An M/M/3 repairable Markovian queueing system with servers on vacation has been taken into consideration. The service rates of the servers in this system differ. Each server is allowed to go on its own vacation when there are no customers in system. At the end of the vacation period, if the system is empty, then servers take another vacation; otherwise, they turn on to the busy state. Besides, a breakdown may occur in a Poisson manner in a particular state and the repair process will start immediately. The steady state probability vector of the number of customers in the system was evaluated as a quasi-birth and death (QBD) process, and the matrix geometric technique was used to derive the stationary condition. Some system performance measures are obtained. The effects of arrival rates, breakdown rates and repair rates on the mean number of customers and the influence of arrival rate on steady probabilities are shown through graphs.
Abstract
Index Terms-Heterogenous servers, Matrix geometric method, Multiple vacation, Quasi birth and death process, Server breakdown, Three servers.
I. INTRODUCTION
A server in a queueing system is said to be on vacation if it becomes unavailable to its customers for a random period of time when the system becomes empty. After completion of vacation, if the servers are found empty, again the servers go on vacation. This type of vacation is called a multiple vacation. Over the last two decades, queuing system with vacations have been made use of in industrial and computer communication systems. Doshi [1], Takagi [2], Tian and Zang [3], Ke et al. [4] and Upadhyaya [5] conducted several many surveys on vacation models.
A number of academics have looked into multi-server queues with vacations. Initially, Levy and Yechiali [6] and Vinod [7] studied the M/M/c exponential queue with server vacation. Early research did not provide precise information on the stationary distributions of variables like waiting time and queue length. The M/M/c queueing system with vacation was thoroughly described by Tian et al.[8], who also provided conditional stochastic decomposition estimates for waiting time and queue length. Numerous scholars, including Kao and Narayanan [9], Igaki [10], Chao and Zhang [11], Zhang and Tian [12], [13], and Houalef et al. [14], have evaluated multi-server queueing systems with vacations. In addition Li et al. [15] analysed about a single server retrial queue with the working vacation.
The studies on multi-server queueing systems described above presume that the servers are all homogeneous, i.e., the service rates of all the servers are equal for the whole system. The above assumption is difficult to realise in a queueing system with human servers. Only a few studies have been carried out on a multi-server queueing system with server vacation and different service rates for different servers. Madan et al. [16] carried out an analysis on a M/M/2 queue with Bernoulli schedules and a single vacation policy, where two servers bring heterogeneous exponential service to customers. They obtained steady-state probability generating functions for the system size for various server states. The two server Markovian queues with balking is discussed by Singh [17], and comparison between heterogeneous and homogeneous servers is also given. Lin and Ke [18] analysed the multi-server queueing system with single working vacation.
In recent decades, interest in queueing systems with server breakdown has increased. This is a result of their use in computer systems, telecommunications and production systems. For instance, in a facility that processes machines, equipment malfunctions could be caused by things like power outages, a lack of preventive maintenance, or the use of poor raw materials. Recently, Kalidass et al.[19], Shengli et al.[20], Shengli [21], Jing and Tao [22], Chakravarthy et al.[23], Seenivasan and Abinaya [24], Dasa et al. [25], Atencia [26], Gupur [27] and Jain and Jain [28], Tsai et al. [29] have analysed the single server queueing system with the server breakdown. Recently, Yang et al. [30] have studied a two server multiple vacation queueing system with server breakdown.
In this paper, we present a model that has several practical applications. For instance, there is a University with three servers to check the exam results for students. If there are no students checking their results, then all the three servers will be at rest. If at least one student is willing to check his results, then any one server can give the response to the student. At that point, the other two servers are at rest. If at least two students checks their results at the time, any two servers will give their response, while the third server is at rest. If at least three students to check their results at the same time, all the three servers will give their response. When all the three servers are used simultaneously, there is a possibility of system breakdown.
In section II, we describe the model and describe a quasibirth-death process. In Section III, we present the steady state solution via the matrix geometric technique. We provided some system performance measures and a numerical analysis is in sections IV and V. A conclusion has been offered.
II. The Model Description
Here we are dealing with a three-servers multiple working vacation queueing system with servers breakdown, and the service rates of the servers are considered to be different. The following is a description of the model's assumptions:
1) Customers arrive to the system in a Poisson manner with rate A. In the order of arrival, customers form a waiting line. It is assumed that both the system capacity and the total number of possible consumers are unlimited.
2) The three servers provides heterogeneous exponential service to clients on a First-Come First-Service (FCFS) basis with rates мь M2 and m3 (p1 > M2 > M3).
3) When there are no customers in the system, each server is permitted to take a vacation on its own. If a customer is present in the system at the end of a vacation time, service will begin. Otherwise, the server immediately takes another vacation and continues in this fashion until the server returns from vacation and finds at least one customer waiting in the system.
4) The vacation periods of the servers follow heterogenous exponential distribution with vacation rates Oj ,j = 1, 2, 3.
5) In addition servers may breakdown and then repair process starts instantly.
6) Here both breakdown and repair process follow exponential distribution with the rates a and ß respectively.
A. The Quasi-Birth-and-Death (QBD) process
At the time t, the number of customers in the systems is consider H(t) and let I(t) be the servers state. Then
...
Then X(t) = {H(t); I(t)}, is a QBD process with a state space denoted by w as follows: w = {(0, 0)}U{(1,Z),Z = 0,1, 2, 3}U{(2,Z),Z = 0,1, 2, 3, 4, 5, 6}U{(i,Z),i >3, Z= 0, 1, 2, 3, 4, 5, 6, 7, 8}
Using lexicographical sequence for the states, Q is the infinitesimal generator of the Markov chain and is given by
...
...
and Ao = AIg
Vo = - (A + 6i + 62 + 63)
Vi = -(a + Mi + 62 + 63)
V2 = -(a + м2 + 61 + 63)
V3 = -(A + м3 + 61 + 62)
V4 = -(A + Mi + M 2)
V = -(A + M2 + M3)
V6 = -(A + Mi + M3)
V7 = Mi + M2
Vs = M2 + M3
V9 = Mi + M3
Vi0 = -(A + Mi + M2 + 63)
V11 = - (A + M2 + M3 + 6i)
Vi2 = -(A + mi + M3 + 62)
v13 = -(A + Mi + M2 + M3 + a)
Vi4 = -(A + в)
V15 = Mi + M2 + M3
III. Steady-state Analysis
In this section we find the condition for the system to reach a steady state. The system's steady-state probability vectors are calculated using a method of matrix-geometric. Also, the rate matrix and the boundary probability vectors are found. Finally, using the steady-state probability vectors, we give several system performance measures.
A. Stationary Condition
To obtain the stationary condition, we first define the matrix A = Ao + A i + A2. Then the matrix A can be written
...
X3 = -(6i + 63)
X4 = -(6i + 62)
It is obvious that A is an irreducible Markov process generator.
Let n = (no,ni, n2,n3,n4, n5, n6,n7,ng) be a stationary probability vector of this Markov process. Then, n satisfies the linear equations:
nA = 0, ne =1
Following Neuts [31], the system is stable if and only if
nAoe < nA2e.
That is the system is stable if and only if p < 1 where
A(a + в)
^ (Mi + M2 + M3)e
B. Matrix geometric Solution
Let H and I be the stationary random variables for the number of customers in the system and the states of servers. We denote the stationary probability by Pn,i = lim P{H(t) = i,I(t) = l}, where (i,1) G w Under the stationary condition p < 1, the stationary probability vector P of the generator Q exists. This stationary probability vector P is partitioned as P = (p0,pi,p2,.)
where po = Poo
Pi = (Pio,Pii ,Pi2,Pi3)
P2 = (Pio,Pii ,Pi2,Pi3,Pi4,Pi5,Pi6)
Pi = (Pio,Pii,Pi2,Pi3,Pi4,Pi4,Pi5,Pi6,Pi7,Pi8) for i > 3. The stationary probability vector P is generated by using Netus [31] matrix-geometric solution approach, which is given by
PoBoo + PiBio =0 (1)
PoBoi + PiBii + p2B2i =0 (2)
PiBi2 + P2B22 + P3B32 =0 (3)
P2B23 + P3Ai + P4A2 =0 (4)
PiAo + Pi+iAi + Pi+2A2 = 0 for i > 3
pi = P3R(i-3) for i > 3 (5)
and the normalizing condition is
Po + Piei ++P2Ē2 + P3(I - R)-ie3 = 1 (6)
where ei , e2 , and e3 are column vectors with elements that are all one and in the proper order, and R is called the Rate matrix, which has the minimal non-negative solution of the matrix quadratic equation as follows:
B2A2 + RAi + Ao = 0 (7)
Theorem 3.1: If p < 1 then (7) has the minimal nonnegative solution R = -[Ao + ñ2A2]A-i.
Proof: Since A is reducible. The analysis present in Netus [31] is not applicable. According to Lucantoni [32], similar reducible matrix is treated for the case when the elements are probabilities. Equation (7), can be written as, AoA-i + RAiA-i + R2A2A-i = 0A-i Since Ai is non-singular, a-1 exists. Therefore
R = - [Ao + R2A2]A- i (8)
Using Netus and Lucantoni [33] the matrix R is numerically computed by using the recurrence relation with R(0) = 0 in equation (8).
C. Boundary Probability Vectors
We should solve equations (1)-(4) and (6) to obtain the stationary boundary probability vectors po, pi, p2 and p3. Additionally, we can define matrices as follows:
Di = Bii - BiOBo0B01, D2 = B22 - B2iD- 1Bi2
D3 = B32D2 iB2iDl i
It is easy to show that matrices Di, D2, D3 are invertible. The following theorem gives the boundary probability vectors po, pi, p2 and p3.
Theorem 3.2: The boundary probability vectors are given by
po = -psDsBioB-O
pi = p3 D3
p2 = -p3B32D-i and the following equations determine
Рз(-B32D-1 B23 + Ai + RA2) = 0 Рз(-D3 BioB-ß + D3ei - B22D-1e2 + (I - R)-1e 3) = 1
Proof: We know B00 is invertible and from (1)-(3) we get
po = -pi BioB1 (10)
pi = p2 B2iDli (11)
p2 = -p3 B32D-1 (12)
using equations (10)-(12) we get
po = -psDsBioB-o pi = psDs
Substituting equation (5) in equation (4) we have
p2B23 + p3 (Ai + RA2) = 0 (13) using equations (6) and (13) we get (9)
This gives the required proof. *
D. Remark
* If p3 = 63 = 0, the current model reduces to M/M/2 multiple vacation queueing system with server breakdown.
* If p2 = p3 = 02 = 63 = 0, our proposed model reduces to M/M/1 multiple vacation queueing system with server breakdown.
* If a = p3 = 63 =0 and в=1, the current model reduces to M/M/2 multiple vacation queueing system, which is studied by Kumar and Madheshwari [34].
* If a = p2 = p3 = 02 = 63 = 0, ß = 1, our proposed model reduces to M/M/1 multiple vacation queueing system.
IV. PERFORMANCE MEASURES
By using the normal calculations, performance measures are calculated as follows:
1) Mean number of customers in the system E(L) = piei +2p2€2 +p3r-2((I-R)-2 -(1+2R))e3
2) The probability that
* all the servers are on vacation (Pv) =J2 pio
* only the server 1 is busy (Pib) = J2 pii
* only the server 2 is busy (P2b) = pi2
* only the server 3 is busy (P3b) = J2 pi3
* the servers 1 and 2 are busy (P(i,2)b) = £ pi4 i=2
* the servers 2 and 3 are busy (P(2,3)b) = £ pi5 i=2
* the servers 1 and 3 are busy (P(i,3)b) = £ pi6 i=2
* all the servers are busy (Pb) = J2 pi7
* all the servers have breakdown (Pbr) = J2 pi8
V. Numerical Results
In this section, we discuss the numerical results we have obtained by the model described in this paper. To gain an understanding of the performance of this system, we study the effects of the parameters on the specific probabilistic descriptions, which means the number of customers in the system. Figures (1)-(4) show the effect of the arrival rate on steady state probabilities. Other values are as follows: pi=15, p2=12, p3=9, di=7, d2=5, d3=3, a=4, ß=6. Figure 1, tells us that if the arrival rate increases, the probability that all the servers will be on vacation state will decrease. As the arrival rate increases, the system will progress to the next state. That is, the system moves on to servers busy states. From Figure 2, we notice that if the arrival rate increases, the probability of a single server being in a busy state (any one server being busy while others are on vacation) decreases. When the arrival rate increases, the system moves on to the state where any two servers out of the three are busy or all the servers busy. From figure 3, we observe that if the arrival rate increases, the probability that any two servers busy states (any two servers are busy, another one is on vacation) increases. And also from Figure 4, we notice that if the arrival rate increases, the probability that all the servers are busy and the servers are in breakdown states increases. If the arrival rates increase, the system will most likely switch to any two or all three busy servers.
Figures (5)-(7) show the effect of the arrival rate on the mean number of customers in the system for different values of service rate. We fix the other parameters as 9i=7, 92=5, θ3=3, a=4, в=6. In figure 5, we fix pi and p2 values as 15 and 12 respectively and draw the graphs for different values (7 ,9, 11) of p3. From figure 5, we observe that if the arrival rate increases, the mean number of customers in the system also increases. Also, we notice that if the service rate of server 3 increases, the mean number of customers in the system decreases. In figure 6, we fix pi and p3 values as 15 and 9 respectively and draw the graphs for different values (12, 14, 16) of p2. From figure 6, we observe that if the arrival rate increases, the mean number of customers in the system also increases. Also, we notice that if the service rate of server 2 increases, the mean number of customers in the system decreases. In figure 7, we fix p2 and p3 values as 12 and 9 respectively and draw the graphs for different values (13, 15, 17) of pi. From figure 7, we observe that if the arrival rate increases, the mean number of customers in the system also increases. Also, we notice that if the service rate of server 1 increases, the mean number of customers in the system decreases.
Figures (8)-(10) show the effect of the breakdown rate on the mean number of customers in the system for different values of service rates. We fix the other parameters as A=13, 0i=7, 02=5, 0з=3, ß=6. In figure 8, we fix p1 and p2 values as 15 and 12 respectively and draw the graphs for different values (7, 9, 11) of р3. From figure 8, we observe that if the breakdown rate increases, the mean number of customers in the system also increases. Also, we notice that if the service rate of server 3 increases, the mean number of customers in the system decreases. In figure 9 we fix p1 and p3 values as 15 and 9 respectively and draw the graphs for different values (10, 12, 14) of р2. From figure 9 we observe that if the breakdown rate increases, the mean number of customers in the system also increases. Also, we notice that if the service rate of server 2 increases, the mean number of customers in the system decreases. In figure 10, we fix p2 and p3 values as 12 and 9 respectively and draw the graphs for different values (13, 15, 17) of р1. From figure 10, we observe that if the breakdown rate increases, the mean number of customers in the system also increases. Also, we notice that if the service rate of server 1 increases, the mean number of customers in the system decreases.
Figures (11)-(13) show the effect of the repair rate on the mean number of customers in the system for different values of service rate. We fix the other parameters as A=13, 01=7, 02 =5, 03=3, a=4. In figure 11 we fix p1 and p2 values as 15 and 12 respectively and draw the graphs for different values (7, 9, 11) of р3. From figure 11, we observe that if the repair rate increases, the mean number of customers in the system decreases. Also, we notice that if the service rate of server 3 increases, the mean number of customers in the system decreases. In figure 12 we fix p1 and p3 values as 15 and 9 respectively and draw the graphs for different values (10, 12, 14) of р2. From figure 12, we observe that if the repair rate increases, then the mean number of customers in the system decreases. Also, we notice that if the service rate of server 2 increases, the mean number of customers in the system decreases. In figure 13, we fix p2 and p3 values as 12 and 9 respectively and draw the graphs for diffent values (13,15,17) of р1. From figure 13, we observe that if the repair rate increases, the mean number of customers in the system decreases. Also, we notice that if the service rate of server 1 increases, the mean number of customers in the system decreases.
A. Comparison of Heterogeneous and Homogeneous service rates
In this section we give comparison between the heterogenous and homogeneous service rates. In tables I-III, for the case heterogenous service rates we take p1 = 0.8р, p2 = 1.1р, р3 = 1.4p and for homogenous service rates we take р1 = р2 = р3 = р also we are varying the values of р as 11, 11.5 and 12.
For table I we take the values of a = 4 and ß = 6 and varying the values of A as 13, 14, 15, 16, 17. From that table we observe that if the arrival rate increase, the mean number of customers in the system increases. For table II we take the values of A = 13 and ß = 6 and varying the values of a as 4.0, 4.1, 4.2, 4.3, 4.4. From that table we observe that if the breakdown rate increase, the mean number of customers in the system increases. For table III we take the values of A = 13 and a = 4.0 and varying the values of ß as 5.8, 5.9, 6.0, 6.1, 6.2. From that table we observe that if the repair rate increase, the mean number of customers in the system decreases. Also we can notice from tables I-III, if the service rate for the servers increases, the mean number of customers in the system decreases. In addition we observe that for any particular values, the mean number of customers in the system for heterogenous service rates is less than the mean number of customers in the homogenous service rates.
VI. Conclusion
In this work, we have analysed a three-server heterogenous Markovian queueing system with multiple vacations and server breakdown. We have provided the stationary condition and boundary probability vectors for our model. The influence of Л on steady state probabilities and mean number of customers in the system for different values of service rates is also graphed in the numerical section. And also we have tested the effects of a and ß on mean number of customers in the system for various service rates. In future we can find the explit expression of the rate matrix R and the conditional stochastic decomposition properties of the stationary queue length and the stationary waiting time for this proposed model. And also this work will be extended to heterogenous multi server multiple vacations queueing model with servers breakdown.
Acknowledgement
The authors appreciate and thank Dr. R. Padma Priya as a recognised research supervisor of English and Foreign Languages and Assistant Professor from SRM Institute of Science and Technology, Chennai, India for proofreading and English language corrections.
Manuscript received September 26, 2022; revised April 08, 2023.
V. Karthick is Research Scholar in Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur - 603 203, Tamil Nadu, INDIA e-mail: kv0994@srmist.edu.in 'Corresponding Author - V. Suvitha is Assistant Professor in Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur - 603 203, Tamil Nadu, INDIA e-mail: suvithav@srmist.edu.in
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Abstract
An M/M/3 repairable Markovian queueing system with servers on vacation has been taken into consideration. The service rates of the servers in this system differ. Each server is allowed to go on its own vacation when there are no customers in system. At the end of the vacation period, if the system is empty, then servers take another vacation; otherwise, they turn on to the busy state. Besides, a breakdown may occur in a Poisson manner in a particular state and the repair process will start immediately. The steady state probability vector of the number of customers in the system was evaluated as a quasi-birth and death (QBD) process, and the matrix geometric technique was used to derive the stationary condition. Some system performance measures are obtained. The effects of arrival rates, breakdown rates and repair rates on the mean number of customers and the influence of arrival rate on steady probabilities are shown through graphs.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer