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80
OPTIMIZATION OF A FEEDBACK WORKING VACATION QUEUE WITH REVERSE BALKING AND
REVERSE RENEGING
K. Jyothsna1'*, P. Vijaya Laxmi2, P. Vijaya Kumar3
•
1'* Department of Basic Sciences and Humanities, Vignan's Institute of Engineering for Women,
Visakhapatnam, Andhra Pradesh, India. mail2jyothsnak@yahoo.co.in, drjyothsnak1984@gmail.com 2Department of Applied Mathematics, Andhra University, Visakhapatnam, Andhra Pradesh, India. vijayalaxmiau@gmail.com 3Department of Mathematics, GITAM (Deemed to be University), Visakhapatnam, Andhra Pradesh, India. vprathi@gitam.edu
Abstract
This paper analyzes a steady-state finite buffer M/M/1 feedback queue with reverse balking, reverse reneging and multiple working vacations. The concept of reverse balking and reverse reneging evolves from investment businesses wherein more the number of customers associated with a firm less the probability of balking of a customer and similar is the case of reverse reneging. Furthermore, if a customer is dissatisfied with the service provided, he or she may chose to rejoin the queue as a feedback customer. The server exits for working vacations whenever the system becomes empty instead of staying idle in the system. Vacation times and service times during working vacations are all independent random variables following exponential distribution. The model's steady-state system length distributions are calculated using the matrix approach. Some performance characteristics and cost optimization using ant colony optimization (ACO) are presented. Sensitivity analysis is performed using numerical results which are shown in the form of tables and graphs.
Keywords: reverse balking, reverse reneging, feedback, multiple working vacations, ACO
1. Introduction
Queueing models with server vacations have been actively researched and successfully used in manufacturing and production systems, service systems, communication systems, and other fields over the last three decades. Working vacations (WV) are a form of vacation policy established by Servi and Finn [13] wherein the server can provide service at a reduced rate rather than shutting down altogether during the vacation period. Wu and Takagi [16] and Baba [2] extended the M/M/1/WV queue to M/G/1 and GI/M/1 queues with working vacations, respectively. Krishnamoorthy and Sreenivasan [8] analyzed an M/M/2 queue with one of the two servers in working vacations. A survey on WV queues has been presented by Chandrasekaran et al. [4].
There is a growing trend to examine queueing systems from an economic perspective in order to address customers' unhappiness with waiting and desire for service. Customer impatience has a damaging influence on businesses since it causes them to lose potential consumers, which has a negative impact on the entire company. Balking and reneging are two queueing concepts that are commonly used to depict customer impatience. In balking, if a customer sees a large
queue ahead of him, he may resist at joining the queueing system. In the case of reneging, the customer joins the queue, waits for his service, and then departs the system without receiving service if the wait time exceeds his expectations. The situation of impatient customers in a server vacation period was investigated by Altman and Yechiali [1]. Yue, Yue and Xu [17] analyzed the single server queueing systems with customer impatience and WV. A Markovian queueing system with balking, reneging and WV has been studied by Vijaya Laxmi et al. [15].
In the above mentioned queueing models, the size of the system or the length of the queue influences balking and reneging. The larger the system, the more balking occurs, and the same is true with reneging. However, in the case of investment enterprizes, the number of customers with a certain firm becomes an intriguing and appealing feature for potential investors. As a result, the likelihood of joining such a company is high. In this scenario, the larger the system size, the greater the number of consumers who join it. As a result, when the system size is high, the chance of balking is low which is referred to as "reverse balking". Furthermore, having a large number of investors with an investment firm instils trust in investors and helps them to complete the term of their policies/bonds. That instance, when a firm has a big number of investing consumers, waiting customers will have more patience. When seen as a queueing system, it is obvious that as the queue becomes longer, fewer consumers would renege, a phenomenon known as "reverse reneging". Jain et al. [7] first incorporated the concept of reverse balking in queueing theory. Kumar and Som [10] developed the concept of reverse reneging and incorporated into an M/M/1 queueing system with reverse balking. A heterogeneous two server queue with reverse balking and reneging has been studied by Bouchentouf and Messabihi [3].
In queueing theory, feedback refers to a dissatisfied client rejoining the queue owing to poor service quality. Rework is another example of a queue with feedback in industrial processes. Tackacs [14] studied a single server queue with feedback to determine the stationary process for the queue size. Shanthakumaran and Thangaraj[12] considered a single server feedback queue with impatient customers. An M/M/1 feedback queueing model with retention of reneged customers and balking has been studied by Kumar and Sharma [9]. Kumar et al. [11] developed an M/M/1/N feedback queueing system with reverse balking.
To the best of our knowledge, the impatient attitude of customers in the reverse view has not been explored in working vacations queues. Therefore, we intend to embed reverse balking and reverse reneging in a feedback WV queue. In this article, we explore a finite buffer feedback WV queue in which customers may balk or renege owing to impatience in the reverse notion. The inter-arrival times, service times during regular service period, during WV period and vacation times are presumed to be exponentially distributed. The matrix form solution of the steady-state probabilities is found by putting the steady-state equations in block matrix form. The model's performance metrics, cost analysis using ACO are obtained. Tables and graphs have been used to demonstrate certain numerical findings.
The rest of the paper is laid out as follows. The queueing model is described in Section 2, followed by the steady-state equations and their solution in Section 3. In Section 4, we offer different model performance metrics as well as a cost model. Section 5 contains the sensitivity analysis followed by conclusions in Section 6.
2. Model description
Consider an M/M/1/N feedback queueing system with reverse balking, reverse reneging and WV. According to a Poisson process with an arrival rate A, customers arrive one at a time. When the system is unoccupied, a new customer has a probability q of joining the system and a p = (1 — q) probability of not joining. When there are i customers ahead of him in the system, let bi indicate the probability that the customer will join the queue or balk with probability 1 — bi. Furthermore, we assume that b0 = q and bN = 0. The assumption of reverse balking has been incorporated with bi+1 > bi, 1 < i < N — 1.
After joining the queue each customer will wait a certain length of time which is exponentially distributed with mean 1/a. When there are i customers in the system, the average rate of
reverse reneging of a customer is given by (N — (i — 1))a, , 1 < i < N.
If a customer receives service and finds it unsatisfactory, it can return to the system as a feedback customer with a probability q1 or depart with a probability p1 = 1 — q1.
A single server serves the customers on a first-come first-served basis with a service rate that follows an exponential distribution with mean 1/p. When the system gets empty, the server takes WV. If there are waiting customers in the line after a vacation expires, the server resumes regular service; otherwise, he departs for another WV. During the vacation time, the server stays active and provides service at a different service rate to the arriving customers. This type of working vacation is called multiple working vacations (MWV).
The vacation times and service times during WV are assumed to follow Poisson distribution with parameter $ and q, respectively. The inter-arrival times, vacation times, service times during regular service and during working vacation are mutually independent.
3. Analysis of the model
In this section, the Markov process is used to build the steady-state probability equations and the matrix technique is adopted to determine steady-state probabilities. Let n0/i,0 < i < N, be the probability that the server is on WV when there are i customers in the system, and ni,i, 1 < i < N, be the probability that there are i customers in the system while the server is in regular service period. The steady-state equations are derived using the Markov process as:
Abo no,o = m 1^0,1 + 01^1,1, (1)
Zi noi = Abi—1 no— + Mi+1 no,i+1, 1 < i < N — 1, (2)
ZN no,N = AbN—1 п0,N—1, (3)
t1 = 02 ^1,2 + $no,1, (4)
tin1,i = Ab— nXi—1 + 0i+1 n1,i+1 + $no,i, 2 < i < N — 1, (5)
0N n1,N = AbN—1 n1,N—1 + $no,N, (6)
where for 1 < i < N, ui = qp1 + (N — i + 1)a; 0i = pp1 + (N — i + 1)a; Zi = Abi + $ + mi; t{ =
Abi + 0i.
3.1. Matrix solution
In this subsection, the steady-state probabilities Kjj, j = 0,1; j < i < N, are obtained by solving the system of equations (1) to (6) using matrices.
Let n = (n0, n1) be the steady-state probability vector, where n0 = (n0,0, n0,1, n0,2,..., n0,N) and n1 = (n1,1, n1,2,..., n1,N). The equations (1) to (6) can be written in matrix form as
nQ = ne
0,
1,
(7)
(8)
where e is a column vector with each component equal to unity and the Markov process's transition rate matrix Q has the block form:
Q
Av
Avv Avb .
V Abv Abb
Abv and Abb are given by
—Abo , if i = j = 1,
Abi—1 , if i = j — 1, j > 2,
Mi—1 , if i = j + 1,
—Zi—1 , if i = j, j > 2,
o , otherwise
Avb Abv
A
bb
Q , if i = j + 1, 0 , otherwise
V1 , if i = j = 1, 0 , otherwise
—ti , if i = j,
Abi , if i = j — 1, j > 2,
Vi , if i = j + 1,
0 , otherwise.
Avb is a (N + 1) x N matrix, Abv is a N x (N + 1) matrix, Avv and Abb are square matrices of orders N + 1 and N, respectively.
Based on the partition n = (n0, ni), equations (7) and (8) can be written as:
no Avv + niAbv = 0, (9)
noAvb + niAbb = 0, (10)
noeo + niei = 1, (11)
where e0, ei are column vectors of order N + 1 and N, respectively, with each component as 1. From (9), we have
n = —niAbvAvv—1. (12)
Using (12) in (10) and (11), we get
ni (i — AbvAvv—1 AvbAbb—^ = 0, (13)
ni (ei — AbvAvv—1 e„) = 1. (14)
The matrices Abv and Avb can be written as
Abv = ( O1 O1 ) , Avb = Q ( I°1 ) ,
V O2 °3 / Nx(N+1) V iNxN ) (N+1)xN
where O1, O2 and O3 are zero matrices of order 1 x N, (N — 1) x 1 and (N — 1) x N, respectively. Let Avv—1 = [«i,j] (n+1) x (N+1) and w denote the first row of Avv—1, i.e., w = (an, «12,..., «1,n+1), then
AbvAvv—1 =( VO*) , (15)
V 04 / Nx (N+1)
where 04 is a zero matrix of order (N — 1) x (N + 1). Now,
AvbAbb—1 = Q ( AO'—1 ) . (16)
From (15) and (16), we have
AbvAvv—1 AvbAbb—1 = V1Q ( W°0b3b 1 ) , (17)
where W0 = («12,«13,...,«1,n+1).
Let us partition n1 as ^1,1, n1 where n1 = [ni,i,2 < i < N]1x(N—1). From (13) and (17), we have
^1,1, ni = ^1,1, ni
V1Qw0Abb
O3
1.
Hence, the system length probabilities of regular service period are given by
ni,i = ni,ivi <£woAbb-1€i,1 < i < N,
where ei is a column vector whose ith component is unity and the remaining components are zero. From (12) and (15), the system length probabilities of server being in WV are given by
( v\ w \
[^0,0/ n0,l/ • • • / n0,N] = -[^1,1/ ni] ( o )
Hence,
no,i = -ni/iviwe,+i,0 < i < N. Using the normalization condition E/=0 Ei=y = 1, the only unknown n1/1 is obtained as
NN
i N N \ -1
v1 ( ^ WoAbb-1€i - v1 £ W£i+1
V i=1 i=0 J
n1,1
i=0
This completes the evaluation of steady-state probabilities.
4. Performance measures
Once the steady-state probabilities are determined, several model performance measures may be calculated. The average number of customers in the system (ls), the probability that the server is busy with regular service (pb) and the probability that the server is in WV (pwv) are given by
N N N
ls = ^i (no,i + n1,i) ; pb = ^ n1,i; pwv = ^ no,i. i=1 i=1 i=0
The average reverse balking rate (br), the average reverse reneging rate (rr) and the average rate of loosing a customer due to impatience (lr) are obtained as
N N N
br = ^ A(1 - bi)no,i + ^ A(1 - bi)nhi; rr = ^ a(N - i + 1) (no,i + nhi) ; lr = br + rr. i=0 i=1 i=1
4.1. Cost model
The total expected cost function per unit time is formulated in this subsection with service rates as the decision variables. Our goal is to figure out the best service rates that minimize the total expected cost function. The cost parameters are assumed to be:
• Cls — holding cost per unit time,
• Clr - cost incurred when a customer is lost due to impatience,
• C^ — cost per service during regular service period,
• Cn — cost per service during WV period,
• Cfp — cost per service for a feedback customer during regular service period,
• Cfn — cost per service for a feedback customer during WV period. The total expected cost (tec) is defined as:
tec = Clsls + Clrlr + + q1 CfV) + n(C + q1Cfn).
Analytical optimization of the aforementioned cost model is a tedious job due to the complexity of the cost function. As a result, we have used the ACO developed by Colorni et al. [5] and Dorigo et al. [6] to find the best values for ^ and q. A brief algorithm of ACO is given below: Algorithm for ACO
Step 1: Consider a suitable number of ants in the colony (B). Assume a set of permissible discrete values for each of the n design variables Xij as xi1, Xi2,..., Xip (i = 1,2,..., n). Assume
equal amounts of pheromone t^ initially along all the arcs. The superscript to Tij denotes the
iteration number. For simplicity, t^ is assumed to be 1. Set the iteration number l = 1.
Step 2: (a) Compute the probabilities (pij) of selecting the arc Xij as
Tj Dj)
P(l) =-ij—ij-• i = 12 n- j = 12 P
ij yp h(i)d(f>)]' i ..... j .....p,
ym=1[ im im ]
where Tij is a pheromone amount between arc i and arc j, Dj is a reciprocal of the distance between arc i and arc j, ft is the parameter that allow a user control on the relative importance of trail versus visibility.
(b) The specific path (or discrete values) chosen by the kth ant can be determined using random numbers generated in the range (0, 1). For this, we find the cumulative probability ranges associated with different paths based on the probabilities given by above equation. The specific path chosen by ant k will be determined using the roulette-wheel selection process in step 3(a). Step 3: (a) Generate B random numbers T\, r2,..., rB in the range (0, 1), one for each ant. Determine the discrete value or path assumed by ant k for variable i as the one for which the cumulative probability range (found in step 2 (b)) includes the value ri.
(b) Repeat step 3 (a) for all design variables i = 1,2,..., n.
(c) Evaluate the objective function values corresponding to the complete paths (design vectors X(k) or values of Xij chosen for all design variables i = 1,2,..., n by ant k, k = 1,2,..., B):
fk = f (X(k)); k = 1,2.....B.
Determine the best and worst paths among the B paths chosen by different ants as follows:
fbest = min n ^ fworst = max fh
Step 4: Test for the convergence of the process. The process is assumed to have converged if all the B ants take the same best path. If convergence is not achieved, assume that all the ants return home and start again in search of food. Set the new iteration number as l = l + 1, and update the pheromone on different arcs as
Tij = Tij
(old) v—> . (k)
+ yA Tf,
where T(old denotes the pheromone amount of the previous iteration left after evaporation, A j is the amount of pheromone deposited on arc i and arc j by the best ant k and are taken as
(old)
ja> = (1 - P) Tj,
A (k) = I ; if (i, j) e global best tour,
ATij — ^ Jworst
0; otherwise,
where p e (0, 1] is the evaporation rate (also known as the pheromone decay factor) and £ is the parameter used to control the scale of the global updating of the pheromone. With the new values of Tij, go to step 2. Steps 2, 3, and 4 are repeated until the process converges. In some cases, the iterative process may be stopped after completing a prespecified maximum number of iterations (l maX)■
The complexity of the algorithm is O(lQ2B) where l is the number of iterations, q is the number of nodes and B is the number of ants.
k
Table 1: Various performance measures of the model for different A and qi
A= 1.0 A= 1.7 A= 2.4
q1 = 0.6 q1 = 0.2 q1 = 0.6 q1 = 0.2 q1 = 0.6 q1 = 0.2
ls 0.038536 0.027681 0.092328 0.052341 0.407215 0.094482
Pb 0.002170 0.001060 0.006109 0.002123 0.036461 0.004367
pwv 0.997830 0.998930 0.993891 0.997876 0.963538 0.995632
br 0.947901 0.948532 1.606110 1.610020 2.237570 2.266750
rr 0.034617 0.025827 0.061203 0.044884 0.092589 0.065255
lr 0.982518 0.974359 1.667310 1.654901 2.330160 2.332010
Table 2: Effect of a on the performance measures
a = 0.5 a = 1.0 a = 1.5
ls 0.015341 0.008061 0.005467
Pb 0.000244 0.000071 0.000033
pwv 0.999756 0.999929 0.999966
br 1.613650 1.614230 1.614530
rr 0.073773 0.079008 0.080914
lr 1.687420 1.693310 1.695440
5. Sensitivity analysis
In this section, tables and graphs have been used to display certain numerical results. We fix the capacity of the system as N = 10 and the balking function is taken as bi = i/N, 1 < i < N — 1, bN = 0. The various parameters of the model are chosen to be A = 1.7, ^ = 2.0, n = 1.2, $ = 0.1, q = 0.05, a = 0.1, q1 = 0.3, unless they are considered as variables or their values are mentioned in the respective tables and figures. For employing the ACO, we have arbitrarily chosen the following: n = 2, B = 3, q = 40, l = 100, fi = 0.5, £ = 2, p = 0.5 and the distances between the arcs are obtained using the RandomReal function of Mathematica software.
Table 1 shows the model's performance metrics for various values of A and q\. All the performance measures, with the exception of pwv and br, drop as qi lowers, whereas pwv and br rise as q1 decreases for fixed A. Further, increase in A results in a drop in pwv, whereas increase in A results in the increase of the remaining performance metrics.
Table 2 shows the influence of a on the model's performance measures. With the rise of a, a rising trend can be noticed in pwv, br, rr and Ir while a declining trend can be found in ls and pb.
Figure 1 shows the influence of ^ on the server's state probabilities for various values of the vacation parameter ($). The picture illustrates that when ^ grows, the probability of the server being busy with regular service (pb) decreases while the probability of the server being on vacation (pwv) increases. Furthermore, as the vacation parameter ($) is increased, pb grows while pwv decreases for any
The impact of A on the average number of customers in the system (ls) in models with and without reverse balking and reverse reneging is shown in Figure 2. From the graph, one may observer that in either of the models ls increases with the increase of the arrival rate A. Further, the queue lengths are lower in models with reverse balking and reverse reneging when compared to models without reverse balking and reverse reneging.
Figure 3 displays the effect of service rates ^ and n on the average rate of loosing a customer (lr). With the increase of both ц and n, the average rate of loosing a customer decreases. We can carefully setup the service rates ц and n in the system in order to ensure the minimum average rate of loosing a customer due to impatience.
Figure 1: y versus pwv and pb
10 9 8 7 6
_W 5 4 3 2 1
with reverse balking and reverse reneging • -*- ■ without reverse balking and reverse reneging
/
/
0.5 1 1.5 2 2.5 3 3.5 4 4.5
Figure 2: Impact of A on ls
X
1.64 0
Figure 3: Effect of y and n on lr
3 1
n
Table 3: Optimum service rates and the corresponding minimum tec.
$ 0.06 0.08 0.1
Case 1 (v*, n* ) (0.606627, 0.393620) (0.648424, 0.295251) (0.679539, 0.213562)
tec* 65.9207 65.3714 64.7454
Case 2 (v*, n* ) (0.613050, 0.539007) (0.656637, 0.446063) (0.689743, 0.366756)
tec* 70.8644 70.5699 70.1348
Case 3 (v*, n* ) (0.608498, 0.390333) (0.650509, 0.291586) (0.681784, 0.209613)
, tec* 57.5689 57.0077 56.3712
Case 4 (v*, n* ) (0.557353, 0.457533) (0.597739, 0.360028) (0.628019, 0.278463)
tec* 68.8265 68.4826 68.0101
Case 5 (v*, n* ) (0.695438, 0.199822) (0.737154, 0.101221) (0.768531, 0.018769)
tec* 67.9667 66.7260 65.5257
Case 6 (v*, n* ) (0.590782, 0.413392) (0.632152, 0.315211) (0.662999, 0.233546)
tec* 66.8187 66.3317 65.7522
Case 7 (v*, n* ) (0.622554, 0.356552) (0.664430, 0.257807) (0.695557, 0.175969)
tec* 66.3707 65.703 64.9789
Table 4: Optimum service rates and the corresponding model characteristics for various model parameters
(v*, n* ) tec* is Pb pwv lr
1.5 (0.560865, 0.053717) 53.0653 0.291992 0.027470 0.972530 1.486520
A = 2.0 (0.856029, 0.449138) 82.0273 0.355744 0.032820 0.967180 1.955930
2.5 (1.145080, 0.832582) 110.272 0.415343 0.037775 0.962225 2.421721
0.1 (0.533546, 0.289963) 55.1871 0.231776 0.021178 0.978822 1.672162
q1 = 0.2 (0.597385, 0.258192) 59.5429 0.269601 0.024895 0.975105 1.673490
0.3 (0.679539, 0.213562) 64.7454 0.318140 0.029674 0.970325 1.674791
0.04 (1.021179, 0.862099) 91.8238 0.332031 0.034453 0.965547 1.625062
a = 0.08 (0.782366, 0.436608) 73.5377 0.320756 0.030862 0.969137 1.659655
0.12 (0.572881, 0.022225) 81.3452 0.304472 0.027703 0.972297 1.687794
Table 3 presents the optimum values of the service rates (v*,n*) that minimize the total expected cost (tec) for different values of $ and for the following cost values:
Case 1: C/s Case 2: C/s Case 3: C/s Case 4: Cis Case 5: C/s Case 6: Cis Case 7: Cis
40, Cir 60, Clr 40, Cir 40, Cir 40, Cir 40, Cir 40, Cir
25, Cn = 20, f = 22,, Cfn = 18 25, Cn = 20, CfF = 22,, Cfn = 18 25, Cn = 20, CfF = 30, Cn = 20, CfF 25, Cn = 27, CfF 25, Cn = 20, CfF
25, Cn = 20, CfF = 22,, Cfn = 22. One may observe from the table that for any set of cost values with the increase of increases while tec* and n* decrease.
The values of the service rates that minimize the total expected cost are presented in Table 4 along with the corresponding performance metrics for A, q1, a and the cost values in Case 1. It is clear from the table that an increase in A or q1 results in the increase of ^*, tec*, ls, pb and lr while pwv decreases with A or q1. One may note that n* increases with A and decreases with q1. On the otherhand increase in a leads to the decrease of all the values except pwv and lr.
15, CF 15, CF 10, CF 15, Cv 15, CF 15, CF 15, Cv
22,, Cfn
22,, Cfn
22,, Cfn
22,, Cfn
22,, Cfn
27,, Cfn
22,, Cfn
6. Conclusions
We investigated a Markovain feedback queue with reverse balking, reverse reneging, and working vacations in this study. Using the matrix technique, we have obtained the steady-state probabilities. Different performance measures, cost analysis using ACO and numerical findings in the form of tables and graphs are sketched out to show the influence of the system parameters. The provided approach has the potential to be utilized in a variety of investment business areas, including insurance, mutual funds, banking and so on. The current model may be expanded to a renewal input feedback queue with working vacations under reverse balking and reverse reneging in future.
References
[1] Altman, E. and Yechiali, U. (2006). Analysis of customers' impatience in queues with server vacation. Queueing Systems, 52(4):261-279.
[2] Baba, Y. (2005). Analysis of a GI/M/1 queue with multiple working vacations. Operations Research Letters, 33(2):201-209.
[3] Bouchentouf, A. A. and Messabihi, A. (2017) . Heterogeneous two-server queueing system with reverse balking and reneging. Opsearch, 55(2):251-267.
[4] Chandrasekaran, V. M., Indhira, K., Saravanarajan, M. C. and Rajadurai, P. (2016). A survey on working vacation queueing models. International Journal of Pure and Applied Mathematics, 106(6):33-41.
[5] Colorni, A., Dorigo, M. and Maniezzo, V. (1991). Distributed optimization by ant colonies. Proceedings of ECAL'91 European Conference on Artificial Life, Elsevier Publishing, 134-142.
[6] Dorigo, M., Maniezzo, V. and Colorni, A. (1996). The ant system: Optimization by a colony of cooperating agents, IEEE Transactions on Systems, Man, and Cybernetics-Part B, 26:29-41.
[7] Jain, N. K., Kumar, R. and Som, B. K. (2014). An M/M/1/N queuing system with reverse balking. American Journal of Operational Research, 4(2):17-20.
[8] Krishnamoorthy, A. and Sreenivasan, C. (2012). An M/M/2 queueing system with heterogeneous servers including one with working vacation. International Journal of Stochastic Analysis, Article ID 145867, 16 pages.
[9] Kumar, R. and Sharma, S. K. (2013). M/M/1 feedback queueing models with retention of reneged customers and balking, American Journal of Operational Reserach, 3(A):1-6.
[10] Kumar, R. and Som, B. K. (2014). An M/M/1/N queuing system with reverse balking and reverse reneging. Advanced Modeling and Optimization, 16(2):339-353.
[11] Kumar, R., Som, B. K. and Jain, S. (2015). An M/M/1/N feedback queuing system with reverse balking. Journal of Reliability and Statistical Studies, 8(1):31-38.
[12] Santhakumaran, A. and Thangaraj, V. (2000). A single server queue with impatient and feedback customers. International Journal of Information and Management Sciences, 11(3):57-70.
[13] Servi, L. D. and Finn, S. G. (2002). M/M/1 queue with working vacations (M/M/1/WV). Performance Evaluation, 50:41-52.
[14] Takacs, L. (1963). A single server queue with feedback, The Bell System Technical Journal, 42(2):134-149.
[15] Vijaya Laxmi, P., Goswami, V. and Jyothsna, K. (2013). Analysis of fnite buffer Markovian queue with balking, reneging and working vacations. International Journal of Strategic Decision Sciences, 4(1):1-24.
[16] Wu, D. and Takagi, H. (2006). M/G/1 queue with multiple working vacations, Performance Evaluation, 63(7):654-681.
[17] Yue, D., Yue, W. and Xu, G. (2012). Analysis of customer's impatience in an M/M/1 queue with working vacations, Journal of Industrial and Management Optimization, 8(4):895-908.
