposed in our study may be useful indetermining the optimal service rate needed for operatingthe system smoothly at optimum cost. The given model canbe generalized to multiple servers with balking, reneging,and working vacations. The important significance of thefindings reported in this paper is that managers of servicesystems can assist customers in making the right decisionby providing them with precise estimates of expected timein the system. Better tradeoff of balking rate and servicerate may improve decision making and satisfaction and italso lowers reneging. The modeling analysis for the discrete-time queueing systems with balking, reneging, and workingvacations may be applied in many congestion situations ofcommunication systems, manufacturing systems, produc-tion/inventory systems, transportation, and many others.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
References
[1] H. Takagi, Queueing AnalysisβA Foundation of PerformanceEvaluation, vol. 3, North Holland, Amsterdam, The Nether-lands, 1993.
[2] L. D. Servi and S. G. Finn, βM/M/1 queues with workingvacations (M/M/1/WV),β Performance Evaluation, vol. 50, no.1, pp. 41β52, 2002.
[3] D.-A. Wu and H. Takagi, βπ/πΊ/1 queue with multiple workingvacations,β Performance Evaluation, vol. 63, no. 7, pp. 654β681,2006.
[4] N. Tian, Z. Ma, andM. Liu, βThe discrete time GEOm/GEOm/1queue with multiple working vacations,β Applied MathematicalModelling, vol. 32, no. 12, pp. 2941β2953, 2008.
[5] J. Li and N. Tian, βAnalysis of the discrete time Geo/Geo/1queue with single working vacation,β Quality Technology &Quantitative Management, vol. 5, no. 1, pp. 77β89, 2008.
[6] P. Liao, βOptimal staffing policy for queueing systems withcyclic demands: waiting cost approach,β in Proceedings of the18th Annual Conference of the Production and OperationsManagement Society (POMS β07), Dallas, Tex, USA, 2007.
[7] D. Yue, Y. Zhang, and W. Yue, βOptimal performance analysisof an M/M/1/N queue system with balking, reneging andserver vacation,β International Journal of Pure and AppliedMathematics, vol. 28, pp. 101β115, 2006.
[8] R. Tian and D. Yue, βOptimal balking strategies in a Markovianqueue with a single vacation,β Journal of Information & Compu-tational Science, vol. 9, no. 10, pp. 2827β2841, 2012.
[9] Y. Ma, W.-q. Liu, and J.-h. Li, βEquilibrium balking behaviorin the πΊππ/πΊππ/1 queueing system with multiple vacations,βApplied Mathematical Modelling, vol. 37, no. 6, pp. 3861β3878,2013.
[10] F. Zhang, J. Wang, and B. Liu, βEquilibrium balking strategiesin Markovian queues with working vacations,β Applied Mathe-matical Modelling, vol. 37, no. 16-17, pp. 8264β8282, 2013.
[11] P. Vijaya Laxmi, V. Goswami, andK. Jyothsna, βAnalysis of finitebuffer Markovian queue with balking, reneging and workingvacations,β International Journal of Strategic Decision Sciences,vol. 4, no. 1, pp. 1β23, 2013.
[12] P. V. Laxmi, V. Goswami, and K. Jyothsna, βOptimization ofbalking and reneging queue with vacation interruption underN-policy,β Journal of Optimization, vol. 2013, Article ID 683708,9 pages, 2013.
[13] M. Lozano and P. Moreno, βA discrete time single-server queuewith balking: economic applications,β Applied Economics, vol.40, no. 6, pp. 735β748, 2008.
[14] P. Vijaya Laxmi, V. Goswami, and K. Jyothsna, βAnalysis ofdiscrete-time single server queue with balking and multipleworking vacations,β Quality Technology and Quantitative Man-agement, vol. 10, no. 4, pp. 441β454, 2013.
[15] J. J. Hunter, Mathematical Techniques of Applied Probability,Discrete-time models: Techniques and applicationss and appli-cations, Academic Press, New York, NY, USA, 1983.
[16] A. Gravey and G. Hebuterne, βSimultaneity in discrete-timesingle server queues with Bernoulli inputs,β Performance Eval-uation, vol. 14, no. 2, pp. 123β131, 1992.
[17] R. L. Rardin, Optimization in Operations Research, PrenticeHall, Upper Saddle River, NJ, USA, 1997.
Submit your manuscripts athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Stochastic AnalysisInternational Journal of