
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12809-5
10.1016/j.heliyon.2024.e36778
e36778
Research Article
Unreliable M[X]/G(P1,P2)/1 feedback retrial queues with combined working vacation
S Bharathy bharathy81kuttys@gmail.com

M.C. Saravanarajan mcsaravanarajan@vit.ac.in
⁎
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore - 632 014, India
⁎ Corresponding author. mcsaravanarajan@vit.ac.in
28 8 2024
15 9 2024
28 8 2024
10 17 e367784 11 2023
22 8 2024
22 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
The study examines M[X]/G(P1,P2)/1 feedback retrial queues coupled with starting failure, repair, delay to repair, working vacation, and general retrial times. Also, it explores how different batch sizes affect performance and how bulk arrival affects system behavior. When the server is not in use, a single customer initiates the system while the remaining customers transition to a state of orbit. A new customer must turn on the server to provide two phases of mandatory service at any time. The server could have starting issues. If the service is successfully started (with likelihood α), the customer receives service immediately. In the absence of that, the likelihood of starting failure happens (with likelihood 1−α=α¯). The server was taken for repair with some delay and that customer was transported to an orbital location. When the server was busy or unavailable, the arriving customers queued by FCFS in the orbit. We also discuss the idea of reworking with probability p, and restarting unsuccessful service attempts to improve customer happiness and service efficiency. We also introduce the concept of working vacation, which permits servers to temporarily stop providing services, affecting system performance and availability at both peak and off-peak times. A supplementary variable technique was adopted for the system's and orbit size's probability-generating function. Various performance measurements were provided with appropriate numerical examples.

MSC

60K25
60K30
90B22
35A01
65L10
Keywords

Bulk arrival
Two phase service
Starting failure
Feedback
Markov chain
Working vacation
==== Body
pmc1 Introduction

Queueing models play an important role in modeling operating systems, network transportation, industrial fields, and communication networks, as well as in the development of applications like the internet's rapid expansion, audio and video data traffic, etc. Once the arriving customers find the server engaged or unreachable, they are gratified to depart from the service station and join the retrial group, known as orbit. Additional industrial applications for trial queues include other production and manufacturing processes, transportation and service systems, multiple access protocols, shared bus local area networks, and packet switching networks that utilize collision avoidance in star local area networks, among others. These queues have a wide range of uses in everyday congested settings. A thorough examination of retrial queues may be established by Falin and Templeton [1] in their book Retrial Queues, as well as by Artalejo and Gomez-Corral [2]. Noteworthy survey papers were done by Falin [3]. Singh et al. [4] discussed the Markovian retrial queueing model for bulk arrivals that combine multiple vacation policies. The research by Jain and Bhargava [5] looked at a M[X]/G/1 model with a retry, two types of non-preemptive priority subscribers, and a server that wasn't always there. Ayyappan and Shyamala [6] investigated an unreliable M[x]/G/1 queueing structure combining Bernoulli feedback vacations and random setup time. While the researchers were on vacation, Meena et al. [7] looked at a non-Markovian machining system with both active and passive machines working together under an N-policy. They used genetic algorithms and metaheuristic particle swarm optimization to find the best parameters at the lowest cost. Malik et al. [8] introduced the G-queue for bulk arrival retrials, which exposed the server to state-dependent and multi-optional services. Niranjan and Indira [9] conducted a review of bulk queues accompanied by vacations to provide analysts, researchers, and businesspeople with the knowledge they need to model traffic issues and determine the best performance metrics for queueing structure. Not every service system runs perfectly at startup. In numerous systems, servers or other equipment may periodically have starting issues that cause delays in the commencement of services. To handle situations when service can be temporarily unavailable, this study incorporates the idea of initiating failures. This study also considers repair delays, accounting for the time technicians need to identify and fix hardware or software issues, schedule their availability, or secure replacement components. Starting failures can mimic initial call routing problems in a call center, equipment failures that require immediate repair before services begin, or software defects that necessitate a system restart before users can interact with the system. In a study by Jain et al. [10], they talked about the model strategic behavior of common and primitive users in cognitive radio systems, including vacation and rework. Singh et al. [11] analyzed a bulk arrival repairable single server queueing system involving starting failure, repair, and delay to repair. Karpagam et al. [12] highlight a bulk retrial queue that includes a potential for failure at the start, repair, vacation, and additional service for faulty batches. Madheswari et al. [13] analyzed a model where a single server provides service in two phases with optional second phase service and taking Bernoulli vacations, which include K− types.

Additionally, we introduced the concept of delay time. In certain real-world congested conditions, the availability of repairmen or other factors may prevent the timely repair of the defective system. The process of delaying the start of a service system's repair is known as delayed repair. Singh et al. [14], Jain and Bhagat [15] have studied a model on bulk arrival retrial queues that incorporates the ideas on delay in repairs. Balking is the practice of consumers opting to exit the line ahead of service because they are dissatisfied with long wait periods or poor service. Managing customer flow, lowering queue desertion rates, and improving service effectiveness all depend on an understanding of balking behavior. The need to understand what influences customers' actions, create efficient queue management techniques, and enhance overall service delivery by reducing customer unhappiness are some of the driving forces behind the research on balking. The balking phenomenon occurs in numerous congestion scenarios when the customer decides to avoid the queue because he thinks he is waiting for a long time to receive the required service for a variety of reasons. Numerous queueing scholars have examined several balking-related obstruction conditions. Sundarapandiyan and Nandhini [16] and Rajadurai et al. [17] looked into M[X]/G/1 queues with the idea that customers could complain about the system by getting two types of service and having to go through a new vacation policy. A strong grasp of two-phase service systems is essential for developing efficient service procedures, reducing service times, and increasing customer satisfaction. Some of the motivations behind the study of two-phase service include the need to improve service systems, minimize waiting times, and allocate resources as efficiently as possible.

If a customer from the batch approaches and finds the server idle, they can initiate the service, allowing the others to join the orbit. The server offers every client two crucial phase services in continuation: opening FPS, the next phase of service. Harini and indira [18] conducted extensive research on the bulk arrival model, integrated with 5G based energy conservation station. In their study, Choudhury et al. [19] looked at how the M[X]/G/1 model behaved by giving clients some optional services for SPS, service interruptions, and random failures while providing offerings. By including general retrial times, Bhagat et al. [20] discussed bulk arrival queueing systems subject to unreliable servers. Xu et al. [21] collision integrated retrial queuing system providing heterogeneous service in two phases with delayed vacation. Well, along with Rajadurai et al., [22] framed a model by including feedback and negative customer arrival. Ayyappan et al. [23] looked at how bulk arrivals were handled by a server with standby and two-phase service in a variety of ways, including a single server that could be fixed, failure at the beginning, and many vacations. Maheshwari et al. [24] looked into a single server retry queue that had two service phases, the second of which was optional. Additionally, the server experiences startup errors and goes on K different kinds of Bernoulli vacations. Srivastava et al. [25] worked on the bulk arrival Markovian queueing model, which consists of two types of services and multiple vacations. Rework refers to situations where consumers must undergo multiple service attempts or iterations due to errors, failures, or quality issues. Recognizing the underlying reasons for service failures, raising the caliber of services, and cutting costs all depend on an understanding of rework procedures. By studying rework, we can improve operational efficiency, eliminate customer annoyance, and streamline service procedures by addressing the root causes.

Many authors have discussed the idea of feedback customers in the literature on retry queueing. Several queueing scenarios may service customers repeatedly for a specific cause. After receiving the essential or optional service, unsatisfied consumers can either depart from the system with the probability q or rejoin to orbit as feedback customers, incorporating the probability p to receive another regular service following the conclusion of each of the two essential phases of service. Many real-world scenarios involve retrial queues incorporating feedback. For instance, in multiple access telecommunication systems, we model the retrial queue with feedback to resend rejected dispatched messages. Also, re-service has numerous practical applications in places like bank desks, operational ATMs, supermarkets, medical facilities, etc. Jain, Madhu, and Sandeep Kaur [26], Ayyappan and Arulmozhi [27], Rajadurai et al. [28] made a significant investigation into feedback retrial queues.

Working vacations are times when servers have less work to do or are available to do maintenance or other non-service-related duties. Understanding the working vacation dynamics is critical to maximizing server usage, achieving workload balance, and maintaining service quality during times of low capacity. The need to create effective scheduling regulations, optimize resource use, and strengthen system resilience through server outage management motivates the study of working vacations. By adding these numerous aspects, this research attempts to close the gap between the complex realities of real-world service systems and the conventional, frequently oversimplified queueing models. In most real-world queueing circumstances, the server was not supposed to remain fully idle during the vacation for a variety of purposes, including loss of revenues or increasing queue size at the time of vacation. In these instances, the server provided service at a slow rate instead of fully stopping during a typical busy period, a concept known as a working vacation (WV). Servi et al. [29] and Jain et al. [30] have established a new class of semi-vacation policies. Dhibar and Jain [31] investigated the working vacation, users' dissatisfaction actions, and servers prone to disasters incorporated into the Markovian retrial queueing model. Gao et al. [32] investigated a model with batch input that includes the policy of jumbled J working vacation.

This study employs a two-phase service model to illustrate the multi-step nature of real-world service methods. The primary goals of this examination are to maximize entropy and provide a comparative study of the system's exact and approximate expected waiting period. The focus of research on “bulk arrival, two-phase service, starting failure, delayed repair, balking, rework, and working vacation” likely centers on the analysis and modeling of complex, sophisticated queueing systems with specific characteristics. Various fields, including online shopping platforms, manufacturing systems, internet services, and public transportation systems, involve the design, optimization, and administration of real-world systems. These contributions can influence telecommunications, production, medical services, and other contexts with comparable complexity. This allows for more accurate assessments of variables that affect metrics such as average waiting time, queue length, and overall system efficiency. Rework and working vacations provide a level of realism that is sometimes absent from more basic models. This makes it possible to investigate methods for reducing the likelihood of rework and maximizing service staff break scheduling. Furthermore, the comparison table may be found in Table 5.

We have created our model using the following segments: The initial section, 1, was accompanied by an introductory discussion that included recent literature contents. In contrast, 2 presents both the system's diagrammatic and mathematical representation. In section 3, we investigated the model's state probability by including stability conditions, mathematical notations and probability, steady-state solutions. Performance estimation and other additional evaluations are shown in section 4. The objects that are made are based on the model's mathematical results and are shown in both two-dimensional and three-dimensional ways are shown in section 5. The conclusion is provided in section 6.

2 Model's diagrammatic representation with transition diagram and mathematical representation

2.1 Mathematical framework

Customers from a variety of batch sizes enter the system through a compound Poisson process. It was assumed that λ`cidτˇ; [i=1,2,3,4,...] as probability in the order first. Here, i represents the customer number (in batch) that joins into the system within a small time interval (τˇ,τˇ+dτˇ). We could use the bulk arrival model to enhance the performance of wireless local area networks. In this scenario, various length messages arrive at the station, where they are split into packets and sent through wireless channels to the target station.

The retrial is termed arriving customers by knowing the busy or unavailable server, leaving that station, and moving to the retrial group termed orbit. After a random interval of time, the customer in the orbit takes another trial to get the required support. The inter-retrial time is determined by an arbitrary distribution fun Aˆ(τˇ), which is associated with its corresponding density fun θ(xˆ). The Laplace-Stieltijes transform (LST) Aˆ⁎(Φ) is used to describe the behavior of the time between trials. It was assumed that xˆ represents the amount of retrial that has already passed, also assuming that θ(xˆ)dxˆ represents the probability conditional completion of retrial within an interval (xˆ,xˆ+dxˆ], such that θ(xˆ)=ϒ(xˆ)1−Aˆ(xˆ). According to specified transmission standards, the wireless channels' availability will be determined before broadcast. The remaining packets are held to be in a buffer (i.e., in orbit) and only one is automatically chosen to be broadcast if the channel is deemed to be open. All the packets must be saved in the buffer if the server is busy; after some time, it will try for transmission again.

An entering customer (primary or from orbit) will activate the free server to begin the service, which takes negligible startup time. Based on FCFS, a single server provides service to the customer one by one in two different and compulsory phases of service, such as the first (FPS) and second (SPS) phases of service (P1) and (P2), respectively. The service time for (P1) takes an arbitrary general distribution incorporated with distribution fun Lˆb(τˇ) along its density fun and LST Lˆb⁎(Φ). Also, the first, second moments are E(Lˆb) and E(Lˆb2). It was assumed that xˆ represents (P1) service time duration that already passed, also assuming that μb(xˆ)dxˆ denotes the conditional completion probability of P1 within the interval (xˆ,xˆ+dxˆ], with μb(xˆ)=Ψb(xˆ)1−Lˆb(xˆ).

Once P1 was completed, the customer was taken to the next phase of service. P2 along its distribution fun Rˆˆb(τˇ), its density fun πb(xˆ) and the LST Rˆˆb⁎(Φ) with the first two moments E(Rˆˆb) and E(Rˆˆb2). Let μsb(xˆ)dxˆ represent the conditional completion probability of P2 within the interval (xˆ,xˆ+dxˆ]. Here, xˆ acts as a time for elapsed service, such that μsb(xˆ)=πb(xˆ)1−Rˆˆb(xˆ). The period of working vacation (WV) occurs when the system is vacant. During WV, the server decides to continue the service rather than being completely idle. The service rate is slow during this period. WV accompanies a RV Wˆv through the probability distribution fun Wˆv(τˇ), along with LST Wˆv⁎(Φ). By assuming the initial and subsequent moments as E(Wˆv) and E(Wˆv2) respectively. Let μv(xˆ)dxˆ represent the conditional completion prob of WV within the interval (xˆ,xˆ+dxˆ]. Here, xˆ acts as a time for elapsed WV, such that μv(xˆ)=℧v(xˆ)1−Wˆv(xˆ).

The concepts of starting failure, feedback customers, and impatience customers, along with their probability, are also considered to be the primary novels of this article. Let us assume either an incoming customer starts the server to be busy with prob α or the server may face starting failure with prob (α¯=1−α). When the starting failure occurred, the customer in the service area swiftly moved back to the front of the queue within the orbit. In reality, a starting failure happens in the mail server process, due to unreliability in the software or for some other purpose (such as a virus). In some random amount of time, the mail server was taken back and repaired). After receiving all the required services, the customer will choose whether to move again to the trial group to get additional services along with prob p instead of permanently quitting the system through prob q=1−p. For instance, feedback plays an important role during the transmission of data, some packets might be returned to the destination from the source. This cycle goes on until the entire packet is sent. Consumers in queueing models frequently exhibit an impatient phenomenon. The arriving customers either join the orbit through prob b or by prob 1−b balks the system on finding the busy or unavailable server. Customers can be impatient in a variety of settings, including a health center, telephone switchboard users, web users, call center agents, computer systems, etc.

Once a starting failure happens, the server is not immediately taken for repair but with a certain delay. The delay time follows distribution fun Eˆf`(xˆ) along the density fun D´(xˆ) and LST Eˆf`⁎(Φ) through initial and the next moments to be E(Eˆf`) and E(Eˆf`2). It was considered η(xˆ)dxˆ as a conditional completion probability for delayed repair within a given interval of (xˆ,xˆ+dxˆ] here xˆ is the time of elapsed delay to repair, such as η(xˆ)=D´n(xˆ)1−Eˆf`(xˆ). The time interval for repair follows the general (arbitrary) distribution together through distribution fun Sˆf`(τˇ), density fun Ωn(xˆ), LST Sˆf`⁎(Φ) with first, next moments be E(Sˆf`) and E(Sˆf`2). Let ϑ(xˆ)dxˆ be assumed as a conditional completion prob towards repair on a given interval (xˆ,xˆ+dxˆ]. xˆ be the elapsed time for repair, so ϑ(xˆ)=Ω(xˆ)1−Sˆf`(xˆ).

The schematic representation of the model and the transition diagram were presented in Fig. 1 and Fig. 2, respectively.Figure 1 Model's diagrammatic representation.

Figure 1

Figure 2 Model's transition diagram.

Figure 2

Real-world implementation of the model

A smart irrigation system for large farms or agricultural fields could be one real-time application of our model in agriculture, specifically in irrigation management. Sophisticated algorithms and sensors could be used in this application by a real-time irrigation management system to overcome these obstacles. It would work in vacation modes to maximize its performance even in times of low demand. It would also continuously monitor conditions, respond to bulk arrivals of irrigation needs, handle starting failures, manage delayed repairs efficiently, prevent balking when possible, and incorporate feedback from various sources. Crop yield, water efficiency, and overall farm productivity could all be greatly increased by such a system. One bulk arrival is the term used to describe the simultaneous arrival of many requests or duties (in this case, irrigation demands). When it comes to irrigation, this can mean a rapid spike in the amount of water needed or the need to irrigate more than one field because of the crop's needs or the weather. Two-phase service in irrigation management could relate to the planning and carrying out phases of water delivery to the field. The method calculates the ideal irrigation schedule during the planning phase (FP) using crop kinds, soil moisture, and weather forecasts. In the execution phase (SP), irrigation is done as planned, meaning that a larger volume is delivered to guarantee adequate saturation.

Starting failure could refer to situations in which the irrigation system fails to start with a likelihood of α because of technical problems like a malfunctioning pump, an outage in electricity, or a malfunctioning sensor. A power outage or a malfunctioning pump, for instance, could stop the irrigation system from starting the watering cycle. In the event of an irrigation system malfunction, such as a broken pump or pipeline, the system can be fixed and brought back online; however, this process may take longer than expected for a variety of reasons, such as a lack of technical know-how or spare parts. Crop productivity and health may be impacted by this delay. Balking is the term for the circumstance in which a client (in this case, a field in need of irrigation) declines service with probability (1−b) because they are not happy with the terms of service for a variety of reasons. For example, certain fields may choose not to participate in the irrigation cycle if the system is not able to supply enough water to all of them at the same time.

Sensors monitoring crop growth phases, weather patterns, soil moisture levels, and other variables are continuously gathered by the irrigation system. Rework with probability (p) describes the situation when you have to redo a task because your first attempt was flawed or insufficient. If water is distributed unevenly over fields during irrigation, rework may occur and further rounds of watering may be required to guarantee adequate coverage. The system uses this feedback loop to guide its decision-making when deciding how much and when not to water. “Working vacation” could be used to describe times in agriculture when there is less demand for irrigation (e.g., during the dormant season or when crops require less water) or when irrigation needs are lower. In these cases, the system may enter a “working vacation” mode and carry out maintenance, update its algorithms, or do diagnostics to ensure optimal performance when demands increase again.

3 An analysis of the stable state probability of the model

An equation describing the difference differential of the framed structure in steady state was formulated by incorporating supplementary variables, including an elapsed period of retrial, busy, working vacation (WV), delayed repair, and repair. The process of obtaining the customer numbers for the system and orbit, along with the PGF: probability-generating function for the current server's state, has been accomplished.

3.1 Stability condition of the model

By taking elapsed time for retrial, P1,P2, WV, delay, and repair as supplementary variables, this section builds difference-differential equations for this steady state system. The PGF towards server states, system, as well as orbit size, were acquired.

During the period of equilibrium, it was widely believed that the following were continuous when xˆ=0,Aˆ(0)=0,Lˆb(O)=0,Rˆˆb(O)=0,Wˆv(O)=0,Eˆf`(O)=0,Sˆf`(O)=0, where O=0, also Aˆ(∞)=1,Lˆb(∞)=1,Rˆˆb(∞)=1,Wˆv(∞)=1,Eˆf`(∞)=1, and Sˆf`(∞)=1. Also, completion rate towards retrial, FPS, SPS, WV, delayed to repair and for repair can be obtained by θ(xˆ), μb(xˆ), μsb(xˆ), μv(xˆ), η(xˆ), ϑ(xˆ) respectively. Apart from these, at time τˇ let Aˆ0(τˇ),Lˆb0(τˇ),Rˆˆb0(τˇ),Wˆv0(τˇ),Eˆf`0(τˇ),Sˆf`0(τˇ) were taken to be elapsed time for retrial, P1,P2, WV, delay, and repair respectively. Additionally, we launch the random variables as,

During τˇ the state for the system was described with bivariate Markov procedure {Ϝ(τˇ),X(τˇ),Aˆ0(τˇ),Lˆb0(τˇ),Rˆˆb0(τˇ),Wˆv0(τˇ),Eˆf`0(τˇ),Sˆf`0(τˇ);τˇ≥0} here Ϝ(τˇ) speaks for server's state (0,1,2,3,4,5) based on above mentioned states. If Ϝ(τˇ)=0, X(τˇ)>0; Aˆ0(τˇ) speaks for completed period for retrial. If Ϝ(τˇ)=1, X(τˇ)≥0; Lˆb0(τˇ) speaks for completed FPS during busy time τˇ. Also Ϝ(τˇ)=2 as well as X(τˇ)≥0; now Rˆˆb0(τˇ) speaks for completed normal SPS duration during busy time of τˇ. If Ϝ(τˇ)=3 with X(τˇ)≥0; on that occasion Wˆv0(τˇ) speaks for finished WV during τˇ, if Ϝ(τˇ)=4 and X(τˇ)>0; on that occasion Eˆf`0(τˇ) speaks for completed period for delayed to repair at τˇ finally if Ϝ(τˇ)=5, X(τˇ)>0; at that moment Sˆf0(τˇ) represents the time duration spent for repair during busy period τˇ.

Ergodicity in the proposed dynamic system

Let {τˇn;n∈N} denote the order of time for either conclusion of service or end of vacation. For the proposed retrial queueing paradigm, The random vector's sequence Z`n={Ϝ(τˇn,+),X(τˇn,+)} exhibits embedded Markov chain. Also S={0,1,2,3,4,5}×N denotes the state space. Theorem 1 Embedded Markov chain{Z`n;n∈N}be ergodic iffC1¯(1−A⁎(λ`))+(αp+αbλ`C1¯(E(Lˆb)+E(Rˆˆb)))+(α¯+α¯λ`bC1¯(E(Sˆf`)+E(Eˆf`)))<1.

Proof The chain {Z`n;n∈N} exhibits properties of both irreducibility and aperiodicity, suggesting its strength as a Markov chain. In accordance with Foster's criterion [33], the ergodicity of an irreducible, aperiodic Markov chain is demonstrated by the existence of a non-negative function, as a required condition. Consider the function f(u´) where u´ is an element of the set of natural numbers, denoted as N. Let ϵ be a positive real number. It is assumed the mean drift given by χu´=E[(f(z˘n+1)+f(z˘n)|z˘n=u´))] noticed to be finite measure also applicable for any values of u´∈N with χu´≤−∈. This statement holds for all values of the variable u´ in the set of natural numbers, except for a finite number of u´ values in theory. In this situation, now we deliberating f(u´)=u´ to getχu´={α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))+α¯(1+λ`bC1¯(E(Sˆf`)+E(Eˆf`)))−1;u´=1,2,…C1¯(1−A⁎(λ`))+α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))+α¯(1+λ`bC1¯(E(Sˆf`)+E(Eˆf`)))−1;u´=0

The necessary along with sufficient condition towards ergodicity is expressed by an inequality α(E(Lˆb)+E(Rˆˆb))λ`+α¯(1+λ`(E(Eˆf`)+E(Sˆf`)))<Aˆ⁎(λ`). In order to satisfy necessary condition, according to Sennott [34], If Markov chain {Z`n;n∈N} satisfies a structure proposed by Kaplan, particularly χu´<∞, for every u´≥0 and also for u´0∈N such as χu´≥0 for every u´>u´0's data. Given the presence of a˘, Kaplan's condition appears to be met, in this instance φu˘,u´=0 for u˘>0 also u´<u˘−a˘ such as R=(φu˘,u´) was regarded as single-step transition matrix of {Z`n;n≥1}. Subsequently, also establishing C1¯(1−A⁎(λ`))+(αp+αλ`bC1¯(E(Lˆb)+E(Rˆˆb)))+(α¯+α¯λ`bC1¯(E(Sˆf`)+E(Eˆf`)))≥1 which represents non-ergodicity Markov chain. □

In favor of Markov process {Z`n;n∈N}, we can define probability when τˇ≥0,xˆ≥0, asϒ0(τˇ)=P{Ϝ(τˇ)=0,X(τˇ)=0}also defining probability densities on asϒn(xˆ,τˇ)dxˆ=P{Ϝ(τˇ)=0,X(τˇ)=n,xˆ≤Aˆ0(τˇ)<xˆ+dxˆ},n≥1Ψn,b(xˆ,τˇ)dxˆ=P{Ϝ(τˇ)=1,X(τˇ)=n,xˆ≤Lˆb0(τˇ)<xˆ+dxˆ},n≥0Πn,b(xˆ,τˇ)dxˆ=P{Ϝ(τˇ)=2,X(τˇ)=n,xˆ≤Rˆˆb0(τˇ)<xˆ+dxˆ},n≥0℧n,v(xˆ,τˇ)dxˆ=P{Ϝ(τˇ)=3,X(τˇ)=n,xˆ≤Wˆv0(τˇ)<xˆ+dxˆ},n≥0D´n(xˆ,τˇ)dxˆ=P{Ϝ(τˇ)=4,X(τˇ)=n,xˆ≤Eˆf`0(τˇ)<xˆ+dxˆ},n≥1Ωn(xˆ,τˇ)dxˆ=P{Ϝ(τˇ)=5,X(τˇ)=n,xˆ≤Sf`0(τˇ)<xˆ+dxˆ},n≥1

3.2 Probability with mathematical notations utilized in the model

ϒn(xˆ,τˇ)	:	The probability that there will be precisely n customers exist in the orbit at τˇ with completed retrial time of customer is xˆ.	
Ψn,b(xˆ,τˇ)	:	The probability were the orbit occupies n customers during τˇ of regular completed first phase service time xˆ.	
Πn,b(xˆ,τˇ)	:	The probabilities of having a specific n number of customers in the orbit at time τˇ with fulfilled normal second phase service periods is represented by the variable xˆ with fulfilled time for repair is xˆ.	
℧n,v(xˆ,τˇ)	:	The probabilities of having a specific n number of customers in the orbit at time τˇ with fulfilled lower speed service time to the customer is xˆ.	
D´n(xˆ,τˇ)	:	The probabilities of having a specific n number of customers in the orbit at time τˇ with fulfilled time to delay is xˆ.	
Ωn(xˆ,τˇ)	:	The probabilities of having a specific n number of customers in the orbit at time τˇ with fulfilled time for repair is xˆ.	

We can establish the limiting probability for τˇ≥0, xˆ≥0 and n≥0 since it is accepted for the stability criteria to be met. Furthermore, we display the random variable asϒ0=limτˇ→∞⁡ϒ0(τˇ),n≥0;ϒn(xˆ)=limτˇ→∞⁡ϒn(xˆ,τˇ),n≥1;Ψn,b(xˆ)=limτˇ→∞⁡Ψn,b(xˆ,τˇ),n≥0;Πn,b(xˆ)=limτˇ→∞⁡Πn,b(xˆ,τˇ),n≥0;℧n,v(xˆ)=limτˇ→∞⁡℧n,v(xˆ,τˇ),n≥0;Ωn(xˆ)=limτˇ→∞⁡Ωn(xˆ,τˇ),n≥1;D´n(xˆ)=limτˇ→∞⁡D´n(xˆ,τˇ),n≥1.

The system's governing equations

Using the supplemental variable method, we arrive to the system of equations that is shown below. This system's behavior is controlled by system's dynamics,(1) λ`ϒ0=q∫0∞℧0,v(xˆ)μv(xˆ)dxˆ

(2) ddxˆϒn(xˆ)+(θ(xˆ)+λ`)ϒn(xˆ)=0,n≥1

(3) ddxˆΨ0,b(xˆ)+(λ`+μb(xˆ))Ψ0,b(xˆ)=(1−a)λ`Ψ0,b(xˆ),n=0

(4) ddxˆΨn,b(xˆ)+(λ`+μb(xˆ))Ψn,b(xˆ)=λ`(1−a)Ψn,b(xˆ)+λ`a∑k=1nCkΨn−k,b(xˆ),n≥1

(5) ddxˆΠ0,b(xˆ)+(μsb(xˆ)+λ`)Π0,b(xˆ)=(1−a)λ`Π0,b(xˆ),n=0

(6) ddxˆΠn,b(xˆ)+(μsb(xˆ)+λ`)Πn,b(xˆ)=(1−a)λ`Πn,b(xˆ)+λ`a∑k=1nCkΠn−k,b(xˆ),n≥1

(7) ddxˆ℧0,v(xˆ)+(λ`+μv(xˆ))℧0,v(xˆ)=λ`(1−a)℧0,v(xˆ),n=0

(8) ddxˆ℧n,v(xˆ)+(λ`+μv(xˆ))℧n,v(xˆ)=λ`(1−a)℧n,v(xˆ)+λ`a∑k=1nCk℧n−k,v(xˆ),n≥1

(9) ddxˆD´n(xˆ)+(λ`+η(xˆ))D´n(xˆ)=λ`(1−a)D´n(xˆ)+λ`a∑k=1n−1CkD´n−k(xˆ),n≥1

(10) ddxˆΩ0(xˆ)=0,n=0

(11) ddxˆΩn(xˆ)+(λ`+ϑ(xˆ))Ωn(xˆ)=λ`(1−a)Ωn(xˆ)+λ`a∑k=1n−1CkΩn−k(xˆ),n≥1

The model's stationary probability solutions are non-negative if C1¯(1−A⁎(λ`))+(αp+αλ`bC1¯(E(Lˆb)+E(Rˆˆb)))+(α¯+α¯λ`bC1¯(E(Sˆf`)+E(Eˆf`)))<1, [possible under the ergodicity condition only.]

When xˆ=0 , boundary conditions for steady state system are given by (12) ϒn(0)=q∫0∞℧n,v(xˆ)μv(xˆ)dxˆ+p∫0∞℧n−1,v(xˆ)μv(xˆ)dxˆ+q∫0∞Πn,b(xˆ)μsb(xˆ)dxˆ+p∫0∞Πn−1,b(xˆ)μsb(xˆ)dxˆ+∫0∞Ωn(xˆ)ϑ(xˆ)dxˆ,n≥1

(13) Ψ0,b(0)=αλ`ϒ0+α∫0∞ϒ1(xˆ)θ(xˆ)dxˆ,n=0

(14) Ψn,b(0)=αλ`∫0∞∑k=1nCkϒn−k+1(xˆ)dxˆ+α∫0∞ϒn+1(xˆ)θ(xˆ)dxˆ,n≥1

(15) Π0,b(0)=∫0∞Ψ0,v(xˆ)μb(xˆ)dxˆ,n=0

(16) Πn,b(0)=∫0∞Ψn,v(xˆ)μb(xˆ)dxˆ,n≥1

(17) ℧0,v(0)=q∫0∞Π0,b(xˆ)μsb(xˆ)dxˆ,n=0

(18) ℧n,v(0)=0,n≥1

(19) D´1(0)=α¯∫0∞ϒ1(xˆ)θ(xˆ)dxˆ+α¯λ`ϒ0,n=1

(20) D´n(0)=α¯∫0∞ϒn(xˆ)θ(xˆ)dxˆ+α¯λ`∫0∞∑k=1n−1Ckϒn−k(xˆ)dxˆ,n≥2

(21) Ωn(0)=∫0∞D´n(xˆ)η(xˆ)dxˆ,n≥1

The normalizing condition was provided by (22) ϒ0+∑n=1∞∫0∞ϒn(xˆ)dxˆ+∑n=0∞∫0∞Ψn,b(xˆ)dxˆ+∑n=0∞∫0∞Πn,b(xˆ)dxˆ+∑n=0∞∫0∞℧n,v(xˆ)dxˆ+∑n=1∞∫0∞D´n(xˆ)dxˆ+∑n=1∞∫0∞Ωn(xˆ)dxˆ=1

3.3 The framed model solution in steady state

PGF for this retrial queueing model is used in to approach the steady state solution. The generating functions utilized to solve the extraneous equations were defined to be when |y˘|≤1,ϒ(xˆ,y˘)=∑n=1∞ϒn(xˆ)y˘n;ϒ(0,y˘)=∑n=1∞ϒn(0)y˘n;Ψb(xˆ,y˘)=∑n=0∞Ψn,b(xˆ)y˘n;Ψb(0,y˘)=∑n=0∞Ψn,b(0)y˘n;Πb(xˆ,y˘)=∑n=0∞Πn,b(xˆ)y˘n;Πb(0,y˘)=∑n=0∞Πn,b(0)y˘n;℧v(xˆ,y˘)=∑n=0∞℧n,v(xˆ)y˘n;℧v(0,y˘)=∑n=0∞Gn,v(0)y˘n;D´(xˆ,y˘)=∑n=1∞D´n(xˆ)y˘n;D´(0,y˘)=∑n=1∞D´n(0)y˘n;Ω(xˆ,y˘)=∑n=1∞Un(xˆ)y˘n;Ω(0,y˘)=∑n=1∞Un(0)y˘n.

Theorem 2 The total count of customers present within a given orbit consistently over a specified period, the values0,1,2,3,4,5indicate, that the server is inactive, doing its first-phase and second-phase regular service, on vacation, experiencing some delay, or undergoing maintenance, respectively. Following is a list of generating functionsϒ(y˘),Ω(y˘),Ψb(y˘),℧v(y˘),D´(y˘),Πb(y˘). Also, the normalizing circumstance(22)revealedϒ0.ϒ0+Ψb(1)+Ω(1)+Πb(1)+ϒ(1)+℧v(1)+D´(1)=1

Proof The set of partial differential equations (PDE) presented above were derived by multiplying equations (2) to (11) by appropriate powers of y˘, followed by summation over the variable n.(23) ∂ϒn(xˆ,y˘)∂xˆ+(θ(xˆ)+λ`)ϒ(xˆ,y˘)=0

(24) ∂Ψb(xˆ,y˘)∂xˆ+(aλ`(1−C(y˘))+μb(xˆ))Ψb(xˆ,y˘)=0

(25) ∂Πb(xˆ,y˘)∂xˆ+(aλ`(1−C(y˘))+μsb(xˆ))Πb(xˆ,y˘)=0

(26) ∂℧v(xˆ,y˘)∂xˆ+(aλ`(1−C(y˘))+μv(xˆ))℧v(xˆ,y˘)=0

(27) ∂D´(xˆ,y˘)∂xˆ+(aλ`(1−C(y˘))+η(xˆ))D´(xˆ,y˘)=0

(28) ∂Ω(xˆ,y˘)∂xˆ+(aλ`(1−C(y˘))+ϑ(xˆ))Ω(xˆ,y˘)=0

The solutions to the PDE (23) to (28) mentioned above is provided by(29) ϒ(xˆ,y˘)=e−λ`xˆ(1−Aˆ(xˆ))ϒ(0,y˘)

(30) Ψb(xˆ,y˘)=e−aλ`(1−C(y˘))xˆΨb(0,y˘)(1−Lˆb(xˆ))

(31) Πb(xˆ,y˘)=e−aλ`(1−C(y˘))xˆΠb(0,y˘)(1−Rˆˆb(xˆ))

(32) ℧v(xˆ,y˘)=e−aλ`(1−C(y˘))xˆ℧v(0,y˘)(1−Wˆv(xˆ))

(33) D´(xˆ,y˘)=e−aλ`(1−C(y˘))xˆD´(0,y˘)(1−Eˆf`(xˆ))

(34) Ω(xˆ,y˘)=e−aλ`(1−C(y˘))xˆΩ(0,y˘)(1−Sˆf`(xˆ))

Similarly, we get solving for (12) to (21)(35) ϒ(0,y˘)=q∫0∞℧v(xˆ,y˘)μv(xˆ)dxˆ+q∫0∞Πb(xˆ,y˘)μsb(xˆ)dxˆ+zp∫0∞℧v(xˆ,y˘)μv(xˆ)dxˆ+zp∫0∞Πb(xˆ,y˘)μsb(xˆ)dxˆ+∫0∞Ω(xˆ,y˘)ϑ(xˆ)dxˆ−λ`ϒ0−℧0,v(0)

(36) Ψb(0,y˘)=αλ`C(z)y˘∫0∞ϒ(xˆ,y˘)dxˆ+αy˘∫0∞ϒ(xˆ,y˘)θ(xˆ)dxˆ+α∫0∞ϒ1(xˆ)θ(xˆ)dxˆ+Ψ0,b(0)

(37) Πb(0,y˘)=∫0∞Ψb(xˆ,y˘)μb(xˆ)dxˆ

(38) ℧v(0,y˘)=℧0,v(0)

(39) D´(0,y˘)=α¯λ`∫0∞C(z)ϒ(xˆ,y˘)dxˆ+α¯∫0∞(ϒ(xˆ,y˘)−ϒ1(xˆ)y˘)θ(xˆ)dxˆ+D´1(0)y˘

(40) ℧0,v(0)=λ`ϒ0qWˆv⁎(aλ`)

(41) Πb(0,y˘)=Lˆb⁎(aλ`(1−C(y˘)))[αy˘C˘(y˘)ϒ(0,y˘)+αλ`ϒ0]

(42) ℧v(0,y˘)=Lˆb⁎(aλ`(1−C(y˘)))Πb(0,y˘)

(43) D´(0,y˘)=α¯ϒ(0,y˘)C˘(y˘)+α¯λ`y˘ϒ0

(44) Ω(0,y˘)=(α¯λ`y˘ϒ0+α¯C˘(y˘)ϒ(0,y˘))Ef`⁎(aλ`(1−C(y˘)))

Let us assume (A(y˘))=(aλ`(1−C(y˘)))

Getting equations (1), (31), (32), and (34) to (35) and then (38), (40), (41) and (44) we get the following equations(45)

In the same way, if we put (29) and (45) into (36) we move to following:(46) Ψb(0,y˘)=λ`αϒ0qWˆv⁎(aλ`){(1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(aλ`))C˘(y˘)−y˘qWˆv⁎(aλ`)(α(q+py˘)Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘))+α¯y˘Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))C˘(y˘)−y˘}

In the same way, substitute equation (30), (46) in (37) we moved to following(47) Πb(0,y˘)=λ`αϒ0Lˆb⁎(A(y˘))qWˆv⁎(aλ`){(1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(aλ`))C˘(y˘)−y˘qWˆv⁎(aλ`)(α(q+py˘)Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘))+α¯y˘Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))C˘(y˘)−y˘}

Solving the equations (43) and (45) then taking to the equation (39), we move to(48) D´(0,y˘)=λ`α¯y˘ϒ0qWˆv⁎(aλ`){(1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(aλ`))C˘(y˘)−y˘qWˆv⁎(aλ`)(α(q+py˘)Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘))+α¯y˘Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))C˘(y˘)−y˘}

Similarly substitute equation (45) in (44) we moved to following(49) Ω(0,y˘)=λ`α¯y˘ϒ0Eˆf`⁎(A(y˘))qWˆv⁎(aλ`){(1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(aλ`))C˘(y˘)−y˘qWˆv⁎(aλ`)(α(q+py˘)Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘))+α¯y˘Sˆf`⁎(A(y˘))Eˆf`⁎(A(y˘)))C˘(y˘)−y˘}

Let C˘(y˘)=C(y˘)+Aˆ⁎(λ`)(1−C(y˘))

Now, aforementioned equations from (45) to (47) in (29) to (31) are used to establish the limiting probability generating functions.(50)

(51) Ψb(xˆ,y˘)=λ`αP0qWˆv⁎(aλ`){(1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(aλ`))C˘(y˘)−y˘qWˆv⁎(aλ`)(α(q+py˘)Rˆˆb⁎(A(y˘))Lˆb⁎(A(y˘))+α¯y˘Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))C˘(y˘)−y˘}(1−Lb(xˆ))e−A(y˘)xˆ

(52) Πb(xˆ,y˘)=λ`αϒ0Lˆb⁎(A(y˘))qWˆv⁎(aλ`){(1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(aλ`))C˘(y˘)−y˘qWˆv⁎(aλ`)(α(q+py˘)Rˆˆb⁎(A(y˘))Lˆb⁎(A(y˘))+α¯y˘Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))C˘(y˘)−y˘}(1−Rˆˆb(xˆ))e−A(y˘)xˆ

Similarly substitute equation (40), (42) in (32) we moved to the following equation (53) also, substitute equation (43), (44) in (35) and then to (33), (34) we moved to the following equation (53) we moved to the following equations (54) and (55)(53) ℧v(xˆ,y˘)=λ`ϒ0qWˆv⁎(aλ`)(1−Wˆv(xˆ))e−A(y˘)xˆ

(54) D´(xˆ,y˘)=α¯y˘λ`ϒ0qWˆv⁎(aλ`){(1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(aλ`))C˘(y˘)−y˘qWˆv⁎(aλ`)(α(q+py˘)Rˆˆb⁎(A(y˘))Lˆb⁎(A(y˘))+α¯y˘Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))C˘(y˘)−y˘}(1−Eˆf`(xˆ))e−A(y˘)xˆ

(55) Ω(xˆ,y˘)=λ`α¯y˘ϒ0Eˆf`⁎(A(y˘))qWˆv⁎(aλ`){(1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(aλ`))C˘(y˘)−y˘qWˆv⁎(aλ`)(α(q+py˘)Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘))+α¯y˘Eˆf`⁎(A(y˘))Sˆf⁎(A(y˘)))C˘(y˘)−y˘}(1−Sˆf(xˆ))e−A(y˘)xˆ

With the aid of following equations, the necessary probability generating functions are turned meanwhile working on the marginal generating functions for the server states and orbit size.ϒ(y˘)=∫0∞ϒ(xˆ,y˘)dxˆ;Ψb(y˘)=∫0∞Ψb(xˆ,y˘)dxˆ;Πb(y˘)=∫0∞Πb(xˆ,y˘)dxˆ;℧b(y˘)=∫0∞℧b(xˆ,y˘)dxˆ;D´(y˘)=∫0∞D´(xˆ,y˘)dxˆ;Ω(y˘)=∫0∞Ω(xˆ,y˘)dxˆ.

Now, by integrating the above equations (50) to (55) to xˆ as well as letting limit 0 to ∞ required equations are (56) to (61).(56)

(57) Ψb(y˘)=αϒ0(1−Lˆb⁎(A(y˘)))qaWˆv⁎(aλ`)(1−C(y˘)){(1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(aλ`))C˘(y˘)−y˘qWˆv⁎(aλ`)(α(q+py˘)Rˆˆb⁎(A(y˘))Lˆb⁎(A(y˘))+α¯y˘Sˆf`⁎(A(y˘))Eˆf`⁎(A(y˘)))C˘(y˘)−y˘}

(58)

(59) ℧v(y˘)=ϒ0(1−Wˆv⁎(A(y˘)))qaWˆv⁎(aλ`)(1−C(y˘))

(60) D´(y˘)=α¯y˘ϒ0(1−Eˆf⁎(A(y˘)))qaWˆv⁎(aλ`)(1−C(y˘)){(1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(aλ`))C˘(y˘)−y˘qWˆv⁎(aλ`)(α(q+py˘)Rˆˆb⁎(A(y˘))Lˆb⁎(A(y˘))+α¯y˘Sˆf`⁎(A(y˘))Eˆf`⁎(A(y˘)))C˘(y˘)−y˘}

(61)

Since by modifying y˘=1 in the normalizing condition, can be used to compute the single unknown, ϒ0, which is the chance of the server during idle while the orbit with no customers.(62) ϒ0+Ψb(1)+ϒ(1)+℧v(1)+D´(1)+Πb(1)+Ω(1)=1

(63) ϒ0=qWˆv⁎(aλ`){1−C1¯(1−A⁎(λ`))−α(p+λ`aC1¯(E(Lˆb)+E(Rˆˆb)))−α¯(1+λ`aC1¯(E(Eˆf`)+E(Sˆf)))}DrDr=qWˆv⁎(bλ`)(1−C1¯(1−Aˆ⁎(λ`)))+p(1−Aˆ⁎(λ`)(1+Wˆv⁎(bλ`)(1−qα¯)))+E(Wv)(αλ`q−λ`C1¯(1−b)(1−Aˆ⁎(λ`)))+(α¯λ`qWˆv⁎(bλ`)(C1¯Aˆ⁎(λ`)(1−b)+1−C1¯)+α¯pλ`)((E(Eˆf`)+E(Sˆf`))+(E(Lˆb)+E(Rˆˆb)))

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Theorem 3 For the model to adhere to the stability criterion, it is necessary thatC1¯(1−A⁎(λ`))+(αp+αλ`bC1¯(E(Lˆb)+E(Rˆˆb)))+(α¯+α¯λ`bC1¯(E(Eˆf`)+E(Sˆf`)))<1. The chance of consumers being present in the intended system as well as in an orbit was determined using the functionsK(y˘)andΘ(y˘)during several server states, including idle, routine busy, lower rate service, delay, and repair. Here(64) K(y˘)=ϒ0+ϒ(y˘)+y˘Ψb(y˘)+y˘℧v(y˘)+D´(y˘)+y˘Πb(y˘)+Ω(y˘)

also,(65) Θ(y˘)=ϒ0+Ψb(y˘)+ϒ(y˘)+Πb(y˘)+D´(y˘)+℧v(y˘)+Ω(y˘).

Proof Additionally, employing equations labeled as (56) through (61) as replacements to (65) Θ(y˘)=ϒ0+ϒ(y˘)+Ψb(y˘)+Πb(y˘)+℧v(y˘)+D´(y˘)+Ω(y˘). Now, the PGF for customer's number in the orbit was identified asΘ(y˘)=ϒ0NrDr

Nr=(α(q+py˘)Rˆˆb⁎(A(y˘))Lˆb⁎(A(y˘))+α¯y˘Sˆf`⁎(A(y˘))Eˆf`⁎(A(y˘)))((C˘(y˘))(qbWˆv⁎(bλ`)(1−C(y˘))+(1−Wˆv⁎(A(y˘))))−qby˘Wˆv⁎(bλ`)(1−Aˆ⁎(λ`))(1−C(y˘)))−y˘b(1−C(y˘))(qAˆ⁎(λ`)Wˆv⁎(bλ`)−(1−Aˆ⁎(λ`))(1−(q+py˘)Wˆv⁎(A(y˘)))−y˘(1−Wˆv⁎(A(y˘))))+(α(1−Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘)))+α¯y˘(1−Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘))))((1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(bλ`))(C˘(y˘))−y˘qWˆv⁎(bλ`))Dr=bqWˆv⁎(bλ`)(1−C(y˘))(((C˘(y˘)))(α(q+py˘)Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘))+α¯y˘Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))−z)

Furthermore, by replacing the equations through (56) to (61) into (64) K(y˘)=ϒ0+ϒ(y˘)+y˘Ψb(y˘)+y˘Πb(y˘)+y˘℧v(y˘)+D´(y˘)+Ω(y˘), we obtain PGF for consumer's number for required model.K(y˘)=ϒ0NrDr

where ϒ0 is given by the equation (63).Nr=(α(q+py˘)Rˆˆb⁎(A(y˘))Lˆb⁎(A(y˘))+α¯y˘Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))((C˘(y˘))(qbWˆv⁎(aλ`)(1−C(y˘))+y˘(1−Wˆv⁎(A(y˘))))−qby˘Wˆv⁎(bλ`)(1−Aˆ⁎(λ`))(1−C(y˘)))−y˘b(1−C(y˘))(qAˆ⁎(λ`)Wˆv⁎(bλ`)−(1−Aˆ⁎(λ`))(1−(q+py˘)Wˆv⁎(A(y˘))))−y˘2(1−Wˆv⁎(A(y˘)))+y˘(α(1−Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘)))+α¯(1−Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘))))((1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(bλ`))(C˘(y˘))−y˘qWˆv⁎(bλ`))Dr=bqWˆv⁎(bλ`)(1−C(y˘))((C˘(y˘))(α(q+py˘)Rˆˆb⁎(A(y˘))Lˆb⁎(A(y˘))+α¯y˘Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))−z)

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4 Discussion

System size's and orbit size's average: Under conditions of steady state within the system, the average size of both the system and its orbit were provided as follows.1. By assessing at y˘=1, the equation (65) was differentiated with respect to y˘ to estimate the typical number of consumers in orbit (Lˆq).Lˆq=limy˘=1⁡(ddy˘Θ(y˘))=D″Nq‴−Nq″D‴3(D″)2.

2. The average of consumers number present in the system (Lˆs) was calculated by differentiating equation (66) with regard to y˘ also by substituting y˘=1.Lˆs=limy˘=1⁡(ddy˘K(y˘))=D″Ns‴−Ns″D‴3(D″)2,

[The description for Nq″,Nq‴,Ns″,Ns‴,D″,D‴ are given in Appendix A]

4.1 Performance estimations

This section covers intriguing probabilities for various model's state. Having the help of L-Hospitals rule system's average number (Ls), along with its average waiting time (Ws), similarly, orbit's average number (Lq), along with its average waiting time (Wq) are calculated. Our findings are calculable, therefore the performance indicators have managerial consequences.

It should be noted that (P0) offers server's probability to be not in use however remains accessible for system's steady state.

System state probabilities: 1. The steady-state probability ϒ where the server won't be active throughout the retrial session will be shown.(66)

2. Probability for server's regular FPS duration in a stable state has been given by(67) Ψb(1)=ϒ0αλ`E(Lˆb)qWˆv⁎(λˆb){p+q(1−C1¯)Wˆv⁎(bλ`)+qC1¯Aˆ⁎(λ`)Wˆv⁎(λ`b)+bC1¯λ`E(Wˆv)1−C1¯(1−A⁎(λ`))−α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))−(1+C1¯λ`b(E(Sˆf`)+E(Eˆf`)))α¯}

3. The likelihood of server's regular SPS duration in stable state was provided by(68) Πb(1)=ϒ0αλ`E(Rˆˆb)qWˆv⁎(λˆb){p+q(1−C1¯)Wˆv⁎(bλ`)+qC1¯Aˆ⁎(λ`)Wˆv⁎(λ`b)+bC1¯λ`E(Wˆv)1−C1¯(1−A⁎(λ`))−α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))−(1+C1¯λ`b(E(Sˆf`)+E(Eˆf`)))α¯}

4. Likelihood of a server being occupied during the working vacation period in a stable state was supplied by(69) ℧v(1)=ϒ0λ`E(Wˆv)qWˆv⁎(λ`b)

5. The chance of server maintenance delays during the regular busy time was provided under steady-state conditions(70) D´(1)=ϒ0α¯λ`E(Eˆf`)qWˆv⁎(λˆb){p+q(1−C1¯)Wˆv⁎(bλ`)+qC1¯Aˆ⁎(λ`)Wˆv⁎(λ`b)+bC1¯λ`E(Wˆv)1−C1¯(1−A⁎(λ`))−α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))−(1+C1¯λ`b(E(Sˆf`)+E(Eˆf`)))α¯}

6. The chance that a server would need to be repaired during a ordinary busy time was provided as(71) Ω(1)=ϒ0α¯λ`E(Sˆf`)qWˆv⁎(λˆb){p+q(1−C1¯)Wˆv⁎(bλ`)+qC1¯Aˆ⁎(λ`)Wˆv⁎(λ`b)+bC1¯λ`E(Wˆv)1−C1¯(1−A⁎(λ`))−α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))−(1+C1¯λ`b(E(Sˆf`)+E(Eˆf`)))α¯}

7. Little's formula says that a customer in the system (Wˆs) and in the orbit (Wˆq) is expected to stay there for long time isWˆs=Lˆsλ`;Wˆq=Lˆqλ`

4.2 Additional evaluations

4.2.1 Reliability measure

The utilization of reliability measures inside a queueing system that incorporates an unreliable server will provide the requisite data for enhancing the overall performance of the system. The availability measure and failure frequency are computed in the following manner in order to substantiate and verify the analytical findings of this model. The probability of the server's steady state availability for the arriving customer, whether during a working or idle period, can be described as follows.

4.2.2 Stochastic breakdown of the system along with certain special cases

Here, stochastic decomposition was used to break down the random variables for the framed model in terms of the sum of more than two independent random variables. For M/G/1 type queues, Fuhrmann and Cooper [35] initially defined the stochastic decomposition property (SDP), by dividing the PGF of queue sizes into two or more independent random variables. Zs(y˘) denotes the decomposed generating function of the system's customer number. Permit χ(y˘) for the generating function of the number of customers inside P1,P2 services and on vacation at random times during steady state, and let ϕ(y˘) stand for orbit's customer number generating function while the server seems to be idle, delayed to repair, then during maintenance at a random time. As soon as the decomposition law was developed, the retry model made use of it. A two-phase queueing system's decomposition is expressed mathematically by Zs(y˘)=ϕ(y˘)×χ(y˘).(72) Zs(y˘)=ϕ(y˘)×χ(y˘)

whereϕ(y˘)=ϒ0+ϒ(y˘)+Ω(y˘)+D´(y˘)ϒ0+ϒ(1)+D´(1)+Ω(1)χ(y˘)=Ψb(y˘)+Πb(y˘)+℧v(y˘)Ψb(1)+Πb(1)+℧v(1)

Substituting the equations (63), (56), (60), (61) also (66), (70), (71) in ϕ(y˘) we have,ϒ0+ϒ(y˘)+Ω(y˘)+D´(y˘)=Nr(1)Dr(1);ϒ0+ϒ(1)+D´(1)+Ω(1)=Nr(2)Dr(2)

Substituting the equations (57), (58), (59) also (67), (68), (69) in χ(y˘).Πb(y˘)+Ψb(y˘)+℧v(y˘)=Nr(3)Dr(3);Πb(1)+Ψb(1)+℧v(1)=Nr(4)Dr(4)

Hence,(73) ϕ(y˘)=Nr(1)Dr(1)×Dr(2)Nr(2)

(74) χ(y˘)=Nr(3)Dr(3)×Dr(4)Nr(4)

whereNr(1)=C˘(y˘)(qαWˆv⁎(aλ`)(1−C(y˘))(q+py˘)Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘))+α¯y˘(1+qWˆv⁎(aλ`)−(q+py˘)Wˆv⁎(A(y˘))))+y˘α¯(1−Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘))(1−aqWˆv⁎(aλ`)(1−C(y˘)))+(1−Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))(qWˆv⁎(aλ`)−(q+py˘)Wˆv⁎(A(y˘))))+y˘b(1−C(y˘))(1−Aˆ⁎(λ`))(1−(q+py˘)Wˆv⁎(A(y˘))+αq(q+py˘)Wˆv⁎(aλ`)Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘))+α¯qy˘Wˆv⁎(aλ`)Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))−y˘b(1−C(y˘))Aˆ⁎(λ`)Wˆv⁎(aλ`)−α¯y˘2qWˆv⁎(aλ`)(1−Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))Dr(1)=qaWˆv⁎(aλ`)(1−C(y˘))(α(q+py˘)Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘))+α¯y˘Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))((C˘(y˘))−y˘)Nr(2)=(1−C1¯(1−A⁎(λ`)))qWˆv⁎(bλ`)+(1−A⁎(λ`))(p+bC1¯λ`E(Wˆv))−qWˆv⁎(bλ`)(α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))+α¯(1+C1¯λ`b(E(Sˆf`)+E(Eˆf`))))+(E(Sˆf`)+E(Eˆf`))α¯λ`(p+q(1−C1¯)Wˆv⁎(bλ`)+qC1¯A⁎(λ`)Wˆv⁎(bλ`)+λ`bC1¯E(Wˆv))

Dr(2)=qWˆv⁎(aλ`)(1−C1¯(1−A⁎(λ`))−α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))−(1+C1¯λ`b(E(Sˆf`)+E(Eˆf`)))α¯)N(3)=C˘(y˘)(α(1−Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘)))(1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(aλ`))+(1−Wˆv⁎(A(y˘)))(α(q+py˘)Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘))+αEˆf`⁎(A(y˘))Sˆf`⁎(A(y˘))y˘))−α(1−Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘)))y˘qWˆv⁎(aλ`)(1−(q+py˘)Wˆv⁎(A(y˘))+qWˆv⁎(aλ`))−y˘(1−Wˆv⁎(A(y˘)))D(3)=qaWˆv⁎(aλ`)(1−C(y˘))((α(q+py˘)Lˆb⁎(A(y˘))Rˆˆb⁎(A(y˘))+α¯y˘Eˆf`⁎(A(y˘))Sˆf`⁎(A(y˘)))(C(y˘)+(1−C(y˘))Aˆ⁎(λ`))−y˘)N(4)=1−(pα+C1¯(1−A⁎(λ`))+α¯(1+bλC1¯(E(Sˆf`)+E(Eˆf`))))+λα(E(Lˆb)+E(Rˆˆb))(p+qWˆv⁎(aλ`)(1−C1¯(1−A⁎(λ`)))−bC1¯(1−λE(Wˆv)))Dr(4)=qWˆv⁎(aλ`)(1−C1¯(1−A⁎(λ`))−α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))−(1+bλ`C1¯(E(Sˆf`)+E(Eˆf`)))α¯)

At last, substitute (73) and (74) in (72) to get Zs(y˘)

4.3 Discussion

Exceptional case study:

In this section, we look at a few of our concept's unusual applications.

Case 1: In the scenario when E(Rˆˆb)=0 and E(Eˆf`)=0,(C1¯,C2¯,b,p)=(1,1,1,0), our model simplifies to a system with a single service retrial queue, featuring a single arrival process. Additionally, the system incorporates a single phase service coupling, which incorporates both WV and failure at the start. The results obtained in this particular case exhibit a high level of concurrence with the conclusions reported by Gowsalya and Arivudainambi [36]. In this context, we consider the idle probability denoted as ϒ0ϒ0=Wˆv⁎(λ`){Aˆ⁎(λ`)−(1+E(Sˆf`)λ`)α¯−αλ`(E(Lˆb))}α(Aˆ⁎(λ`)Wˆv⁎(λ`)+λ`E(Wˆv))

Case 2: The existing model was simplified to a solitary service retrial waiting structure providing service in a single phase as well as low rate WV, assuming that E(Rˆˆb) and E(Eˆf``) to 0, (C1¯,C2¯,b,α,p)=(1,1,1,1,0), in this context, the symbol ϒ0 refers to the probability of idleness. The result coincides towards Arivudainambi et al. [37].ϒ0=Wˆv⁎(λ`){Aˆ⁎(λ`)−(λ`(E(Lˆb)))}λ`E(Wˆv)+Wˆv⁎(λ`)Aˆ⁎(λ`)

Case 3: Under the above assumptions, the organized model can be simplified into retrial queueing system with a single server with service in single phase and low working rate vacation. The constraints E(Rˆˆb)=E(Eˆf`)=0 and Aˆ⁎(λ`)=1,(C1¯,C2¯,b,α,p)=(1,1,1,1,0) hold in this scenario. The variable ϒ0 represents the probability of being idle. Thus, the final results align with those presented by Zang and Hou [38].ϒ0=Wˆv⁎(λ`){1−(λ`(E(Lˆb)))}Wˆv⁎(λ`)+λ`E(Wˆv)

Case 4: The constructed model was transformed into a queue with M/G/1 structure and service given in single phase by considering (E(Rˆˆb),E(Eˆf`),E(Wˆv))=(0,0,0),Aˆ⁎(λ`)=1,(C1¯,C2¯,b,α,p)=(1,1,1,1,0) were satisfied. The idle probability, denoted as ϒ0. These adjustments are in accordance with the PK formula proposed by Gross and Harris [39].ϒ0=1−(λ`(E(Lˆb)))

5 Numerical illustration

We present this section with some examples based on the numerical values. MATLAB software was used to demonstrate the different parameters and results of our model's performance measures. Here, retrial and service duration, working vacation, delay to repair and time for repair were exponentially distributed, given by f(xˆ)=ve−vxˆ,xˆ>0. We make arbitrary assumptions for the parameters to ensure that the steady state requirement is met. The estimated values of model characteristics, such as idle of server (P0), the mean size of the orbit, the likelihood of server's idle during the retrial, busy on P1,P2 phases, also in slow rate vacation, delayed to repair together with repair, are provided in the tables below. Table 1, Table 2, Table 3, Table 4 give the effect of arrival rate, FPS rate, retrial rate, and repair rate on some performance measures. Fig. 3 illustrates the anticipated upward trend of the surface in relation to the increasing repair rate ϑ and first phase service μb values with respect to the idle of system P0 for the values θ=0.5,μsb=20,μv=5,η=20,λ`=0.2,α=0.8,αb=0.2,p=0.5,q=0.5,b=0.1. Fig. 4 illustrates for μb=5,μsb=5,μv=0.3,η=45,λ`=0.3,α=0.5,αb=0.5,p=0.4,q=0.6,b=0.3, a surface that exhibits an anticipated upward trend in response towards increase for both the repair (ϑ) as well as retrial θ rates, as compared to system's idealness P0. Fig. 5 illustrates the anticipated rapid decline in the surface for θ=0.5,μsb=20,μv=5,η=20,λ`=0.2,α=0.8,αb=0.2,p=0.5,q=0.5,b=0.1, indicating an increasing in the rate of repair ϑ also first phase service μb in relation to the mean system size Ls. Fig. 6 illustrates by considering μb=20,μsb=20,μv=5,η=20,λ`=0.2,α=0.8,αb=0.2,p=0.5,q=0.5,b=0.1, the anticipated dramatic decline trend observed on the surface when the values of the rate for retrial θ also for repair ϑ increase in relation to the mean orbit size Lq.Table 1 Effects of rate of arrival (λ`) about P0,Lq,Wq,Ψb. The Table reveals on increasing the values for rate of arrival (λ), the probability of server's idleness (P0) with server occupying FPS (ψb) decreases, whereas the average size of orbit (Lq) exhibits an increase in size as its mean period of waiting, represented by (Wq, also increases for the given quantities of θ=0.5;μb=20;μsb=20;μv=5;η=20;ϑ=10;α=0.8;α¯=0.2;q=0.5;p=0.5;b=0.1.

Table 1λ`	P0	Lq	Wq	Ψb	
10	0.632848	18.27343	23.7411	0.4342	
12	0.441516	21.22651	28.47367	0.40771	
14	0.339038	24.37355	33.24398	0.39916	
16	0.275107	27.70208	38.05741	0.39921	
18	0.231393	31.20457	42.92086	0.40402	
20	0.199608	34.87605	47.84138	0.41174	

Table 2 Effects of FPS rate (μb) on P0,Lq,Wq,Ls,Ws,Ω. Also it reveals that on increasing the values for (μb), the probability of idle server (P0) along with server undergoing repair (Ω) increases, whereas the average size of orbit and system (Lq,Ls) fall off towards the respective average wait time (Wq,Ws), for the given quantities of θ=0.5;μsb=20;μv=5;η=20;λ`=0.2;ϑ=10;α=0.8;α¯=0.2;p=0.4;q=0.6;b=0.1.

Table 2μb	P0	Lq	Wq	Ls	Ws	Ω	
20	0.383104	11.90227	19.09678	59.51133	95.48390	0.2795	
22	0.383249	11.86495	19.05609	59.32475	95.28046	0.2796	
24	0.383370	11.83390	19.02223	59.16948	95.11117	0.2797	
26	0.383472	11.80765	18.99362	59.03824	94.96810	0.2798	
28	0.383560	11.78517	18.96912	58.92586	94.84559	0.2799	
30	0.383636	11.76571	18.94790	58.82854	94.73951	0.2799	

Table 3 Effects of retrial rate (θ) on P0,Lq,Wq,Ls,Ws,πb. It reveals that on increasing the values of rate of retrial θ, the probability of idle of server (P0) and probability of server occupying SPS increases (πb), whereas the average size of orbit and system (Lq,Ls) fall off according to the respective average waiting time (Wq,Ws), towards μb=20;μsb=20;μv=5;η=20;λ`=0.2;α=0.7;ϑ=10;α¯=0.3;p=0.4;q=0.6;b=0.2.

Table 3θ	P0	Lq	Wq	Ls	Ws	πb	
5	0.772573	2.079076	4.426503	10.39538	22.13251	0.01524	
7	0.797350	1.942753	4.106431	9.713765	20.53215	0.01583	
9	0.811434	1.871349	3.936652	9.356743	19.68326	0.01616	
11	0.820518	1.827442	3.831466	9.137209	19.15733	0.01638	
13	0.826864	1.797722	3.759911	8.988610	18.79956	0.01654	
15	0.831547	1.776273	3.708083	8.881365	18.54042	0.01665	

Table 4 Effects towards repair rate (ϑ) for P0,Wq,Lq,Ls,Ws,Ψb. It reveals that on increasing the values for repair rate ϑ, probability of idle server (P0) and server occupying FPS (ψb) increases, whereas the average size of orbit and system (Lq,Ls) falls off towards the average wait time, (Wq,Ws) for the given quantities of θ=0.5;μb=20;μsb=20;μv=5;η=20;λ`=0.2;α=0.7;α¯=0.3;p=0.4;q=0.6;b=0.2.

Table 4ϑ	P0	Lq	Wq	Ls	Ws	Ψb	
5	0.243034	26.51821	48.20001	132.5910	241.0000	0.00406	
6	0.243791	26.11709	47.66864	130.5854	238.3432	0.00408	
7	0.244331	25.83327	47.29283	129.1663	236.4642	0.00409	
8	0.244737	25.62187	47.01299	128.1093	235.0650	0.00410	
9	0.245052	25.45830	46.79653	127.2915	233.9826	0.00411	
10	0.245304	25.32799	46.62410	126.6399	233.1205	0.00411	

Figure 3 Impact of ϑ & μb rate over P0.

Figure 3

Figure 4 Impact of ϑ & θ over P0.

Figure 4

Figure 5 Impact of μb & ϑ over Ls.

Figure 5

Figure 6 Impact of ϑ & θ over Lq.

Figure 6

Fig. 7 clearly demonstrates that when the FPS (μb) grows, for θ=0.5μsb=20,μv=5,η=20,ϑ=10,λ`=0.2,α=0.8,αb=0.2,p=0.5,q=0.5,b=0.1, the system's idle state (P0) also exhibits a progressive increase. Additionally, length for system (Ls), also for queue (Lq), with waiting time (Wq) consistently decrease. Fig. 8 illustrates the inverse relationship between the retrial rate (θ) and various performance metrics of the system, includes the measurement of the system also queue length (Lq, Ls) as well as wait time (Wq) on μb=20,μsb=20,μv=5,η=20,ϑ=10,λ`=0.2,α=0.8,αb=0.2,p=0.5,q=0.5,b=0.1. Fig. 9 illustrates the increase of arrival rate, with corresponding increase of measurement of the system and queue length (Lq, Ls) as well as wait time (Wq) of the model. The system's idle state is responsible for replicating the decline in the idle state (P0) when θ=0.5,μb=20,μsb=20,μv=5,η=20,ϑ=10,α=0.8,αb=0.2,p=0.5,q=0.5,b=0.1.Figure 7 Impact of μb over P0,Ls,Lq,Wq.

Figure 7

Figure 8 Impact of θ over P0,Ls,Lq,Wq.

Figure 8

Figure 9 Impact of λ` over P0,Ls,Lq,Wq.

Figure 9

Figure 10, Figure 11, Figure 12, Figure 13 represent the effect of repair rate (ϑ) over Ls,Lq,Ws,Wq on increasing the repair rate. It was observed that Ls,Lq,Ws,Wq get decreased for the parameters μb=20,μsb=20,μv=5,η=20,λ=0.2,α=0.8,αb=0.2,p=0.5,q=0.5,b=0.1.Figure 10 Repair rate (ϑ) Vs Ls.

Figure 10

Figure 11 Repair rate (ϑ) Vs Lq.

Figure 11

Figure 12 Repair rate (ϑ) Vs Ws.

Figure 12

Figure 13 Repair rate (ϑ) Vs Wq.

Figure 13

The Figure 14, Figure 15 show how the arrival rate λ changes Ls,Lq. As the arrival rate went up, Ls,Lq also went up for the given parameters μb=20,μsb=20,μv=10,η=20,θ=5.ϑ=10,α=0.8,αb=0.2,p=0.5,q=0.5,b=0.1.Figure 14 Arrival rate (λ`) Vs Ls.

Figure 14

Figure 15 Arrival rate (λ`) Vs Lq.

Figure 15

Figure 16, Figure 17, Figure 18, Figure 19 show, on changing the chance of a successful start, α, and it affects Ls,Lq,Ws, and Wq. As α goes up, Ls,Lq,Ws,andWq go down for the given parameters μb=20,μsb=20,μv=5,λ=7,η=20,θ=5,ϑ=10,p=0.5,q=0.5,b=0.1.Figure 16 Successful start (α) Vs Ls.

Figure 16

Figure 17 Successful start (α) Vs Lq.

Figure 17

Figure 18 Successful start (α) Vs Ws.

Figure 18

Figure 19 Successful start (α) Vs Wq.

Figure 19

The Figure 20, Figure 21, Figure 22, Figure 23 show how the retrial rate θ changes Ls,Lq,Ws, and Wq as θ goes up. It was seen that Ls,Lq,Ws,andWq go down when μb=20,μsb=20,μv=5,η=20,λ=0.2,ϑ=10,α=0.8,αb=0.2,p=0.5,q=0.5,b=0.1.Figure 20 Retrial rate (θ) Vs Ls.

Figure 20

Figure 21 Retrial rate (θ) Vs Lq.

Figure 21

Figure 22 Retrial rate (θ) Vs Ws.

Figure 22

Figure 23 Retrial rate (θ) Vs Wq.

Figure 23

As b (probability of customers joins the orbit without balking) goes up, it was seen that Ls,Lq,Ws,andWq go up in Figure 24, Figure 25, Figure 26, Figure 27 for the parameters μb=50,μsb=50,μv=5,η=15,λ=6,θ=5,ϑ=3,α=0.2,αb=0.8,p=0.4,q=0.6.Figure 24 No-balking (b) Vs Ls.

Figure 24

Figure 25 No-balking (b) Vs Lq.

Figure 25

Figure 26 No-balking (b) Vs Ws.

Figure 26

Figure 27 No-balking (b) Vs Wq.

Figure 27

These pictures Figure 28, Figure 29, Figure 30, Figure 31 show how FPS μb affects Ls,Lq,Ws,andWq. As μb went up, Ls,Lq,Ws,andWq went down for the values μsb=50,μv=5,η=20,λ=0.2,θ=0.5,ϑ=10,α=0.8,αb=0.2,p=0.5,q=0.5,b=0.1.Figure 28 FPS (μb) Vs Ls.

Figure 28

Figure 29 FPS (μb) Vs Lq.

Figure 29

Figure 30 FPS (μb) Vs Ws.

Figure 30

Figure 31 FPS (μb) Vs Wq.

Figure 31

The Figure 32, Figure 33, Figure 34, Figure 35 show how SPS μsb affects Ls,Lq,Ws,Wq. As μsb went up, Ls,Lq,Ws,Wq went down for the given parameters (μb=20,μv=5,η=20,λ=0.2,θ=0.5,ϑ=10,α=0.8,αb=0.2,p=0.5,q=0.5,b=0.1.Figure 32 SPS (μsb) Vs Ls.

Figure 32

Figure 33 SPS (μsb) Vs Lq.

Figure 33

Figure 34 SPS (μsb) Vs Ws.

Figure 34

Figure 35 SPS (μsb) Vs Wq.

Figure 35

The pictures 36 to 37 show what happens to Ls,Lq,Ws, and Wq when the low-rate service μv is raised. It was seen that Ls,Ws went down and Lq,Wq went up for the conditions μsb=50,μb=50,η=15,λ=1.7,θ=3.ϑ=33,α=0.2,αb=0.8,p=0.4,q=0.6,b=0.1.Figure 36 Low speed service (μv) Vs Ls.

Figure 36

Figure 37 Low speed service (μv) Vs Ws.

Figure 37

The pictures Figure 38, Figure 39, Figure 40, Figure 41 show what happens to Ls,Lq,Ws,andWq when η is increased. It was seen that Ls,Lq,Ws,andWq got bigger when the following conditions were met: μb=20,μsb=20,μv=5,λ=0.2,θ=0.9,ϑ=20,α=0.2,αb=0.8,p=0.5,q=0.5,b=0.6.Figure 38 Delay (η) Vs Ls.

Figure 38

Figure 39 Delay (η) Vs Lq.

Figure 39

Figure 40 Delay (η) Vs Ws.

Figure 40

Figure 41 Delay (η) Vs Wq.

Figure 41

The pictures Figure 42, Figure 43, Figure 44, Figure 45 show what happens when the chance of a starting failure αb goes up. As αb went up, it was seen that Ls,Ws went down and Lq,Wq went up for the values μb=50,μsb=50,μv=5,λ=0.2,θ=0.3,η=15,ϑ=3,p=0.4,q=0.6,b=0.7.Figure 42 Starting failure (α¯) Vs Ls.

Figure 42

Figure 43 Starting failure (α¯) Vs Lq.

Figure 43

Figure 44 Starting failure (α¯) Vs Ws.

Figure 44

Figure 45 Starting failure (α¯) Vs Wq.

Figure 45

Figure 46, Figure 47 represent the effect of the probability of feedback p on Ls and on Lq on increasing the probability for feedback p. Ls,Lq were increased for the parameters Lq,Wq were increased for the parameters μb=20,μsb=20,μv=5,λ=0.2,θ=0.5,η=20,ϑ=10,α=0.8,α=0.2,b=0.1.Figure 46 Feedback (p) Vs Ls.

Figure 46

Figure 47 Feedback (p) Vs Lq.

Figure 47

6 Conclusions

In summary, the field of queueing theory has greatly benefited from this study, especially when it comes to arrival in bulk with balking along with working vacation. The current model comes close to most of the reality queueing framework, since it incorporates the ideas of two-phase mandatory service and repair as well as delaying maintenance. This paper's significant contributions come from its thorough examination of a complicated queuing system that takes into account these numerous variables, many of which are seen in real-world service systems. This study provides a more accurate and applicable depiction of service systems in real-world contexts, increasing the usefulness and accuracy of queueing theory. Furthermore, examining system performance indicators like throughput, waiting times, and reliability can offer important insights into the workings of intricate service systems. Through the integration of several modeling concepts, this research broadens the theoretical foundations of smart irrigation systems and offers a comprehensive framework for system design and optimization. The results of this study provide managers with practical advice on how to best manage service systems that deal with scenarios involving bulk arrival, two-phase service, starting failure, delayed repair, balking, rework, and working vacation.

Managers can apply measures to improve overall system efficiency, decrease waiting times, and improve service quality by understanding how these factors affect key performance indicators. The study's findings can be used by stakeholders in agriculture and water resource management to improve the layout and functionality of smart irrigation systems, which will increase agricultural output, resource efficiency, and water conservation. Additionally, our concept is well-suited to smart irrigation systems. The current research can be furthered by enhancing the model with the notions of two categories of customers, such as priority customers with regular customers, immediate feedback, and bulk service to make it more broad and efficient.

Research Limitations: A strong foundation for examining wait times and resource distribution is provided by the queueing theory. Mass arrival is perhaps not a true representation of real-world situations where arrivals are dispersed across time rather than in bulk. Two-phase service may necessitate sophisticated mathematical methods, which could restrict the findings' usefulness. Research on starting failures may fail to consider the interactions between starting failures and other elements of the system. Furthermore, real-world starting failure scenarios may differ and be more complex than those that are simulated. It is challenging to identify the best repair techniques inside the model when there is delayed repair.

Preconceptions about the behavior of balkers could not correspond with actual customer preferences and decision-making procedures. The model's accuracy may be hampered by the paucity of actual data on balking behavior. Working vacation draws generalizations about employee availability and productivity that might not hold in all situations. While simplifications, assumptions, and the difficulties of effectively modeling real-world dynamics may pose barriers to further research in this field, overall, this model offers insightful information about the intricacies of smart irrigation systems. However, because agricultural contexts are distinct, there are various obstacles to its direct application to smart irrigation systems.

Future Directions: Several directions for further investigation are revealed by building on the results of this study. Studying the best way to distribute resources in various operating scenarios like fluctuating arrival rates or service demands may provide insightful information on how to build and run service systems. In general, future studies should keep expanding our knowledge of queueing systems in intricate service settings, which will benefit both theoretical development and real-world implementation. In dynamic agricultural landscapes, investigating more sophisticated optimization methods and adaptive algorithms may improve the effectiveness and versatility of smart irrigation systems. Examining the possible integration of cutting-edge technologies, like the Internet of Things (IoT) and artificial intelligence (AI), may open up new possibilities for smart irrigation control and management innovation. To better capture the distinct dynamics of irrigation systems and agricultural ecosystems, future research should investigate model modifications or other options.

Additional information

No additional information was used.

Ethics statement

Review or approval by an ethics committee was not needed for this study because no data on patients or experimental animals was used in the article.

Informed consent was not required for this study because no clinical data was produced in the review article.

CRediT authorship contribution statement

Bharathy S: Writing – original draft, Software, Methodology, Investigation, Formal analysis, Conceptualization. Saravanarajan M.C.: Writing – review & editing, Visualization, Validation, Supervision.

Declaration of Competing Interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: MC Saravanarajan reports was provided by 10.13039/501100004728 Vellore Institute of Technology . MC Saravanarajan reports a relationship with Vellore Institute of Technology that includes: employment. MC Saravanarajan has patent pending to no. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A D″=(−2qbWˆv⁎(bλ`)C1¯)C1¯(1−Aˆ⁎(λ`))+α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))+(1+λ`bC1¯(E(Sˆf`)+E(Eˆf`)))α¯−1.D‴=3qbWˆv⁎(bλ`)(−2C1¯C2¯(1−Aˆ⁎(λ`))−(C2¯+2C1¯(1−Aˆ⁎(λ`)))(α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))+(1+λ`bC1¯(E(Sˆf`)+E(Eˆf`))))α¯−C1¯α(2pλ`bC1¯(E(Lˆb)+E(Rˆˆb))+(λ`bC1¯)2(E(Lˆb2)+E(Rˆˆb2))+2(λ`bC1¯)2E(Lˆb)E(Rˆˆb)+bλ`C2¯(E(Lˆb)+E(Rˆˆb)))−C1¯α¯(2bλ`C1¯(E(Sˆf`)+E(Eˆf`))+(λ`bC1¯)2(E(Sˆf`)+E(Eˆf`))+2(λ`bC1¯)2E(Eˆf`2)E(Sˆf`2)+λ`bC2¯(E(Sˆf`)+E(Eˆf`))).Nq″=2(α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))+(1+bC1¯λ`(E(Sˆf`)+E(Eˆf`)))α¯)(bqC1¯Wˆv⁎(aλ`)Aˆ⁎(λ`ˆ)−λ`bC1¯E(Wˆv))+2C1¯(1−Aˆ⁎(λ`ˆ))(−bqC1¯Wˆv⁎(aλ`)−λ`bC1¯E(Wˆv))+2C1¯bqWˆv⁎(bλ`)−bC1¯(1−Aˆ⁎(λ`ˆ))(2pλ`bC1¯E(Wˆv)+λ`bC1¯E(Wˆv)+(λ`bC1¯)2E(Wˆv))−bC1¯(1−Aˆ⁎(λ`ˆ))(p+λ`bC1¯E(Wˆv))+2λ`bC1¯E(Wˆv)−2(αλ`bC1¯(E(Lˆb)+E(Rˆˆb))+α¯λ`bC1¯(E(Eˆf`)+E(Sˆf`)))(C1¯(1−Aˆ⁎(λ`ˆ))qWˆv⁎(bλ`)−(p+λ`bC1¯E(Wˆv))−qWˆv⁎(bλ`)).Nq‴=3(bαλ`(E(Lˆb)+E(Rˆˆb))(2pC1¯+C2¯)+α(λ`bC1¯)2(E(Lˆb)+E(Rˆˆb))2+α¯bλ(E(Sˆf`)+E(Eˆf`))(2C1¯+C2¯)+(λ`bC1¯)2α¯(E(Sˆf`)+E(Eˆf`))2)(bqC1¯Wˆv⁎(bλ`)(1−Aˆ⁎(λ`))−λbC1¯(Wˆv⁎(bλ`)+E(Wˆv)))−3(α(p+λ`bC1¯(E(Lˆb)+E(Rˆˆb)))+α¯(1+λ`bC1¯(E(Sˆf`)+E(Eˆf`))))(2C2¯(1−Aˆ⁎(λ`ˆ))bC1¯(qWˆv⁎(aλ`)−λ`E(Wˆv))+bC2¯(qWˆv⁎(bλ`)−λ`E(Wˆv))+bC2¯(qWˆv⁎(bλ`)+λE(Wˆv))+(λbC1¯)2E(Wˆv2)+bqWˆv⁎(aλ`)(2C1¯+C2¯)(1−Aˆ⁎(λ`)))+3(α{λ`bC2¯(E(Lˆb)+E(Rˆˆb))+(λ`bC1¯)2(E(Lˆb)+E(Rˆˆb))2}+α¯{bλ`(E(Sˆf`)+E(Eˆf`))(2C1¯+C2¯)+(bλC1¯)2(E(Sˆf`)+E(Eˆf`))2})(p+bC1¯λE(Wˆv)+qWˆv⁎(bλ`)(1−C1¯(1−Aˆ⁎(λ`))))+3(λ`bC1¯((E(Lˆb)+E(Rˆˆb)))+α¯(E(Eˆf`)+E(Sˆf`)))(−qWˆv⁎(bλ`)C2¯(1−Aˆ⁎(λ`))+2C1¯(1−Aˆ⁎(λ`))(p+λ`bC1¯E(Wˆv))+λ`bE(Wˆv)(2pC1¯+C2¯)+(λ`bC1¯)2E(Wˆv2))−3(C1¯+C2¯)(1−Aˆ⁎(λ`))(qbC2¯Wˆv⁎(bλ`)+bλ`C2¯E(Wˆv)+(λ`bC1¯)2E(Wˆv2))+3bqC2¯Aˆ⁎(λ`)Wˆv⁎(bλ`)+3b(1−Aˆ⁎(λ`))(2C1¯+C2¯)(p+λ`bC1¯E(Wˆv))−3bC1¯(1−Aˆ⁎(λ`))(λ`bE(Wˆv)(2pC1¯+C2¯)+(λ`bC1¯)2E(Wˆv2))Ns″=2(αp+αλ`bC1¯(E(Lˆb)+E(Rˆˆb))+α¯+α¯λ`bC1¯(E(Sˆf`)+E(Eˆf`)))(−λ`bC1¯E(Wˆv)−C1¯bqWˆv⁎(aλ`)Aˆ⁎(λ`))−2C1¯(1−Aˆ⁎λ`)(qbC1¯Wˆv⁎(aλ`)+λ`bC1¯E(Wˆv))−2λ`bC1¯E(Wˆv)+2C1¯bqWˆv⁎(aλ`)+bqWˆv⁎(aλ`)Aˆ⁎(λ`)(1−C1¯)+2bC1¯(1−Aˆ⁎(λ`))(p+λ`bC1¯E(Wˆv))+4λ`bC1¯E(Wˆv)−2(αλ`bC1¯(E(Lˆb)+E(Rˆˆb))+α¯λ`bC1¯(E(Sˆf`)+E(Eˆf`)))(C1¯(1−Aˆ⁎(λ`))qWˆv⁎(aλ`)−(p+λ`bC1¯E(Wˆv))−qWˆv⁎(aλ`))

Ns‴=6αλ`bC1¯(E(Lˆb)+E(Rˆˆb))(λ`bC1¯E(Wˆv)(1−p)+C1¯qWˆv⁎(aλ`)Aˆ⁎(λ`)(1−pb)+p+qWˆv⁎(aλ`)(1−C1¯))+3λ`b(p+qWˆv⁎(aλ`)(1−bC1¯Aˆ⁎(λ`))−qC1¯Wˆv⁎(aλ`)(1−Aˆ⁎(λ`)))(λ`bC¯12(E(Lˆb)+E(Rˆˆb))+2λ`bC¯12E(Lˆb)E(Rˆˆb)+C2¯(E(Lˆb)+E(Rˆˆb))+(2C¯1+C2¯)(E(Sˆf`)+E(Eˆf`))+λ`bC¯1(E(Sˆf`)+E(Eˆf`))+2λ`bC¯12E(Eˆf`)E(Sˆf`))+3(α(p+λ`C¯1b(E(Lˆb)+E(Rˆˆb)))+α¯(1+λ`bC¯1(E(Sˆf`)+E(Eˆf`))))(−2C1¯λ`bE(Wˆv)(1−Aˆ⁎(λ`))−2bqC¯1Wˆv⁎(aλ`)Aˆ⁎(λ)(1−C¯1)−λ`bE(Wˆv)(2C¯1+C¯1)−(λ`bC¯1)2E(Wˆv2)−C¯2bqWˆv⁎(aλ`)Aˆ⁎(λ`))+3b(1−Aˆ⁎(λ`))(−2qC2¯Wˆv⁎(aλ`)−2λ`C1¯C2¯E(Wˆv)(1−b)−λ`2bC1¯3E(Wˆv2)(1−b)−2λ`C1¯E(Wˆv)(1−pb)+p(1+C1¯+C2¯))+3(−αλ`bC1¯(E(Lˆb)+E(Rˆˆb))−α¯λ`bC1¯)(E(Sˆf`)+E(Eˆf`))(C2¯qWˆv⁎(aλ`)(1−Aˆ⁎(λ`))−2C1¯(1−Aˆ⁎(λ`))(p+λ`bC1¯E(Wˆv)−λ`bC2¯E(Wˆv)−(λ`bC1¯)2E(Wˆv2)))

Table 5 Comparison table.

Table 5	Existing	Proposed Model	
Bulk Arrival	Ayyappan and Shyamala [6] investigated unreliable M[x]/G/1 queueing structure combining Bernoulli feedback vacations, and random setup time.	To improve the allocation of resources and examine system performance under varying arrival patterns, incorporate bulk arrival models.	
Meena et al. [7] analyzed bulk arrival system that consisted of both active and passive machines working jointly under N-policy. Minimum cost utilizing genetic algorithm and metaheuristic particle swarm optimization approaches were used.	Introduce bulk arrival distributions like batch Poisson. Examine the effects on system performance and queue dynamics.	
Malik et al. [8] brought about the G-queue for bulk arrival retrials, exposing the server to state-dependent and multi-optional services.	Take weather forecasts or agricultural water demand bulk arrival patterns into account. When predicting times of high water demand, use statistical models or historical data to optimize irrigation schedules.	
Niranjan and Indira [9] made a review on bulk queues accompanied by vacations to give the analysts, researchers, and businesspeople the knowledge they need to model traffic issues and determine the best performance metrics for queueing structure.		
	
Two-Phase Service	The server offers every client two crucial phase services in continuation: opening FPS, the next phase of service. Madan [18] conducted extensive research on the bulk arrival model in which two different phases of services.		
Choudhury et al. [19] examined the behavior of the M[X]/G/1 model by providing some optional services for SPS, interruption in service, and random failure while providing services to the clients.	To analyze workload balance, service time distributions, and system performance metrics, queueing models should be designed with discrete phases.	
Bhagat et al. [20] discussed bulk arrival queueing systems by including general retrial times.	Provide models that specifically take into account two different service phases with different parameters. Examine phase relationships, best practices for sequencing, and methods for allocating resources.	
Xu et al. [21] collision integrated retrial queuing system providing heterogeneous service in two phases with delays in vacation.	Provide two phases of service for smart irrigation systems: the data collecting phase (such as measuring soil moisture) and the irrigation phase (such as water distribution).	
Rajadurai, et al. [22] framed a model by including feedback and negative customers arrival.		
Ayyappan et al. [23] analyzed bulk arrival using a server with standby and two-phase service in a heterogeneous manner provided by a repairable single server, failure at the start, and many vacations.		
Maheshwari et al. [24] looked into a single server retry queue that had two service phases, the second of which was optional with K different kinds of Bernoulli vacations.		
Srivastava et al. [25] worked on a bulk arrival model consisting of two types of services along with multiple vacations.		
	
Starting Failure	Jain et al. [10] described the model with common users and primitive users in cognitive radio systems along with rework and vacation.	System dependability, downtime, and the best maintenance strategies can be evaluated by incorporating initial failure probability and repair times into queueing models.	
Singh et al. [11] analyzed a bulk arrival repairable single server queueing system involving starting failure, repair, and delay to repair.	Examine how watering decisions are affected when sensors malfunction. Create fault detection algorithms or redundancy measures to guarantee reliable data collection for the best irrigation scheduling and to lessen the impact of starting failures.	
Karpagam et al. [12] highlight bulk retrial queue that includes a potential for failure at the start, repair, vacation, also additional service to faulty batches.		
	
Delayed Repair	Singh et al. [14], Jain and Bhagat [15] have studied a model on bulk arrival retrial queues that incorporates the ideas on delay in repairs.	Consider repair time distributions, spare component availability, and economical maintenance techniques when modeling queueing systems with delayed repair procedures.	
	
Balking	Sundarapandiyan and Nandhini [16] and Rajadurai et al. [17] made an investigation on M[X]/G/1 queues accompanied by the presumption that customers could object to the system by having two different kinds of service, undergoing revised vacation policy.	To assess the effect of balking behavior on waiting times, revenue, and service quality, include it in queueing models and develop solutions to reduce the incidence of balking.	
Rework	Jain, Madhu, and Sandeep Kaur [26], Ayyappan and Arulmozhi [27], and Rajadurai et al. [28] made significant investigations into feedback retrial queues.	Rework procedures should be included in queueing models. Their impact on system performance, quality, and resource usage should be assessed, and solutions to lower the frequency of rework should be suggested. Examine the effects of adding rework procedures to queueing models on system throughput, effectiveness, and quality. Examine methods for lowering the rate of rework, locating the source of the problem, and enhancing process dependability. Examine the effects of reworking irrigation decisions on agricultural yield results. Use adaptive control algorithms or feedback systems to reduce the need for rework and continuously enhance irrigation management techniques.	
	
Working Vacation	Servi et al. [29], and Jain et al. [30], have established a new class of semi-vacation policies.	To balance service continuity, server rest, and operational efficiency, queueing models with working vacation intervals for servers should be developed.	
Dhibar and Jain [31] investigated the working vacation along with the users' dissatisfaction actions and servers prone to disasters incorporated into the Markovian retrial queueing model.	Examine how workforce numbers and resource availability affect irrigation system upkeep. Include working vacation times in queueing models and optimize schedules to strike a balance between system performance and server relaxation. Examine the impact on operational effectiveness, consumer wait times, and service dependability.	
Gao et al. [32] investigated a model with batch input that includes the policy of jumbled J working vacation.	Identify methods for effectively managing irrigation professionals' working vacations so that timely maintenance and assistance are provided for continuous smart irrigation operations.	

Data availability

Not applicable.
==== Refs
References

1 Falin G. Templeton J.G. Retrial Queues, vol. 75 1997 CRC Press
2 Artalejo J.R. Gómez-Corral A. Retrial queueing systems Math. Comput. Model. 30 1999 3 4 10.1007/978-3-540-78725-9
3 Falin G. A survey of retrial queues Queueing Syst. 7 1990 127 167 10.1007/BF01158472
4 Singh Sadhna Srivastava R.K. Markovian queuing system for bulk arrival and retrial attempts with multiple vacation policy IJMTT 67 2021 21 28 10.14445/22315373/IJMTT-V67I1P504
5 Jain M. Bhargava Charu Bulk arrival retrial queue with unreliable server and priority subscribers Int. J. Oper. Res. 5 2008 242 259
6 Ayyappan G. Shyamala S. Transient solution of an M [X]/G/1 queueing model with feedback, random breakdowns, Bernoulli schedule server vacation and random setup time Int. J. Oper. Res. 25 2016 196 211 10.1504/IJOR.2016.073956
7 Meena R.K. Jain M. Assad A. Sethi R. Garg D. Performance and cost comparative analysis for M/G/1 repairable machining system with N-policy vacation Math. Comput. Simul. 200 2022 315 328 10.1016/j.matcom.2022.04.012
8 Malik G. Upadhyaya S. Sharma R. Particle swarm optimization and maximum entropy results for mx/g/1 retrial g-queue with delayed repair Int. J. Math. Eng. Manag. Sci. 6 2021 541
9 Niranjan S.P. Indhira K. A review on classical bulk arrival and batch service queueing models Int. J. Pure Appl. Math. 106 2016 45 51 10.12732/ijpam.v106i8.7
10 Wang Jinting Zhou Peng-Feng Batch arrival retrial queue with starting failures, feedback and admission control J. Syst. Sci. Syst. Eng. 19 2010 306 320 10.1007/s11518-010-5140-z
11 Singh Charan Jeet Jain Madhu Kaur Sandeep Performance analysis of bulk arrival queue with balking, optional service, delayed repair and multi-phase repair Ain Shams Eng. J. 9 2018 2067 2077 10.1016/j.asej.2016.08.025
12 Karpagam S. Ayyappan G. Somasundaram B. A bulk queueing system with rework in manufacturing industry with starting failure and single vacation Int. J. Appl. Math. Comput. 6 2020 174 10.1007/s40819-020-00922-2
13 Madheswari S. Pavai S. Suganthi P. Retrial queue with unreliable server and second optional service under K types of Bernoulli vacations Int. J. Serv. Oper. Manag. 40 4 2021 502 533 10.1504/IJSOM.2021.120058
14 Singh C.J. Jain M. Kaur S. Performance analysis of bulk arrival queue with balking, optional service, delayed repair and multi-phase repair Ain Shams Eng. J. 9 2018 2067 2077 10.1016/j.asej.2016.08.025
15 Jain M. Bhagat A. Unreliable bulk retrial queues with delayed repairs and modified vacation policy J. Ind. Eng. Int. 10 2014 1 9 10.1007/s40092-014-0063-9
16 Sundarapandiyan S. Nandhini S. Non-Markovian feedback retrial queue with two types of customers and delayed repair under Bernoulli working vacation Contemp. Math. 2024 2093 2122 10.37256/cm.5220243940
17 Rajadurai P. Saravanarajan M.C. Chandrasekaran V.M. Analysis of an M [X]/(G1, G2)/1 retrial queueing system with balking, optional re-service under modified vacation policy and service interruption Ain Shams Eng. J. 5 2014 935 950 10.1016/j.asej.2014.02.003
18 Harini R. Indhira K. Dynamical modelling and cost optimization of a 5G base station for energy conservation using feedback retrial queue with sleeping strategy Telecommun. Syst. 2024 1 30 10.1007/s11235-024-01155-0
19 Choudhury Gautam Tadj Lotfi Deka Kandarpa A batch arrival retrial queueing system with two phases of service and service interruption Comput. Math. Appl. 59 2010 437 450 10.1016/j.camwa.2009.06.021
20 Bhagat Amita Jain Madhu Unreliable MX/G/1 retrial queue with multi-optional services and impatient customers Int. J. Oper. Res. 17 2013 248 273 10.1504/IJOR.2013.053614
21 Xu Wei Li Linhong Fan Wentao Liu Liwei Optimal control of a two-phase heterogeneous service retrial queueing system with collisions and delayed vacations J. Appl. Math. Comput. 2024 1 28 10.1007/s12190-024-02074-8
22 Rajadurai P. Chandrasekaran V.M. Saravanarajan M.C. Steady state analysis of batch arrival feedback retrial queue with two phases of service, negative customers, Bernoulli vacation and server breakdown Int. J. Math. Oper. 7 2015 519 546 10.1504/IJMOR.2015.071276
23 Ayyappan G. Nirmala M. Karpagam S. Analysis of repairable single server bulk queue with standby server, two phase heterogeneous service, starting failure and multiple vacation Int. J. Appl. Math. Comput. 6 2020 1 22 10.1007/s40819-020-00805-6
24 Madheswari S. Pavai Suganthi P. Retrial queue with unreliable server and second optional service under K types of Bernoulli vacations Int. J. Serv. Oper. Manag. 40 2021 502 533 10.1504/IJSOM.2021.120058
25 Srivastava R.K. Singh Sadhna Singh Amendra Bulk arrival Markovian queueing system with two types of services and multiple vacations Int. J. Math. Comput. Appl. 8 2020 2130 2136 10.47191/ijmcr/v8i8.04
26 Jain Madhu Kaur Sandeep Bernoulli vacation model for MX/G/1 unreliable server retrial queue with Bernoulli feedback, balking and optional service RAIRO Oper. Res. 55 2021 S2027 S2053 10.1051/ro/2020074
27 Ayyappan G. Arulmozhi N. Analysis of M, MAP/PH1, PH2/1 non-preemptive priority Queueing model with Delayed working vacations, immediate feedback, impatient customer, differentiate breakdown and phase type repair Reliab. Theory Appl. 18.4 76 2023 771 790
28 Rajadurai P. Saravanarajan M.C. Chandrasekaran V.M. A study on M/G/1 feedback retrial queue with subject to server breakdown and repair under multiple working vacation policy Alex. Eng. J. 57 2018 947 962 10.1016/j.aej.2017.01.002
29 Servi D. Leslie Finn Steven G. M/M/1 queues with working vacations (m/m/1/wv) Perform. Eval. 50 2002 41 52 10.1016/S0166-5316(02)00057-3
30 Jain Madhu Jain Anamika Working vacations queueing model with multiple types of server breakdowns Appl. Math. Model. 34 2010 1 13 10.1016/j.apm.2009.03.019
31 Dhibar S. Jain M. Particle swarm optimization and FM/FM/1/WV retrial queues with catastrophes: application to cloud storage J. Supercomput. 3 Apr 2024 1 35 10.1007/s11227-024-06068-y
32 Gao Shan Yao Yunfei An MX/G/1 queue with randomized working vacations and at most J vacations Int. J. Comput. Math. 91 2014 368 383 10.1080/00207160.2013.790964
33 Pakes Anthony G. Some conditions for ergodicity and recurrence of Markov chains Oper. Res. 17 1969 1058 1061 10.1287/opre.17.6.1058
34 Sennott L.I. Humblet P.A. Tweedie R.L. Mean drifts and the non-ergodicity of Markov chains Oper. Res. 31 1983 783 789 10.1287/opre.31.4.783
35 Fuhrmann W. Steve Cooper Robert B. Stochastic decompositions in the M/G/1 queue with generalized vacations Oper. Res. 33 1985 1117 1129 10.1287/opre.33.5.1117
36 Gowsalya M. Arivudainambi D. Stochastic analysis of an M/G/1 retrial queue subject to working vacation and starting failure AIP Conference Proceedings 2095, vol. 1 2019 AIP Publishing LLC 030009
37 Arivudainambi D. Godhandaraman P. Rajadurai P. Performance analysis of a single server retrial queue with working vacation Oper. Res. 51 2014 434 462 10.1007/s12597-013-0154-1
38 Zhang M. Hou Z. M/G/1 queue with single working vacation J. Appl. Math. Comput. 39 2012 221 234 10.1007/s12190-011-0532-x
39 Shortle J.F. Thompson J.M. Gross D. Harris C.M. Fundamentals of Queueing Theory, vol. 399 2018 John Wiley and Sons
