HETEROGENEOUS SERVER RETRIAL QUEUEING MODEL WITH FEEDBACK AND WORKING VACATION USING ARTIFICIAL BEE COLONY OPTIMIZATION ALGORITHM

: This research delves into the dynamics of a retrial queueing system featuring heterogeneous servers with intermittent availability, incorporating feedback and working vacation mechanisms. Employing a matrix geometric approach, this study establishes the steady-state probability distribution for the queue size in this complex heterogeneous service model. Additionally, a range of system performance metrics is developed, alongside the formulation of a cost function to evaluate decision variable optimization within the service system. The Artiﬁcial Bee Colony (ABC) optimization algorithm is harnessed to determine service rates that minimize the overall cost. This work includes numerical examples and sensitivity analyses to validate the model’s eﬀectiveness. Also, a comparison between the numerical ﬁndings and the neuro-fuzzy results has been examined by the adaptive neuro fuzzy interface system (ANFIS).


Introduction
In our modern, fast-paced society, it is crucial to prioritize the optimization of service systems due to the ever-changing needs and demands of diverse customers.Although traditional queueing models are valuable, they often fail to address the complexities of modern service environments.This research presents a queueing model that effectively addresses these challenges.This model fundamentally recognizes that service tasks can vary in nature and importance.It acknowledges the significance of selecting the appropriate service provider for a particular task.The concept of 'heterogeneous servers' is relevant here.Some servers specialize in handling routine requests, while others are particularly skilled at addressing complex issues.
Furthermore, we recognize that servers are not able to be available around the clock.Instead, they alternate between performing routine tasks and dedicating their attention to more specialized, secondary jobs.Intermittent availability optimizes resource allocation by ensuring that highly skilled servers are readily available when they are most needed.Finally, we have implemented a "working vacation" feature to guarantee uninterrupted service during periods of downtime.This feature ensures that customers are not left unattended, minimizing disruptions in service even when servers are on a break.This research goes beyond being a mere theoretical innovation; it tackles the actual challenges that service industries encounter in the real world.Our model provides a practical and competitive advantage in the dynamic landscape of modern service provision by enhancing efficiency, boosting customer satisfaction, and optimizing resource utilization.
The novelty of this work lies in its pioneering approach to designing service systems that can adapt to the multifaceted demands of contemporary industries.Unlike traditional queueing models, which rely on uniform servers and predictable service patterns, our model introduces heterogeneity, recognizing that not all service tasks are equal, which is shown in Fig. 1.
The remaining sections of this research paper: Section 2 outlines the model's development and the quasi-birth-death process framework.Moving on to Section 3, we delve into the matrix geometric approach, demonstrating the process of compute steady-state probabilities.In Section 4, we explore various performance metrics derived from the model and their practical implications.Section 5 is dedicated to a comprehensive discussion of sensitivity analysis and a cost assessment for the considered paradigm.Section 6 offers graphical representations of ANFIS and presents the numerical outcomes.The subsequent Section 7, delves into cost optimization strategies.Finally, Section 8 serves as the conclusion, where we summarize our investigation by highlighting the notable characteristics and real-world applications of our study.

Survey of literature
Our model incorporates two types of servers: one that handles routine tasks (server 1) and another that periodically shifts (server 2) its focus to secondary, more specialized tasks.The presence of heterogeneity enables a service delivery that is more customized and effective.In this heterogeneous queueing model, the servers provide service at a different rate.Morse [14] was the first to propose the notion of service heterogeneity.A queueing model with two classes and two servers is being discussed.A non-preemptive priority structure that is heterogeneous has been studied by Leemans [12].According to [3], a heterogeneous two-server queueing system with feedback, reverse balking, and reneging and retaining renege customers can be analyzed.Markovian queueing model with discouraged arrivals, reneging customers, and retention of reneged customers was studied by [11] based on two heterogeneous servers finite capacities.A study presents an investigation of the heterogeneous queueing system M/M/2 with two types of server failures and catastrophes, along with their respective restoration processes, as conducted by the [16].A queueing model with MAP arrivals and heterogeneous phase-type group services was researched by [4].
Agarwal [1] initially introduced the concept of a server with intermittent availability, where server 1 is consistently accessible while server 2 is periodically accessible.In this scenario, server 2 is responsible for executing a range of peculiar and unconventional tasks.Service interruptions may occur for a variable duration, but they are limited to instances when the ongoing task has been completed.This particular service is referred to as an intermittently available service.Sharda [19] investigated a queuing issue involving a server that is intermittently accessible, with entries and exits occurring in batches of varying sizes.
In recent times, queueing systems featuring server vacations have become increasingly popular.These "vacations" can arise from server outages or when the server is tasked with other responsibilities.Our model acknowledges the critical importance of maintaining continuous service, even during these working vacation periods.It is designed to ensure that customers are never left unattended, thus minimizing any disruptions in the quality of service provided.A recent trend in vacation queues has been working vacation, where service is provided at a lower rate during vacation periods than it is normally provided; i.e., while on vacation, the server provides service at a slower rate instead of ceasing completely.An initial proposal for a working vacation model has been made by Servi and Finn [13].Madhu Jain [7] conducted a study on a single server working vacation queueing model that incorporates multiple types of server breakdowns.Sudhesh et al. [21] investigated the time-dependent dynamics of a single server queueing model featuring slow service.The researchers examined the effects of both single and multiple working vacations, as well as customers' impatience during periods of slow service.Krishnamoorthy et al. [9] discussed a queueing system with two heterogeneous servers.One server is always accessible, while the other takes vacations when no users are waiting.Laxmi et al. [23] examined a queuing system with several working vacations, incorporating elements of renewal input, balking, reneging, and heterogeneous servers.Two types of Working Vacations (WVs) and impatient clients were handled with in a multi-server queueing system by Yohapriyadharsini et al. [25].Kumar et al. [10] examined a unreliable Markovian queueing model with two stage service, incorporating hybrid vacation.
The Matrix Geometric Method is a key technique in queueing theory.It simplifies the analysis of systems with varying service rates and transitions by using matrices, providing efficient solutions for steady-state probabilities and performance metrics.Initially introduced by Neuts [8], matrix-geometric models are the basis for stochastic computations.Recently, Divya [5] conducts an investigation on a Markovian queueing model that incorporates heterogeneous, intermittently available servers with feedback, operating under a hybrid vacation policy.ANFIS is an Adaptive Neuro Fuzzy Inference System.This ANFIS computer model uses fuzzy logic and neural networks to analyze and make decisions.ANFIS is famous for tackling complicated issues in numerous industries because it provides a foundation for building hybrid systems that can learn and adapt from data.ANFIS was established in the early 1980s by Professor Lotfi A. Zadeh [26].Ahuja et al. [2] presented a comprehensive analysis of a single server queueing model with multiple stage service and functioning vacation, focusing on transient behavior.Additionally, they employed ANFIS computing techniques to enhance their analysis.Sethi et al. [17] conducted a study on the application of ANFIS in analyzing the performance of an unreliable M/M/1 queueing system.The study specifically focused on the impact of customers' impatience under N-policy.The ANFIS concept has garnered attention from a multitude of researchers in diverse fields of study [6], [18], [20], [22].Wu and Yang [24] conducted optimization of a bi-objective queueing model that incorporates a two-phase heterogeneous service.1. Arrival process.In the RQ system, customers arrive according to a Poisson process with a rate of ω.

Model description and assumptions
3. Retrial process.The retrial queuing mechanism enables customers to opt to orbit in the event that the servers are occupied upon their arrival.After a constant retrial rate φ, individuals may make another attempt to receive service following an exponentially distributed time.
4. Vacation process.A queuing system with working vacation is analyzed, If there are no customers in the orbits when the server 2 finishes servicing, it goes on a working vacation, where the server 2 serves customers with a reduced service rate γ v during such periods, which follow an exponential distribution.When a vacation ends and there are customers waiting for service, the server 2 switches to regular service with retrieval rate τ .If there are no customers waiting, the server 2 retains in the same working vacation.
5. Feedback rule.During WV there are two possible outcomes for customers receiving service during working vacation: they may receive satisfactory service with probability p or unsatisfactory service with complementary probability p(1 − p).In the event of unsatisfactory service, customers must undergo supplementary service, which follows an exponential distribution.
All stochastic processes in the system are independent of one another.The structure of the models transition diagram is depicted in the below Fig. 2. At time t, let χ(t) be the state of the server, which is defined as , the server 2 is in busy, 3, the server 2 is in intermittently obtainable and φ(t) be the number of customers in the system.The bi-variate process {(φ(t), χ(t)), t ≥ 0} that operates on a state space of {0, 1, 2, . . .} × {0, 1, 2, 3}.
The state space of a Markov process is arranged in a lexicographical manner, as described below.

Steady-state equation
To solve this problem and obtain effective and mathematically accurate model solutions, we employ the matrix-geometric method described in the following section.The matrix-geometric method is an effective method for obtaining steady-state probabilities when the state-space expands very quickly. ) )

Matrix-geometric solution
To calculate the steady-state probabilities of the model using the matrix-geometric approach, we utilize a system of equations denoted as (2.1) to (2.8).These equations help determine the probabilities at a steady state.The transition rate matrix Q, which represents the Markov chain in this model, is structured as a block tridiagonal matrix.The matrix Q is subdivided into sub matrices.

Stability criteria
Theorem 1.The inequality is the necessary and sufficient condition for the system to be stable.
P r o o f.Let us define the matrix E = T 0 + V 0 + U 0 given by Where where e = [1, 1, 1, 1] T .Using Theorem 3.1.1 of Netus [8], the necessary and sufficient condition for the stability of the system is as follows: ΠT 0 e < ΠU 0 e.
(3.4) Equation (3.4) can be written as Based on the matrix-geometric method [8,15], we obtain and Π 0 satisfies the set of equations Where R is referred to as the rate matrix which satisfies and 0 denotes a zero squared matrix of an appropriate order.
The rate matrix can be approximate iteratively by considering one sequence with initialization R 0 = 0 and calculating Thus, lim i→∞ R i is an approximate solution of the rate matrix R. From the normalization condition, we obtain the following: Combining (3.6) with the normalization condition yields Once the steady-state probability vector Π 0 is available, then Π i (i ≥ 1) can be determined using (3.5).

Performance measures
Based on the steady-state probabilities, we give numerous performance metrics for the model under evaluation.
• Prob that the servers are in idle • Prob that the server is in busy state • Prob that the server 2 is in WV state • Prob that the server 2 is in IO state π l,3 .
• Average system length

Practical application
The proposed queueing model is applicable to a semi-attended self-checkout system in retail stores.
Server 1 -Always Available (Self-Checkout Machine): This server is designed to operate continuously and is accessible to customers at all times, similar to the self-checkout machines commonly found in retail shops.Customers have the freedom to use it at any time without any interruptions.
Server 2, also known as the Intermittently Obtainable (Human Billing Counter), functions similarly to a human cashier at a retail store.The service is available intermittently, which means that it serves customers but may take breaks or go on working vacations.A working vacation refers to planned breaks or vacations for server 2, during which it is temporarily unavailable to serve customers.For instance, a cashier may take a lunch break or have a scheduled time off during their shift.There are numerous systems similar to the queueing model, such as call centers, healthcare triage, banking services, online customer support, and restaurant service.
All values assigned to the system parameters in our numerical analysis satisfy the stability condition as described by (3.3).The curves depicting the performance measures against the parameters are illustrated in Fig. 3 to 5. Table 1 presents numerical results showing that increasing arrival rate (ω) corresponds to increasing system size (ASL), queue size (AQL) and server busy state probability (P b ).leads to, decreases the probabilities of other states.Similarly, as we increase the service rates (γ 1 and γ 2 ), the server busy probability (P b ), queue size (AQL), and system size (ASL) decreases, the probabilities of other states increase.

ANFIS implementation and results
An Adaptive Neuro-Fuzzy Inference System (ANFIS) is a computational model that combines the principles of neural networks and fuzzy logic to perform complex tasks, such as pattern recognition and system modeling.ANFIS utilizes a hybrid approach that blends the adaptability of neural    networks with the interpretability of fuzzy logic.It consists of a layered architecture where input data is passed through a series of nodes, each representing a fuzzy membership function.These nodes calculate membership values based on the input data's similarity to predefined linguistic terms.ANFIS learns and adjusts its parameters using a combination of gradient descent and leastsquares methods.This enables it to fine-tune the strengths of its fuzzy rules and the connection weights between nodes to accurately model intricate relationships within the data.The model is particularly useful when dealing with non-linear and uncertain data, making it suitable for applications in various fields, including control systems, prediction, and optimization.By incorporating both neural networks and fuzzy logic, ANFIS provides a balance between the strengths of both approaches, offering a powerful tool for researchers and practitioners to tackle complex problems effectively.ANFIS was founded in the first few years of the 1980s by Professor Lotfi A. Zadeh [26].The architecture of a two-input (p, ω, γ 1 , γ 2 ), one-output (ASL) Adaptive Neuro-Fuzzy Inference System (ANFIS) model with five rules is illustrated in Fig. 7. Five fuzzy rules were created and the resulting Gaussian-shaped membership functions (MFs) of the inputs are displayed in Fig. 6.It is noteworthy to emphasize that each colored MF depicted in the curve represents a distinct cluster inside the input space.In ANFIS methodology, the parameters p, ω, γ 1 , γ 2 and ASL are regarded as linguistic variables and subjected to training for a total of ten epochs.Three linguistic values have been utilized for the variables p, ω, γ 1 , γ 2 and ASL, namely very low, low, medium, high, and very high.Gaussian membership functions were employed to represent the linguistic variables, as illustrated in Table 2.
Fig. 8 displays the analytical results using continuous lines, while the results acquired using ANFIS for ω, γ 1 , and γ 2 are shown by a dotted marker point.Furthermore, it was observed that the analytical and ANFIS outcomes had a high degree of concurrence, displaying a significant overlap in their respective trends.Based on the data presented, it can be noted that there is a positive correlation between the system length (ASL) and the arrival rate (ω) while considering different values of p.As the arrival rate increases, the system length also increases.Conversely, the system length lowers as the service rates (γ 1 and γ 2 ) decrease, while considering different values of p.

Cost model and optimization
The suggested queueing model has the potential to be implemented in both self-checkout systems and retail stores with human cashiers.In such scenarios, the primary objective of the manager is to minimize operational expenses.An essential consideration related to the self-checkout process pertains to the determination of the appropriate quantity of self-checkout machines for the service.Another crucial matter to consider is the necessity of upholding a satisfactory service rate in order to ensure the quality of service and customer satisfaction.It is feasible to improve the calibre of service rendered by the staff through comprehensive training.Moreover, it is not possible.To ascertain the customer's familiarity with the operation of a particular system.Self-checkout machines.Therefore, we proceed to formulate a cost function per unit of time in a manner that aligns with our expectations.
The variables in the equation are defined as follows: F h represents the holding cost for each customer in the system, F v represents the cost per unit of time when the server 2 in the working vacation service, F b represents the cost per unit of time when server 2 is busy, F i represents the cost per unit of time when server 2 is intermittently obtainable, F 1 represents the cost of providing a mean service rate γ 1 through server 1, and F 2 represents the cost of providing a mean service rate γ 2 through server 2. It is noteworthy to mention that (7.1) represents a mathematical function that depends on two continuous decision variables, denoted as γ 1 and γ 2 .We set the cost elements as given in Table 3.ABC optimization, also known as Artificial Bee Colony optimization, is a meta-heuristic algorithm inspired by the foraging behavior of honey bees.It is a swarm-based optimization technique that can be applied to solve various optimization problems.The ABC optimization technique, initially introduced by Karaboga in 2005, has garnered significant recognition and acclaim in the field of optimization.The system employs worker bees, observer bees, and scout bees to explore the search space, exchange information, and discover improved solutions.The historical effect in ABC ensures that the algorithm strategically priorities regions of the search space that have demonstrated favorable outcomes, thus leveraging past successes.This enables ABC to efficiently converge towards optimal solutions and effectively address complex optimization problems.
In order to optimize ABC, the following default values are taken into account: ω = 3, γ 1 = 1.5, γ 2 = 0.6, γ v = 0.5, φ = 0.05, τ = 0.5, β = 0.5, p = 0.4, p 1 = 0.6 with a colony size of 100, a maximum of 100 iterations, an acceleration coefficient upper and lower bound are 1 and 5, and number of Onlooker bee 50, an abandonment limit parameter of 60.Table 4 show the effect of cost elements F h , F b , F v , F i , F 1 and F 2 on the optimal service rates (γ * 1 , γ * 2 ) and optimal total cost (F * ) for all two cost sets.The pseudo code of ABC algorithm is given in Table 1.

Convergence
Convergence is a crucial aspect of meta-heuristic optimization algorithms.It signifies the process of gradually refining the candidate solutions toward an optimal or near-optimal solution.The convergence behavior of an algorithm is indicative of its ability to effectively search the solution space and approach the global optimum.In ABC, particles move towards the best-known solution, converging when their movements become limited and the best solution stabilizes.Fig. 9 (a) and (c) shows that ABC reach optimal cost convergence, Fig. 9 (b) and (d) shows the convexity and optimal of the cost function with respect to cost sets which are considered in the optimization analysis.

Conclusion
In this research has tackled the complexities of a retrial queueing system with heterogeneous servers, intermittent availability, feedback, and working vacation mechanisms.Employing a matrix geometric approach, we established the steady-state probability distribution and formulated performance metrics and a cost function.Leveraging the Artificial Bee Colony optimization algorithm, we optimized service rates effectively.Furthermore, we compared our numerical findings with ANFIS results, highlighting the potential synergy between traditional methods and advanced machine learning approaches in queueing theory research.In future, it is possible to expand the proposed model to include additional factors such as different server vacations, server breakdowns, and customer impatience.

Figure 2 .
Figure 2. Transition diagram of the model.

Figure 3 .
Figure 3.The curves depicting the performance measures against the parameter ω.

Figure 4 .Figure 5 .
Figure 4.The curves depicting the performance measures against the parameter γ 1 .

Figure 8 .
Figure 8.Comparison of 2D numerical and ANFIS values for different values of p.

Table 2 .
Values of the MF for the linguistics based on input parameters.

Table 3 .
Cost set values for various cost aspects.

Table 4 .
The estimated optimal solutions (γ * 1 , γ * 2 ) and their corresponding expected cost F * (ω, γ 1 , γ 2 ) γ * Algorithm 1 Artificial Bee Colony Input: Objective function F (γ 1 , γ 2 ), Maximum number of iterations Output: The best solution found Initialization; Initialize employed bees with random solutions; Evaluate the fitness of each solution; Set the best solution as the solution with the best fitness; while Termination condition not met do for each employed bee do Select a solution randomly from the population; Generate a new solution by modifying the selected solution; Evaluate the fitness of the new solution; If the new solution is better, replace the old solution; end for Update the best solution if a better solution is found; end while return The bestsolution found;