Abstract
A working vacation queueing model is described in this work, with three different classes of customers: regular, priority, and disaster. The regular server serves all arriving customers, whereas the optional reservice is only provided to those who request it. The Bernoulli working vacation (BWV) schedule is considered. In WV time, the server serves at a slower rate. The generating functions (GF) technique is used to determine the system capacity of various server states. Different system performances, reliability indices, and cost optimization values are numerically shown. For the current COVID19 pandemic situation, the motivation for this approach is presented in a telephonic communication system.
1. Introduction
Queueing theory is an area of mathematics that investigates and models the act of waiting in lines. Many researchers have studied the queueing models using vacations and different patterns of arrivals. Wireless communication networks, telephone systems, file transfer services, mail systems, and other systems, all use these types of queues. Researchers have recently proposed an alternate service mechanism with various service rates during vacation time. If no customers are in orbit, the server begins a working vacation (WV) mode. During the WV period, the server serves at a slower rate. If any customer is available in orbit at the end of the WV, the server will change the inferior service to the typical busy period. This research developed a model for the Bernoulli schedule (M/G/1/Gq/BWV) with working vacations. In the BWV schedule, the server either waits for a new customer (called a single WV) or moves on to the next WV (called multiple WV).
1.1. Related Literature Review
During the last two decades, the topic of retrial queues in queueing theory has been a fascinating research area. Retrial queues differ from common queueing mechanisms. When the server is busy, customers wait in a virtual pool called “orbit” and retry for service until it is received. For more information about the retrial queues, refer Artalejo and Corral [1] and Artalejo [2].
In the last few decades, many researchers have widely studied two different types of customers in retrial queues. The highpriority customers are formed into a waiting line or not and served according to the discipline of preemptive or nonpreemptive. Furthermore, in some models, an arriving higherpriority customer may push out the lowerpriority customers whose service is currently in progress. Priority retrial queues are used in many reallife applications like realtime systems, operating systems, and manufacturing systems. In 2015, Gao [3] developed a model retrial queueing system with two classes of customers: preemptive resumes and general retrial times, where priority customers do not queue and have an exclusive preemptive priority to receive their services over ordinary customers. Recently, many researchers developed models in priority disciplines like Wu and Lian [4] and Rajadurai [5].
In 1975, Levy and Yechialli [6] introduced the vacation queueing system. The researchers can read ([7–9]) to learn more about the vacation queueing approach. Chandrasekaran et al. [10] have provided a short survey on WV queueing models. In 2002, an M/M/1 queueing system with working vacations was first introduced by Servi and Finn [11]. Later, Wu and Takagi [12] extended the M/M/1/WV queue to an M/G/1/WV queue. Arivudainambi et al. [13] introduced the M/G/1 retrial queue with a single working vacation. Furthermore, during the working vacation period, if there are customers at a lower service completion instant, the server can stop the vacation and return to the usual busy state. This policy is called “vacation interruption”. In 2014, Gao et al. [14] first developed a model of a continuoustime general retrial queue with working vacations and vacation interruption by using the supplementary variable technique. Bouchentouf et al. [15] have discussed a single server Markovian queueing model with N and D policy working vacations and vacation interruption. Seenivasan and Abinaya [16] have recently studied a Markovian queueing model with a single working vacation and catastrophic. Some researchers have discussed models in the presence of WVs, such as Gao and Liu [17], Rajadurai et al. [18], Kasim [19], and Bouchentouf [20].
Gelenbe [21] was the first to investigate disasters, also known as negative arrivals or GQueues, where it is applicable in neural networks. Do [22] has presented a survey on queueing systems with Gnetworks, negative customers, and applications. Choudhury and Deka [23], Yang et al. [24], Rajadurai et al. [25], and Ammar and Rajadurai [26] have recently looked at various queues in unreliable queueing systems. During the service time, a negative arrival occurs and the system will get breakdown and send for repair.
1.2. Motivation of the Model
The advantages and contributions of this work are as follows:
Model: A novel continuoustime M/G/1 retrial queue with three separate classes of customers and optional reservice during working vacations in the Bernoulli schedule (M/G/1/Gq/BWV) is presented. In the presence of retrial queues, the primary innovation of this work is a generalization of both single (q = 0) and multiple () working vacation models. The mathematical conclusions of this model have a specific and exciting application in the telecommunication system and telephone consultation of medical service systems. They also apply to wireless sensor networks (WSNs) [27].
Methodology and results: We adopt all the elapsed times for retrial, service of priority and ordinary customers, working vacation, and repair as the supplementary variables to create a Markovian process of the system and use the generating function approach to obtain the performance measures of the system.
Numerical illustrations: The impacts of parameters in system performance measures and cost optimization are displayed numerically.
2. Nomenclature and Model Description
2.1. Nomenclature
(i) Customers present in the orbit at time t(ii) Server’s state at time t(iii)λ _{1} = Regular arrival rate(iv)λ _{2} = Priority arrival rate(v)α = Disaster rate(vi)θ = Vacation rate(vii)r = Reservice probability(viii) Elapsed retrial time(ix) Priority customers’ elapsed service time(x) Preemptive priority customers’ elapsed service time(xi) Regular customers’ elapsed service time(xii) Regular customers’ elapsed reservice time(xiii) Elapsed working vacation time(xiv) Elapsed repair time(xv) All of the states’ hazard rates. (i = 1,2,3,4,5,6, 7)(xvi)“(Laplace Stieltjes Transform of F (x))”(xvii)“.”(xviii)LST of the retrial state = .(xix)LST of the priority busy states = .(xx)LST of the regular busy states = .(xxi)LST of the WV state = .(xxii)LST of the repair state = .(xxiii)First and second moments for the corresponding states = for state i.(xxiv)T _{b} = Expected busy period(xxv)T _{c} = Expected busy cycle(xxvi)T _{0} = Expected length of empty system
2.2. Description of a Mathematical Model
We addressed an M/G/1 retrial queue with three different classes of customers and optional reservice during WVs under the Bernoulli schedule (M/G/1/Gq/BWV) in Figure 1. This model’s fundamental assumption is as follows:(i)The arrival process: Assume that three types of customers arrive from outside the system using independent Poisson processes: regular, priority, and disaster. Preemptive priority customers have dominated the regular busy server during the service time of regular customers.(ii)The retrial process: The general retrial policy was proposed in Reference [5]. Regular customers who arrive find the server accessible and get its service immediately; otherwise, they are joined into the orbit and try its request later with FCFS discipline.(iii)Priority service process: When priority customers arrive, they find the server accessible and immediately begin its service. Assume that if the server provides service to another priority customer, the newly arriving priority customer will exit the system without receiving service.(iv)The regular service process: The regular busy server serves an ordinary customer; if priority customers arrive, then the regular service will be disrupted. A priority customer takes precedence over an ordinary customer and the server begins its service immediately. The typical customer must wait in the service area for the remainder of the service to be completed.(v)The reservice process: We assume that a single server provides an optional reservice. If the service is successful, the ordinary customer may rejoin the system immediately to receive the same service (optional reservice) or leave the system.(vi)The removal rule: Negative (failure) customers arrive only during regular service hours. This kind of arrival will push regular customers (priority or ordinary) out of the service area and cause the server to fail. When a server fails, it is immediately sent to be repaired. When the repair process is finished, the server will function as if it were brand new.(vii)The Bernoulli working vacation process: When no customers are in orbit, the server initiates a working vacation. During the WV period, the server serves at a slower service rate. If any customer is available in orbit at the end of the WV, the server will switch from the lesser service to the normal busy period. Otherwise, the server either waits with probability p to serve a new customer (a single working vacation) or takes another working vacation with probability q = 1 p (a multiple working vacations). If customers are in orbit at the end of the vacation, the server returns to its normal working state.(viii)The times of all server states are mutually independent.
2.3. A Practical Example of the Model
We consider a telecommunication system one of the best examples of our model. Telecommunication, banks, call centers, and mobile network customers play a significant role in this COVID19 pandemic. All the network operators work only on emergency cases with a limited number of workers. People can contact the center by communicating with a person at the help desk or customer service consultant (CSC) through the telephone (regular server). Customers can contact the center via fax, email, or live chat sessions in addition to calling (priority customers). The CSC or agent picks them both up. If the agent is engaged at the time of the call, the arriving call will lose service. However, if the agent is concerned about emails or faxes, the call takes precedence over an email service. The preempted service will remain in place to complete its mission. If customers are dissatisfied with the service, they will be reserviced as soon as possible (reservice). The working server will be impacted by a weak signal (negative customers), causing the system to fail during the service period and need to be repaired immediately. If CSC gets no voice calls, faxes, or emails, it will execute some maintenance jobs to enhance the computer performance (working vacation). If an arriving message is discovered while the CSC is engaged in a voice call, the messages are temporarily stored in a retrial buffer (orbit) and served after some time (retrial time). This queueing discipline is a good approximation of this type of a telecommunication processing system.
3. SteadyState Analysis
For the model M/G/1/Gq/BWV, the Markov process is defined as . The various server states of {X (t) = 0,1,2,3,4,5,6, and 7} are free, busy (with priority, preemptive priority, and normal customers), reservice, working vacation, and repair.
The probabilities for the process are defined as and. ,
3.1. Ergodicity Condition of the Model
We analyze the ergodicity of the embedded Markov chain at departure or vacation epochs. Let {t_{n}; n = 1,2,…} be the sequence of epochs at which either a normal service for priority/ordinary or a lower service completion occurs or a repair period completion. The sequence of random vectors forms an embedded Markov chain in the retrial queueing system. Its state space is.
Theorem 1. The embedded Markov chain is ergodic if and only if , “where and .
Proof. To prove the sufficient condition of ergodicity, it is very convenient to use Foster’s criterion (see Pakes [28]), which states that the chain is an irreducible and aperiodic Markov chain is ergodic if there exists a nonnegative function f (j), j ∈ N and ε > 0, such that mean drift is finite for all j ∈ N and for all j ∈ N, except perhaps for a finite number j’s. In our case, we consider the function f (j) = j; then, we have the following equation:Inequality is a sufficient condition for ergodicity.
To prove the necessary condition, as noted in the study of Sennott et al. [29], if the Markov chain satisfies Kaplan’s condition, namely, ψ_{j} < ∞ for all j ≥ 0, and there exits j_{0} ∈ N such that ψ_{j} ≥ 0 for j ≥ j_{0}. Notice that, in our case, Kaplan’s condition is satisfied because there is a k such that m_{ij} = 0 for j < i  k and i > 0, where Μ = (m_{ij}) is the onestep transition matrix of Then, implies the nonergodicity of the Markov chain.
3.2. SteadyState Probability Equations
The method of “supplementary variable technique” (SVT) is used to generate the following different state equations,
The boundary conditions are at x = 0 and y = 0.
The system normalized condition is as follows:
3.3. Probability Solutions of the Model
We define the GFs for all of the states to solve the given system of equations.
Multiplying by z^{n} and summing over n to get GFs for equations of (5)–(19), making calculations,where
Using the equations (12)(19) in (21)(27), we obtain as follows:
Using the equations (3)–(19) to (21)–(27) to make some manipulations, we get the partial GFs for all the states.
Lemma 1. The following are the PGFs for several states:where where , Using the normalizing condition , we obtain P_{0} and Q_{0}, where P_{0} is the probability that the server is idle and Q_{0} is the probability that the server is empty in WVs.
Theorem 2. Under the stability condition , PGF of the number of customers in the system and orbit size distribution at a stationary point of time is as follows:where P_{0} and Q_{0} are as shown in equations (37) and (38).
Proof. The probability generating function of the number of customers in the system K_{s} (z) and the probability generating function of the number of customers in the orbit K_{o} (z) are obtained by using the following equations:Substituting (29)–(38) in the above results, the equations (39) and (40) can be obtained by direct calculation.
4. Performance Indices
Some significant system characteristics of the priority M/G/1/Gq/BWV are described in this section.
4.1. Probabilities of Different States
Using LHospital’s rule and limit functions in (29)(35); we obtain the following probabilities for various states,(i)(ii)(iii)(iv)(v)(vi)(vii)
4.2. Computation of L_{s} and L_{q}
We can derive the mean number of units in the system (L_{s}) and Orbit (L_{q}) by differentiating the system size (39) and orbit size (40) and applying the limiting case,
and.
The system’s mean waiting time (W_{s}) and orbit (W_{q}) are calculated using Little’s formula.
and.
4.3. Busy Cycle and Busy Period
The mean busy time and busy cycle are as follows, based on the idea of an alternate renewal process:
From the above results, we obtain as follows.
and
4.4. Reliability Indices
The following methods are used to calculate system availability measures () and failure frequency (F_{f}).
5. Special Cases
Some important special cases of our model are as follows: Case (i): If (λ_{2}, α, θ, r) ⟶ (0, 0, 0, 0), then the model reduces to a single WV retrial queue, and the findings match those of Arivudainambi et al. [13]. Case (ii): If λ_{2} = r = α = 0, the model approaches a retrial queue system with WVs and agrees with Gao et al. [14] findings. Case (iii): If (α, r, θ) ⟶ (0, 0, 0), then the model can be reduced to a priority retrial queue with WVs and compared to Gao’s [3] results.
6. Reliability and Sensitivity Analysis
Using the MATLAB software, the effect of system parameters on different performance measures is presented numerically and graphically.
6.1. Cost Analysis
The impact of the cost function on the system parameters is discussed. The optimal design of the priority retrial on the M/G/1/Gq/BWV is addressed. Here, the cost structure and notation are followed by wellknown results (Tadj and Choudary [30], Yang and Wu [31]), and the arbitrary values are chosen for the parameters such that the steadystate condition is satisfied.
Let the assumes of cost analysis be as follows: H_{c} = the holding costs O_{c} = the cost for the server is in operations S_{c} = the setup cost per busy cycle C_{a} = the startup cost of the server before starting the service.
From the above list, under the linear cost procedure, the expected cost function is given as follows:
Consider the following values for the different parameters like: λ_{1} = λ_{2} = ν_{1} = ν_{5} = 2; ν_{2} = 5; ν3 = 3; ν_{4} = θ = 3; α = p = r = 0.5; H_{c} = 5, O_{c} = 60, S_{c} = 550 and C_{a} = 90. From the parameter values in estimated total cost per unit of time, we obtain TC = 355.94. The server’s steadystate availability is = 98.61% and failure frequency is F_{f} = 2.08%.
The impact of system parameters on cost functions is presented numerically in Tables 1–3 and Figures 2–5. Based on the findings, we conclude that the expected values of TC increase linearly with increasing cost parameters. The procedure for finding the Lq, Ff, Av, and TC are presented in Algorithm 1.

6.2. Sensitivity and Reliability Analysis
For numerical illustration, all the distribution functions are assumed to be exponentially, Erlang 2stage, and hyperexponentially distributed. The parameter values are chosen arbitrarily to satisfy the stability condition. The impact of system performance measurements on the retry, priority, and failure rate parameters is shown in Tables 4–6. We find that (i) as the retrial rate (ν_{1}) rises, L_{s}, L_{q,} and H_{1} decrease; (ii) P_{0}, H_{4}, H_{6}, and F_{f} rise as the retrial rate (a) rises in Table 4. As seen in Table 5, increasing the priority arrival rate (λ_{2}) decreases the value of P_{0} while raising the other metrics L_{q}, H_{2}, H_{5}, and H_{6}. As shown in Table 6, the failure rate (α) has an increasing tendency for P_{0}, L_{s}, L_{q}H_{6}, and F_{f}, while the failure rate (α) has a decreasing trend for other measures such as H_{1} and H_{4}.
The performances such as the mean orbit size (L_{q}) and idle probability (P_{0}) are presented in increasing and decreasing trends against the parameters λ_{1}, α, ν_{1}, and θ in Figures 6–13. Figures 14–17 show the effect of different parameters on the system performance graphically. Figure 14 shows a rising trend in the value of the arrival rate of regular (λ_{1}) and priority (λ_{2}) arrivals concerning L_{q}. We used the nature of F_{f} which increases in Figure 15 to enhance the value of the failure rate (α) and vacation rate (θ). Figure 16 shows that F_{f} decreases as the failure rate (α) and priority rate (λ_{2}) increase. Figure 17 shows the increased trends in idle probability (P_{0}) for increasing the value of reservice probability (r) and vacation rate (θ).
7. Conclusion
This work addresses a new model in priority retrial queues with the presence of reservice and Bernoulli WVs (priority retrial policy of M/G/1/Gq/BWV). The system size and orbit of PGFs are determined using the SVT and PGF approaches. The orbit and system average queue lengths have been determined. Various useful queueing and reliability measures, sensitivity analysis, system busy period, and mean waiting times are obtained. The analytical results are validated numerically and graphically using MATLAB software.
The present investigation includes features simultaneously, such as the following:(i)Retrial queues(ii)Preemptive priority customers(iii)Negative customers (Gqueue)(iv)Working vacations(v)Vacation interruption(vi)Reservice(vii)Bernoulli schedule(viii)Breakdowns and repairs
This model is used in wireless sensor networks (WSNs) in the PRIN MAC protocol, telecommunication systems, and phone counseling of medical service systems.
This work can be further extended in many directions by incorporating the following concepts:(i)Batch arrival(ii)Optional type service(iii)Working breakdowns(iv)Immediate feedback and many others
This study finds other applications in telephone systems, electronic mail services on the Internet, network, and software designs of various computer communications systems, production lines, and mail systems.
Data Availability
The data used to support the findings of this study are included in the article. The reader can directly access the data at work.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of this paper.