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 𝑀/𝑀/1/𝑁 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 𝑀/𝑀/1queue with working vacations. Laxmi et al. [11] analyzedan𝑀/𝑀/1/𝑁 queueing system with balking, reneging, andworking vacations. Laxmi et al. [12] studied an optimizationof balking and reneging queue with vacation interruptionunder𝑁-policy.
The study of the discrete-time queueing systems withimpatient customers’ 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/𝑁queue 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, . . . , 𝑡, . . .. Here, a potential arrival takesplace in (𝑡−, 𝑡) and a potential departure occurs in (𝑡, 𝑡+).More specifics on discrete-time queue have been reported inHunter [15] and Gravey and Hebuterne [16].
We assume that the interarrival times 𝐴 of customers areindependent and geometrically distributed with probabilitymass function (p.m.f.) 𝑎
𝑛= 𝑃(𝐴 = 𝑛) = 𝜆
𝑛−1
𝜆, 𝑛 ≥ 1, 0 <𝜆 < 1, where for any real number 𝑥 ∈ [0, 1], we denote𝑥 = 1 − 𝑥. If on arrival customer finds the system busy, thearriving customer either decides to join the queue or balk.Let 𝑏𝑛represent the probability, when the system size is 𝑛
with which the customer decides to join the queue in orderto be served or balk with probability 1 − 𝑏
𝑛, when there are
𝑛 customers in the system upon arrival (𝑛 = 0, 1, . . . , 𝑁 − 1),where the capacity of the system is finite𝑁. Furthermore, weassume that 0 ≤ 𝑏
𝑛+1≤ 𝑏𝑛< 1, 1 ≤ 𝑛 ≤ 𝑁 − 1, 𝑏
0= 1, 𝑏𝑁= 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.): 𝑃(𝑆 = 𝑛) = 𝜇
𝑛−1
𝜇, 𝑛 ≥ 1, 0 <𝜇 < 1. The service times 𝑆V in a working vacation periodare independent and geometrically distributed with commonp.m.f.: 𝑃(𝑆V = 𝑛) = 𝜂
𝑛−1
𝜂, 𝑛 ≥ 1, 0 < 𝜂 < 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 𝑉 are independent and geometricallydistributed with common p.m.f.: 𝑃(𝑉 = 𝑛) = 𝜙
𝑛−1
𝜙, 𝑛 ≥ 1,0 < 𝜙 < 1.
After joining the queue each customer will wait a certainlength of time 𝑇 for service to begin. If it has not begunby then, he will get impatient and leave the queue withoutgetting service.This time 𝑇 is independent and geometricallydistributed with common p.m.f.: 𝑃(𝑇 = 𝑛) = 𝛼𝑛−1𝛼, 𝑛 ≥ 1,0 < 𝛼 < 1. Since the arrival and the departure of theimpatient customers without service are independent, theaverage reneging rate of the customer can be given by 𝑟(𝑛) =(𝑛 − 1)𝛼, 1 ≤ 𝑛 ≤ 𝑁.
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 (𝛿) as follows: 𝛿 = 1 yields the results for the SWVmodel and 𝛿 = 0 gives the results for the MWVmodel.
International Journal of Stochastic Analysis 3
At steady-state, let us define 𝜋𝑖,0, 0 ≤ 𝑖 ≤ 𝑁, to be the
probability that there are 𝑖 customers in the system when theserver is in working vacation and 𝜋
𝑖,1, 1−𝛿 ≤ 𝑖 ≤ 𝑁, to be the
probability that there are 𝑖 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= 𝜆 (1 − 𝜙𝛿) 𝜋
0,0+ s1(𝜂) 𝜋1,0
+ t2(𝜂) 𝜋2,0+ s1(𝜇) 𝜋1,1+ t2(𝜇) 𝜋2,1,
(1)
𝜋𝑖,0= 𝜙r𝑖(𝜂) 𝜋𝑖,0+ 𝜙q𝑖−1(𝜂) 𝜋𝑖−1,0
+ 𝜙s𝑖+1(𝜂) 𝜋𝑖+1,0
+ 𝜙t𝑖+2(𝜂) 𝜋𝑖+2,0, 1 ≤ 𝑖 ≤ 𝑁 − 2,
(2)
𝜋𝑁−1,0
= 𝜙r𝑁−1(𝜂) 𝜋𝑁−1,0
+ 𝜙q𝑁−2(𝜂) 𝜋𝑁−2,0
+ 𝜙s𝑁(𝜂) 𝜋𝑁,0,
(3)
𝜋𝑁,0= 𝜙r𝑁(𝜂) 𝜋𝑁,0+ 𝜙q𝑁−1(𝜂) 𝜋𝑁−1,0
, (4)
𝜋0,1= 𝜆𝜋0,1+ 𝜙𝜆𝛿𝜋
0,0, (5)
𝜋1,1= r1(𝜇) 𝜋1,1+ s2(𝜇) 𝜋2,1+ t3(𝜇) 𝜋3,1
+ q0(𝜇) 𝛿𝜋
0,1+ 𝜙r1(𝜂) 𝜋1,0+ 𝜙q0(𝜂) 𝜋0,0
+ 𝜙s2(𝜂) 𝜋2,0+ 𝜙t3(𝜂) 𝜋3,0,
(6)
𝜋𝑖,1= r𝑖(𝜇) 𝜋𝑖,1+ q𝑖−1(𝜇) 𝜋𝑖−1,1
+ s𝑖+1(𝜇) 𝜋𝑖+1,1
+ t𝑖+2(𝜇) 𝜋𝑖+2,1
+ 𝜙r𝑖(𝜂) 𝜋𝑖,0+ 𝜙q𝑖−1(𝜂) 𝜋𝑖−1,0
+ 𝜙s𝑖+1(𝜂) 𝜋𝑖+1,0
+ 𝜙t𝑖+2(𝜂) 𝜋𝑖+2,0,
1 ≤ 𝑖 ≤ 𝑁 − 2,
(7)
𝜋𝑁−1,1
= r𝑁−1(𝜇) 𝜋𝑁−1,1
+ q𝑁−2(𝜇) 𝜋𝑁−2,1
+ s𝑁(𝜇) 𝜋𝑁,1+ 𝜙r𝑁−1(𝜂) 𝜋𝑁−1,0
+ 𝜙q𝑁−2(𝜂) 𝜋𝑁−2,0
+ 𝜙s𝑁(𝜂) 𝜋𝑁,0,
(8)
𝜋𝑁,1= r𝑁(𝜇) 𝜋𝑁,1+ q𝑁−1(𝜇) 𝜋𝑁−1,1
+ 𝜙r𝑁(𝜂) 𝜋𝑁,0+ 𝜙q𝑁−1(𝜂) 𝜋𝑁−1,0
,
(9)
where
q𝑖(𝑥) = {
𝜆 : 𝑖 = 0,
𝜆𝑏𝑖𝑥 (1 − (𝑖 − 1) 𝛼) : 𝑖 = 1, . . . , 𝑁 − 1,
r𝑖(𝑥) =
{{{{{{{
{{{{{{{
{
(1 − 𝜆𝑏1) 𝑥 + 𝜆𝑏
1𝑥 : 𝑖 = 1,
(1 − 𝜆𝑏𝑖) 𝑥 (1 − (𝑖 − 1) 𝛼)
+ 𝜆𝑏𝑖(𝑥 (1 − (𝑖 − 1) 𝛼)
+ 𝑥 (𝑖 − 1) 𝛼) : 𝑖 = 2, . . . , 𝑁 − 1,
𝑥 (1 − (𝑁 − 1) 𝛼) : 𝑖 = 𝑁,
s𝑖(𝑥)
=
{{{{{{{
{{{{{{{
{
(1 − 𝜆𝑏1) 𝑥 : 𝑖 = 1,
(1 − 𝜆𝑏𝑖) (𝑥 (1 − (𝑖 − 1) 𝛼)
+ 𝑥 (𝑖 − 1) 𝛼)
+ 𝜆𝑏𝑖𝑥 (𝑖 − 1) 𝛼 : 𝑖 = 2, . . . , 𝑁 − 1,
𝑥 (1 − (𝑁 − 1) 𝛼) + 𝑥 (𝑁 − 1) 𝛼 : 𝑖 = 𝑁,
t𝑖(𝑥) = {
(1 − 𝜆𝑏𝑖) 𝑥 (𝑖 − 1) 𝛼 : 𝑖 = 2, . . . , 𝑁 − 1,
𝑥 (𝑁 − 1) 𝛼 : 𝑖 = 𝑁.
(10)
The steady-state probabilities 𝜋𝑖,0(0 ≤ 𝑖 ≤ 𝑁) and 𝜋
𝑖,1, (1 −
𝛿 ≤ 𝑖 ≤ 𝑁) are computed recursively by solving the system of(1) to (9). Solving (2) to (4) recursively, it can be simplified as
𝜋𝑖,0= 𝜉𝑖𝜋𝑁,0, 0 ≤ 𝑖 ≤ 𝑁, (11)
where
𝜉𝑁= 1, 𝜉
𝑁−1= (
1 − 𝜙r𝑁(𝜂)
𝜙q𝑁−1(𝜂)
) 𝜉𝑁,
𝜉𝑁−2
=
(1 − 𝜙r𝑁−1(𝜂)) 𝜉
𝑁−1− 𝜙s𝑁(𝜂) 𝜉𝑁
𝜙q𝑁−2(𝜂)
,
𝜉𝑖=
(1 − 𝜙r𝑖+1(𝜂)) 𝜉
𝑖+1− 𝜙s𝑖+2(𝜂) 𝜉𝑖+2− 𝜙t𝑖+3(𝜂) 𝜉𝑖+3
𝜙q𝑖(𝜂)
,
𝑖 = 𝑁 − 3, . . . , 1, 0.
(12)
Using (11) in (5) and (7)–(9) yields 𝜋𝑖,1, (1 − 𝛿 ≤ 𝑖 ≤ 𝑁), in
terms of 𝜋𝑁,0
and 𝜋𝑁,1
as
𝜋𝑖,1= 𝜁𝑖𝜋𝑁,1+ 𝜔𝑖𝜋𝑁,0, 0 ≤ 𝑖 ≤ 𝑁, (13)
where 𝜁𝑁= 1, 𝜔
𝑁= 0,
𝜁𝑁−1
=1 − r𝑁(𝜇)
q𝑁−1(𝜇)
,
𝜔𝑁−1
= −𝜙(r𝑁(𝜂) 𝜉𝑁+ q𝑁−1(𝜂) 𝜉𝑁−1
q𝑁−1(𝜇)
) ,
𝜁𝑁−2
=(1 − r
𝑁−1(𝜇)) 𝜁
𝑁−1− s𝑁(𝜇) 𝜁𝑁
q𝑁−2(𝜇)
,
𝜔𝑁−2
= ((1 − r𝑁−1(𝜇)) 𝜔
𝑁−1
− 𝜙 (q𝑁−2(𝜂) 𝜉𝑁−2+ r𝑁−1(𝜂) 𝜉𝑁−1+ s𝑁(𝜂) 𝜉𝑁))
× (q𝑁−2(𝜇))−1
𝜁𝑖−1=(1 − r
𝑖(𝜇)) 𝜁
𝑖− s𝑖+1(𝜇) 𝜁𝑖+1− t𝑖+2(𝜇) 𝜁𝑖+2
q𝑖−1(𝜇)
,
𝑖 = 𝑁 − 2, . . . , 1,
4 International Journal of Stochastic Analysis
𝜔𝑖−1=(1 − r
𝑖(𝜇)) 𝜔
𝑖− s𝑖+1(𝜇) 𝜔𝑖+1− t𝑖+2(𝜇) 𝜔𝑖+2
q𝑖−1(𝜇)
− 𝜙 ( (r𝑖(𝜂) 𝜉𝑖+ q𝑖−1(𝜂) 𝜉𝑖−1
+ s𝑖+1(𝜂) 𝜉𝑖+1+ t𝑖+2(𝜂) 𝜉𝑖+2)
× (q𝑖−1(𝜇))−1
) , 𝑖 = 𝑁 − 2, . . . , 2,
𝜁0= 0, 𝜔
0= (
𝜙𝜆𝛿𝜉0
𝜆) .
(14)
Now, using (11) and (13) in (6), the probability 𝜋𝑁,1
can beexpressed in terms of 𝜋
𝑁,0as
𝜋𝑁,1= Υ𝜋𝑁,0, (15)
where
Υ
= 𝜙(r1(𝜂) 𝜉1+ q0(𝜂) 𝜉0+ s2(𝜂) 𝜉2+ t3(𝜂) 𝜉3
(1 − r1(𝜇)) 𝜁
1− s2(𝜇) 𝜁2− t3(𝜇) 𝜁3
)
+ (s2(𝜇) 𝜔2+ t3(𝜇) 𝜔3+ 𝛿q0(𝜇) 𝜔0− (1 − r
1(𝜇)) 𝜔
1
(1 − r1(𝜇)) 𝜁
1− s2(𝜇) 𝜁2− t3(𝜇) 𝜁3
) .
(16)
Finally, the only unknown 𝜋𝑁,0
is obtained from the normal-ization condition∑𝑁
𝑖=0𝜋𝑁,0+ ∑𝑁
𝑖=1−𝛿𝜋𝑁,1= 1 and is given by
𝜋𝑁,0= [
𝑁
∑
𝑖=0
𝜉𝑖+
𝑁
∑
𝑖=1−𝛿
(𝜔𝑖+ Υ𝜁𝑖)]
−1
. (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) 𝐿
𝑠(𝐿𝑞) are given by
𝐿𝑠=
𝑁
∑
𝑖=1
𝑖𝜋𝑖,0+
𝑁
∑
𝑖=1
𝑖𝜋𝑖,1,
𝐿𝑞=
𝑁
∑
𝑖=1
(𝑖 − 1) 𝜋𝑖,0+
𝑁
∑
𝑖=1
(𝑖 − 1) 𝜋𝑖,1.
(18)
The busy probability of the server (𝑃𝐵), probability that the
server is in working vacation (𝑃wv), and probability that theserver is idle (𝑃id) are given as
𝑃𝐵=
𝑁
∑
𝑖=1
𝜋𝑖,1, 𝑃wv =
𝑁
∑
𝑖=0
𝜋𝑖,0, 𝑃id = 𝛿𝜋0,1. (19)
When there are 𝑖 customers in the system upon arrival, theprobability that a customer balks from the system is 1−𝑏
𝑖, and
the immediate balking rate is 𝜆(1 − 𝑏𝑖). The average balking
rate (B.R.) is given by
B.R. =𝑁
∑
𝑖=1
𝜆 (1 − 𝑏𝑖) 𝜋𝑖,0+
𝑁
∑
𝑖=1
𝜆 (1 − 𝑏𝑖) 𝜋𝑖,1. (20)
If there are 𝑖 customers in the system and server is available,then there are (𝑖 − 1) waiting customers in the queue. Sinceany one of the (𝑖 − 1) customers in the queue may renege, theinstantaneous reneging rate is (𝑖 − 1)𝛼. The average renegingrate (R.R.) is given by
R.R. =𝑁
∑
𝑖=1
(𝑖 − 1) 𝛼𝜋𝑖,0+
𝑁
∑
𝑖=1
(𝑖 − 1) 𝛼𝜋𝑖,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,𝜇∗, and the optimum total expected cost, 𝐹(𝜇∗). Let us define
the following cost elements:
𝐶1≡ service cost per unit time when the server is in
regular busy period,𝐶2≡ service cost per unit time when the server is on
working vacation period,𝐶3≡ cost per unit time when a customer joins the
queue and waits for service,𝐶4≡ 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
𝐹 (𝜇) = 𝐶1𝜇 + 𝐶
2𝜂 + 𝐶3𝐿𝑞+ 𝐶4L.R., (23)
where 𝐿𝑞and L.R. are given in the previous section. The
first two costs are obtained by the server, the third one bythe customer’s waiting in the queue, and the fourth one bythe customer loss. The objective is to determine the optimalservice rate 𝜇∗ to minimize the cost function 𝐹. As theexpected cost function is highly complex, it is a difficulttask to develop analytic results for the optimum value of 𝜇.We use the quadratic fit search method to solve the aboveoptimization problem as given in Rardin [17]. Given a 3-pointpattern, the unique optimum 𝑥 of the quadratic functionagreeing with 𝑓(𝑥) at (𝑥
0, 𝑥1, 𝑥2) occurs at
𝑥
=1
2
𝑓 (𝑥0) (𝑥2
1− 𝑥2
2) + 𝑓 (𝑥
1) (𝑥2
2− 𝑥2
0) + 𝑓 (𝑥
2) (𝑥2
0− 𝑥2
1)
𝑓 (𝑥0) (𝑥1− 𝑥2) + 𝑓 (𝑥
1) (𝑥2− 𝑥0) + 𝑓 (𝑥
2) (𝑥0− 𝑥1).
(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 𝜙, 𝜂, 𝑏
𝑖, and 𝛼.
Case 1. Consider𝛼 → 0.That is, the customers never renege.The model reduces to Geo/Geo/1/𝑁 queue with balking andmultiple working vacations (for 𝛿 = 0) and single workingvacation (for 𝛿 = 1). For 𝛿 = 0, our results match the resultsof Geo/Geo/1/𝑁 queue with balking and multiple workingvacations of [14].
Case 2. Consider 𝜂 → 0. That is, there is no service duringvacation. The model reduces to Geo/Geo/1/𝑁 queue withbalking, reneging, and multiple vacations (for 𝛿 = 0) andsingle vacation (for 𝛿 = 1).
Case 3. 𝛼 → 0, 𝑏𝑖= 1, 0 ≤ 𝑖 ≤ 𝑁 − 1; that is, the customers
neither balk nor renege. The model reduces to Geo/Geo/1/𝑁queue with working vacations. The steady-state probabilitiesare
𝜋𝑖,0= 𝜉𝑖𝜋𝑁,0, 0 ≤ 𝑖 ≤ 𝑁,
𝜋𝑖,1= 𝜁𝑖𝜋𝑁,1+ 𝜔𝑖𝜋𝑁,0, 0 ≤ 𝑖 ≤ 𝑁,
(25)
where q0(𝑥) = 𝜆, q
𝑖(𝑥) = 𝜆𝑥, 1 ≤ 𝑖 ≤ 𝑁 − 1, r
𝑖(𝑥) = 𝜆𝑥 + 𝜆𝑥,
1 ≤ 𝑖 ≤ 𝑁−1, r𝑁(𝜇) = 𝑥, s
𝑖(𝑥) = 𝜆𝑥, 1 ≤ 𝑖 ≤ 𝑁−1, s
𝑁(𝑥) = 𝑥,
and t𝑖(𝑥) = 0 ∀𝑖.
Case 4. Consider 𝜂 → 0, 𝜙 → 1. That is, the server nevertakes any vacation. The model reduces to Geo/Geo/1/𝑁queue with balking and reneging. In this case the vacationprobabilities (𝜋
𝑖,0, 1 ≤ 𝑖 ≤ 𝑁) do not exist. If we take 𝜋
0,0as
𝜋0and 𝜋
𝑖,1as 𝜋𝑖for 1 ≤ 𝑖 ≤ 𝑁, we get the results of balking
and reneging Geo/Geo/1/𝑁 queue without vacations.
Case 5. Consider 𝜂 → 0, 𝜙 → 1, 𝛼 → 0. That is, the servernever takes any vacation.Themodel reduces to Geo/Geo/1/𝑁queue with balking and without vacation. To obtain relationssteady-state probabilities, let us define 𝜋
0,0= 𝜋0and 𝜋
𝑖,1= 𝜋𝑖
for 1 ≤ 𝑖 ≤ 𝑁. In this case, the vacation probabilities (𝜋𝑖,0,
1 ≤ 𝑖 ≤ 𝑁) do not exist. Taking q1(𝜇) = 𝜆, q
𝑖(𝜇) = 𝜆𝑏
𝑖𝜇, 1 ≤
𝑖 ≤ 𝑁− 1, r𝑖(𝜇) = (1 − 𝜆𝑏
𝑖)𝜇 + 𝜆𝑏
𝑖𝜇, 1 ≤ 𝑖 ≤ 𝑁− 1, r
𝑁(𝜇) = 𝜇,
s𝑖(𝜇) = (1 − 𝜆𝑏
𝑖)𝜇, 1 ≤ 𝑖 ≤ 𝑁− 1, s
𝑁(𝜇) = 𝜇, and t
𝑖(𝜇) = 0 ∀𝑖,
we get steady-state probabilities after simplification as
𝜋𝑖= 𝜁𝑖𝜋𝑁, 0 ≤ 𝑖 ≤ 𝑁,
𝜁𝑖=
𝜇𝜁𝑁
𝜆𝑏𝑁−1𝜇
𝑁−2
∏
𝑗=𝑖
𝜇 (1 − 𝜆𝑏𝑗+1)
𝜆𝑏𝑗𝜇
,
𝜁𝑁= [
[
1 +𝜇
𝜆𝑏𝑁−1𝜇
𝑁−1
∑
𝑖=0
𝑁−2
∏
𝑗=𝑖
𝜇(1 − 𝜆𝑏𝑗+1)
𝜆𝑏𝑗𝜇
]
]
−1
.
(26)
Our results match analytically the results posed in [13].
Case 6. Consider 𝜂 → 0, 𝜙 → 1, 𝛼 → 0, and 𝑏𝑖= 1, 0 ≤
𝑖 ≤ 𝑁−1. Themodel reduces to Geo/Geo/1/𝑁 queue without
balking, reneging, and vacation and our results match theresults available in the literature. It is to be noted that 𝜋
𝑖,0,
1 ≤ 𝑖 ≤ 𝑁, 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 𝑏𝑛= 1 − (𝑛/𝑁
2
), 1 ≤ 𝑛 ≤ 𝑁 − 1, 𝑏0= 1, 𝑏
𝑁= 0, and
the various cost elements are 𝐶1= 25, 𝐶
2= 20, 𝐶
3= 21,
and 𝐶4= 18. We fix the capacity of the system as 𝑁 =
30. Table 1 shows the minimum cost 𝐹(𝜇∗), the optimumvalue of 𝜇∗, and some performance measures for MWV andSWV policies. It demonstrates a numerical analysis of someperformance measures such as 𝐿
𝑞, 𝐿𝑠, L.R., 𝑃
𝐵, 𝑃wv, and 𝐹(𝜇)
for different values of reneging rate (𝛼). The parameters aretaken as 𝜆 = 0.2, 𝜙 = 0.4, 𝜂 = 0.2, 𝑁 = 10, and 𝑏
𝑖=
1 − 𝑒−𝑖. We observe that as 𝛼 increases (i) the optimum 𝜇∗
decreases; (ii) the optimum cost and the 𝑃wv 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 𝐹(𝜇∗), the optimumvalue of 𝜇∗, 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 𝐿
𝑞, 𝐿𝑠, L.R., 𝑃
𝐵, 𝑃wv, and 𝐹(𝜇) for different
values of service rate during vacation (𝜂). The parametersare taken as 𝜆 = 0.2, 𝜙 = 0.4, 𝛼 = 0.01, 𝑁 = 10, and𝑏𝑖= 1−𝑖/𝑁
2. It can be seen that as 𝜂 increases, (i) the optimum𝜇∗ decreases; (ii) the optimum cost and the 𝑃wv 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 𝜂 on the average rate of customer loss (L.R.) for variousvacation parameters 𝜙withMWV policy.The parameters aretaken as 𝜆 = 0.2, 𝜇 = 0.5, 𝑁 = 10, and 𝛼 = 0.1. We observethat the average rate of customer loss in the system decreasesas service rate during vacation increases. When 𝜇 > 𝜂, L.R.decreases with the increase of vacation rate (𝜙), but when𝜇 < 𝜂, L.R. increaseswith the increase of vacation rate.Hence,a better performance result can be achieved by choosing thevalue of 𝜂 less than 𝜇, 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 (𝜌) on theaverage queue length (𝐿
𝑞) for different service rates during
vacation period 𝜂. The parameters are taken as 𝜆 = 0.2, 𝜇 =0.7, 𝑁 = 10, 𝜙 = 0.1, and 𝛼 = 0.05. We observe that theexpected queue length 𝐿
𝑞monotonically increases as traffic
intensity (𝜌) increases for both MWV and SWV models.When 𝜌 is kept fixed, 𝐿
𝑞decreases with the increase of 𝜂.
Furthermore, MWV policy outperforms the SWVmodel.
6 International Journal of Stochastic Analysis
Table 1: Effect of 𝛼 on performance characteristics for MWV and SWVmodels.
Figure 3 plots the impact of 𝜂 on the state probabilitiesof the server for various values of vacation parameter 𝜙 withSWV policy. It is observed that as 𝜂 increases, the probabilitythat the server is busy (𝑃
𝐵) decreases, the probability that
the server is in working vacation (𝑃wv) increases, and theprobability that the server is idle (𝑃id) increases. It is alsoseen that as vacation parameter 𝜙 increases both 𝑃
𝐵and 𝑃id
0.14
0.5
1.5
2.5
0.34 0.54 0.74 0.94 1.140
1
2
3
Lq
𝜌
𝜂 = 0.1, MWV
𝜂 = 0.3, MWV𝜂 = 0.5, MWV𝜂 = 0.5, SWV
𝜂 = 0.1, SWV𝜂 = 0.3, SWV
Figure 2: Effect of 𝜌 on 𝐿𝑞.
increase. But, with the increase of vacation parameter 𝜙, 𝑃wvdecreases.
The impact of 𝜂 on the state probabilities of the server forvarious values of vacation parameter 𝜙 for MWV model isshown in Figure 4. It is observed that the probability that theserver is busy (𝑃
𝐵) decreases, whereas the probability that the
server is in working vacation (𝑃wv) increases as 𝜂 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, 𝜂
PB
𝜙 = 0.4
𝜙 = 0.4𝜙 = 0.4
𝜙 = 0.25
𝜙 = 0.25
𝜙 = 0.25
Pwv
Pid
Figure 3: Impact of 𝜂 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, 𝜂
𝜙 = 0.4
𝜙 = 0.4
𝜙 = 0.6
𝜙 = 0.6
𝜙 = 0.8
𝜙 = 0.8
Pwv
Figure 4: Impact of 𝜂 on state probability of the server, MWV.
As expected, 𝑃𝐵increases as vacation parameter 𝜙 increases
but 𝑃wv decreases with the increase of vacation parameter 𝜙.The impact of𝜙 on the average rate of customer loss (L.R.)
with MWV policy for different balking functions (i) 𝑏𝑖= 1 −
(𝑖/𝑁2
), (ii) 𝑏𝑖= 1−(𝑖/𝑁), (iii) 𝑏
𝑖= 1/(𝑖+1), and (iv) 𝑏
𝑖= 1−𝑒
−𝑖
is shown in Figure 5. It is evident from the figure that as 𝜙increases 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 (𝐹(𝜇)) as afunction of regular service rate (𝜇) 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.
𝜙
bi = 1 − (i/N2)bi = 1 − (i/N)
bi = 1/(i + 1)
bi = 1 − e−i
Figure 5: Impact of 𝜙 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(𝜇)
Regular service rate, 𝜇
𝜂 = 0.1
𝜂 = 0.2
𝜂 = 0.3
Figure 6: Effect of 𝜇 on total expected cost for SWV.
model. The parameters are taken as 𝑁 = 10, 𝜙 = 0.1, 𝜆 =0.2, 𝜂 = 0.1, 𝛼 = 0.01, 𝐶
1= 25, 𝐶
2= 20, 𝐶
3= 18, and
𝐶4= 15. One may observe that for fixed 𝜂, the total expected
cost decreases when service rate 𝜇 ≤ 0.4 and increases for𝜇 > 0.4. However, when the service rate during vacation 𝜂 isincreased the total expected cost reduces substantially. Thisinfers that increasing the service rate 𝜇 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.
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