This paper examines a finite-capacity machine repair system with M parallel operating machines, W warm standby machines, and C cold standby machines. It incorporates retrial and Bernoulli feedback queueing characteristics to investigate reliability and optimization issues. The system is equipped with an unreliable repairman responsible for repairing failed machines. During operation, the repairman may experience working breakdown, and provides repair services at a lower rate during the failure. Firstly, based on Markov process theory, matrix analysis method, and Cramer’s rule, transient and steady-state analyses are conducted on the system. By solving the balance equations in matrix form, the queueing and reliability indicators of the system are obtained. Furthermore, the closed form solution for the system’s transient probabilities is derived using the Laplace transform and the eigenvalue method. Subsequently, numerical experiments and numerical simulations are presented to illustrate the effects of system parameters on performance indicators, followed by a sensitivity analysis of reliability measures. Finally, from the decision-maker’s perspective, a bi-objective optimization model is formulated to maximize steady-state availability and minimize total cost, using the Non-dominated Sorting Genetic Algorithm II (NSGA-II) and Multi-Objective Particle Swarm Optimization (MOPSO) algorithms to seek the Pareto fronts. By using Bootstrap method combined with similarity measurement, the stability of the obtained Pareto solution sets are evaluated to ensure the robustness and feasibility of optimization results. The results indicate that both algorithms can achieve effective optimization outcomes, but NSGA-II outperforms MOPSO in terms of optimization performance and solution set stability, exhibiting superior robustness and reliability, thereby providing a theoretical reference for decision-making in practical maintenance systems.

This paper appears to investigate the optimal joining strategies of arriving customers under different scenarios in the single-server Markovian queueing systems with three types of retrial orbits: standard, deluxe, and business class. Customers who arrive at the system and find the server busy can join one of the retrial orbits based on their preference and paying capacity. The standard orbit is assumed to be the cheapest, and customers who want to pay less would choose this option. Customers whose paying capacity is higher would join the deluxe orbit, while those who can afford the high cost would join the business-class orbit. The probability-generating function technique is applied to solve the steady-state difference equations and derive the steady-state probabilities and other important system metrics for a probability distribution. The cost-reward function is framed and analyzed based on the joining strategies of the arriving customers. The arriving customers would decide whether to join the queue or not based on the cost-reward structure. The cost-reward function may reflect the trade-off between the profit for those joining the queue and the waiting cost in the queue. The numerical experiment is also done by taking a numerical example and comparing the results for the fully unobservable, partially unobservable, partially observable, and fully observable. The development of equilibrium and social strategies enabled to make a detailed study on the behavior of customers subject to profit. This study can provide insights into the behavior of customers in queuing systems and inform the design and management of such systems in various applications, such as service operations, transportation, manufacturing, or production, etc.

Queueing systems with retrial and unreliable server are common in communication and manufacturing environments, where service interruptions significantly affect performance. This study analyzes an M 1 , M 2 / G 1 , G 2 / 1 retrial queue in which the server is prone to random failures while serving both primary and secondary customers. With the help of differential–difference equations, we determine probability generating functions and derive the orbit length distribution at random epoch using the roots of the corresponding characteristic equation. Furthermore, the orbit length distribution at post-departure epoch is derived using the Poisson Arrivals See Time Averages principle, which eliminates the need to construct a transition probability matrix. Key performance measures are evaluated under various parameter settings. An additional contribution of this study is a comprehensive cost analysis aimed to optimise the overall system cost. The findings provide useful insights for designing reliable service systems with retrial behaviour and random server failures.

Under the classical retrial policy, we consider a single-server M/G/1 queue with multiple input streams and orbits. Different types of customers have corresponding arrival rates, general distributions of service time and retrial rates. Assume that the retrial rates for different types of customers linearly converge to zero. We firstly derive the first-order asymptotics of the orbit queue lengths. Subsequently, we find that the joint asymptotic distribution of the number of retrials follows a multidimensional geometric distribution. Finally, we obtain the joint asymptotic distribution of waiting times, which follows a multidimensional exponential distribution. This result indicates that the waiting times for different types of customers are independent of each other.

The dynamic behavior of the retrial queueing system following the incorporation of Deep Q-Network Reinforcement Learning in 6G mobile communication services is examined in this study. The proposed method lies in analyzing the DQN-RL agent’s learning convergence by using the first- and second-order Markov chain method. By simulating the temporal evolution of reward sequences as Markov and second-order Markov chains, we can quantify convergence characteristics through mixing time analysis. To capture a wide operational landscape, a thorough simulation framework with 120 independent parameter combinations is created. The obtained results indicate that Markov chain analysis confirms 10 training episodes are more than sufficient for policy convergence, and in some cases, as few as 5 episodes allow the agent to enhance the mobile network performance while maintaining low energy consumption. To assess learning stability and system responsiveness, the mixing time of DQN RL rewards is calculated for every episode and configuration. A deeper understanding of the temporal dependencies in the reward process can be gained by incorporating higher-order Markov models. This paper concentrates on studying the learning convergence using an analysis of the Markov model’s spectral gap properties as an indicator. The results provide a rigorous foundation for optimizing 6G queueing strategies under uncertainty by highlighting the sensitivity of DQN convergence to system parameters and retrial dynamics.

The timely improvement of systems, based on the feedback from the service data, is becoming an increasingly common practice to enhance system reliability. This generates a novel predictive maintenance policy: after each service is completed, the server performs predictive maintenance to reduce the server's failure rate at the next service. We study an unreliable M/G/1 retrial queue with predictive maintenance. First, the stationary distribution and performance measures are analyzed using the supplementary variable method. Then, the threshold strategy of the predictive maintenance is proposed from the perspective of customers' waiting time. Based on the linear reward-cost structure, the customer's equilibrium strategy and socially optimal strategy are derived. Next, the optimal pricing strategy is obtained, in order to eliminate the difference between equilibrium and socially optimal strategies. Finally, numerical examples are provided to illustrate how system parameters affect customers' strategic behavior. These examples also demonstrate the accuracy of the closed-form solution for the socially optimal joining probability using a Particle Swarm Optimization (PSO) algorithm and a Genetic Algorithm (GA).

In this paper, we consider retrial queueing systems with an input flow rate that depends on the number of calls in an orbit and with a limited number of retrials. According to threshold and hysteresis strategies, optimization problems for the effective operation of the system are formulated and solved. The quality functionals are built in terms of the stationary probabilities of the underlying Markov process describing the system. By solving the optimization problems, we obtain optimal policies that ensure effective and stable system operation under varying load conditions and retrial limitations.

This paper analyzes a discrete-time single-server retrial queue with preemptive service, Bernoulli arrivals, and adaptive retrial times, tailored to telecommunications systems. In call centers, the model captures caller retries and priority interruptions, while in cellular networks, it represents user channel access attempts with preemption for emergency calls. Using a Markov chain framework, we derive the stationary distribution, establish a stability condition, and compute performance metrics, including the mean number of retrying callers or users and orbit size probabilities. The model incorporates a novel retrial time adaptation probability, reflecting dynamic retry behaviors in telecommunications. Numerical results demonstrate the impact of arrival rates, preemption probabilities, and retrial adaptations on system performance, offering insights for optimizing call center staffing and cellular network protocols. Applications to slotted ALOHA and TDMA systems highlight the model’s practical relevance.

This paper presents study of a M/M/1 retrial system incorporating differentiated vacation, failure and repair. Customer arrives according to Poisson process with rate λ. In regular state, if server is busy in serving customers, incoming customers who cannot be served immediately enter an orbit (Pool or virtual queue) of infinite capacity. When server becomes free during regular state, customers waiting in orbit reattempt for service according to classical retrial policy with rate nχ, where n shows orbit or pool size otherwise customer will have to wait for the server to be free. Customers get served with rate μ during regular busy state. Server join complete vacation if server is idle in free regular state, where no service will be provided to customer. If customer arrives during this state, then server transition to the working vacation (WV) state where despite not serving customers completely, now get served with some slow rate ω, (ω˂μ). If no customer remains during WV, server may return to complete vacation. During WV completion instant, if customers are still present in the system, then server resumes regular busy state for serving customers otherwise continue working vacation. Additionally, the server is subject to random breakdowns during its regular busy state. In such cases, it is sent for immediate repair and, upon completion, resumes service in the regular state. By using Probability generating function (PGF) approach, steady state analysis of model, analytical expression of distinct metrics of the system have been derived. The model’s analytical results were further supported by numerical simulations and visualizations implemented using Python, an open-source scientific computing language. The analysis provides insights into how system parameters affect the operational efficiency and quality of service.

Effective operation of any service system requires optimal organization of the sharing of resources between the users (customers). To this end, it is necessary to elaborate on the mechanisms that allow for the mitigation of congestion, i.e., the accumulation of many users requiring service. Due to the randomness of the user’s arrival process, congestions can occur even when an arrival rate is constant, e.g., the arrivals are described by the stationary Poisson process, which is assumed in the majority of existing papers. However, congestions can be more severe if the possibility of fluctuation of the instantaneous arrival rate exists. Such a possibility is an inherent feature of many systems and can be taken into account via the description of arrivals by the Markov arrival process (MAP). This makes the problem of congestion avoidance drastically more challenging. In many real-world systems, there exists the possibility of customer admission control via dynamic pricing. We propose a novel predictive mechanism of dynamic pricing. Decision moments coincide with the transition moments of the underlying process of the MAP. A customer may join or balk the system or postpone joining the system depending on the current cost. We illustrate the application of this mechanism in a multi-server retrial queueing model with dynamic service pricing. The behavior of the system is described by a multidimensional Markov chain with state-inhomogeneous transitions. Its stationary distribution is computed and may be used for solving the various problems of system revenue maximization via the choice of the proper pricing strategy.

In this study, we investigate a single-server retrial queueing system that incorporates balking, immediate feedback, and a Bernoulli working vacation policy. Customers arriving to find the server busy, under repair, or on working vacation may balk; otherwise, they either join the orbit or receive immediate service if the server is available. Upon completion of the service, the customer can request a finite number of immediate feedback services. When the orbit is empty after the completion of a service, the server initiates a working vacation, serving at a reduced rate. If customers are present in the orbit at the end of the vacation, the server resumes normal operation. If the system is empty, the server remains idle or continues the vacation. We analyze the ergodicity conditions to ensure system stability and derive the stationary distribution of the underlying Markov process. Several key performance measures are computed. Furthermore, a comprehensive cost function is developed and optimized using metaheuristic approaches, including particle swarm optimization and the genetic algorithm. The convergence behavior and optimization results are illustrated through graphical analysis, offering insight into improving the efficiency of complex retrial queueing systems.

As queueing theory and modeling deal with queue length, waiting time and busy period, that all affect costs for an in institution and/or a busing corporation, the optimization plays a crucial role in such models. This paper focuses on the performance modeling and optimal configuration of a single-server retrial queue with recurrent customers and a standby server, operating under Bernoulli working vacation conditions. The primary aim of the paper is to analyze the dynamics of this queueing model to achieve minimal operational costs while ensuring high performance. Using the supplementary variable technique (SVT), the probability generating functions (PGFs) and steady-state probabilities for the system's states, have been obtained enabling the development of comprehensive performance measures. These measures were rigorously validated through numerical examples. To complement the performance analysis, a cost function was formulated and optimized using advanced techniques, including the grey wolf optimizer (GWO), bat algorithm (BA), whale optimization algorithm (WOA), and cat swarm optimization (CSO). The results revealed that these algorithms successfully minimized operational costs while maintaining optimal system efficiency.

This study presents a comprehensive model for the steady-state behavior of an M/G/1 retrial queue with fault correction and hybrid vacation policies, tackling real-world challenges such as system breakdowns, repair delays and cost optimization. By integrating retrial mechanisms, fault-repair processes, and hybrid vacations, the model captures the complexity of systems such as telecommunications networks managing call redials or manufacturing facilities balancing machine downtime and auxiliary tasks. Using the supplementary variable technique (SVT), the study derives probability generating functions (PGFs) and steady-state probabilities, enabling the evaluation of critical performance metrics like queue length, orbit size, and server utilization. Additionally, the work introduces a cost optimization framework that minimizes operational expenses through advanced metaheuristic algorithms such as grey wolf optimizer (GWO), bat algorithm (BA), and whale optimization (WO). Practical applications include optimizing customer call centers to manage retries during peak hours or improving repair strategies for industrial equipment to reduce downtime and costs. This research bridges theoretical advances with real-world applicability, offering robust solutions for enhancing efficiency and costeffectiveness in queueing systems across industries.

This paper deals with a retrial queue with non-preemptive customers and synchronous vacation. Customers arrive in the system, among which non-preemptive customers do not form any queue. If a customer in the first class discovers that there are no available servers, he will be dropped directly from the system without receiving service. A customer in the second class who is unable to receive service immediately may or may not wait for a period and retry his luck. Servers follow a vacation policy: some of the servers will go on vacation at the same time when they become idle, while the service is completed and system conditions satisfy the management policy. The matrix-geometric method is applied to calculate the stationary distribution. Some system performance measures are established to help determine the efficiency of the presented non-preemptive priority retrial queue. Numerical examples illustrate the influence of system parameters on some performance measures. Finally, the optimization problem is constructed to help the manager make decisions.

Analyzing M/M/1 Working Vacation Model with Dual Server Breakdowns, Retrial Queue, and Interruptions: A Study on Classical Retrial Policies

Inspired by call centers, this paper models them as a constant retrial queue, with feedback and delayed vacations to balance high efficiency and low cost for service agents. After completing the service, the server randomly waits for an idle period. If customers arrive during this period, the service is provided immediately, otherwise, the server will take a vacation. We first derive steady-state probabilities and key performance measures. Then, the system cost is modeled. Particle Swarm Optimization (PSO), Ant Colony Algorithm (ACA) and Sparrow Search Algorithm (SSA) are applied to obtain the minimum system cost, respectively. To verify the correctness of the theoretical results of the system model, we simulate the model using Monte Carlo simulation to obtain the probabilities of different server states and the expected number of customers in the system, and then compare them with the theoretical values. At the same time, the sensitivity of the performance measures obtained by Monte Carlo simulation to the system parameters is also analyzed. Finally, customer behavior is analyzed, and equilibrium and socially optimal arrival rates are derived. In addition, the efficiency of the system is evaluated by examining efficiency indicators such as throughput and price of anarchy.

Hyper-exponential Approximation in the Analysis of a Multi-server Retrial Queue with Non-exponential Service Time Distribution

This paper presents a comprehensive mathematical analysis of an unreliable single-server retrial queue with general retrial times, serving two types of customer arrivals: high-patience and low-patience customers. Customers arrive in the system following two Poisson processes with different service rates. In addition, the model incorporates essential features such as service times, reserved times, and repair times, all following general distributions. The proposed model has practical applications in diverse domains, including healthcare systems, web traffic management, and call centers. Using the supplementary variable technique, we carry out an extensive analysis of the model. This approach allows us to derive the ergodicity condition for this Markov chain and compute its stationary distribution. The main performance measures of the system are expressed through the stationary state probabilities. Numerical illustrations are presented. Finally, we conduct an economic study to assess the impact of various system parameters on performance measures and total cost, offering a visual overview of the system's effectiveness and profitability. A comparative analysis with existing models shows how our approach generalizes traditional retrial queue models, which typically consider a single type of customer arrival, by considering two distinct customer classes. This contributes to the advancement of queueing theory and provides insight into optimizing real-world systems.

This research examines a multi-server retrial queueing system with a batch Markov arrival process and a phase-type service time distribution. The system’s distinguishing feature is its ability to control the admission of retrial customers. An attempt by a customer to retry is successful only if the number of busy servers does not exceed certain threshold values, which may depend on the state of the fundamental process of the primary customer’s arrival. Impatient retrying customers may abandon the system without obtaining service. A group of primary customers that arrives while the number of available servers is fewer than the group size is either entirely rejected or occupies all available servers, while the remainder of the group transitions to the orbit. The system’s behavior, under a defined set of thresholds, is characterized by a multidimensional Markov chain classified as asymptotically quasi-Toeplitz. This enables the acquisition of the ergodicity condition and the computation of the steady-state distribution of the Markov chain and the system’s performance measures. The presented numerical examples demonstrate the impact of threshold value variation. An example of solving an optimization problem is presented. The importance of the account of the batch arrivals is shown.

Optimizing energy harvesting in web of things services: a metaheuristic approach with retrial queue under working vacation