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This study examines Bernoulli scheduling approach, customers(clients) priority shift during server’s busy state and explores customers intolerance during server’s working vacation (WV) state with complete vacation in a M/M/1/∞ queueing model. During the busy state, customers might join the queue with a probability d or leave with a probability s=1-d. Some customers might enter the system but do not join the queue during the busy state are assumed to result from changing priorities or other pressing matters, rather than due to long wait times or frustration. The server enters a working vacation in free time with an exponentially distributed rate θ, providing service at a slower pace. During this phase, customers may become impatient if faced with extended waits, abandoning the queue with an exponentially distributed parameter α if their patience expires. Upon WV completion, the server either resumes a busy state with probability p if any customer present, or join complete vacation with probability m and rate β. If new clients arrive during complete vacation, the server returns to busy state at a rate ψ otherwise availing complete vacation. This paper employs the PGF method to derive equilibrium-state probabilities and assess system measures, including average queue lengths during busy state, WV states and complete vacation, mean sojourn time, and abandonment rate. Additionally, the influence of specific parameters on system metrics is illustrated graphically.

Recent technological advances have equipped vehicles with significant computational capacity, enabling Vehicular Fog Computing (VFC) for decentralized processing at the network edge. This study presents a novel queuing theory-based analytical model that addresses the challenges of breakdowns and server vacations in VFC. The model incorporates factors like intermittent connectivity, energy constraints, and vehicle failures. Numerical results show that optimizing repair rates can reduce average delay by 16.67 % , while decreasing vacation time reduces delays by 33.33 % . Conversely, higher breakdown rates increase delays by 50 % , emphasizing the need for effective maintenance strategies. These results offer insights into Quality of Service (QoS) improvements by adjusting decision parameters. The study concludes by highlighting open areas for future research to further enhance VFC performance. [ABSTRACT FROM AUTHOR]

This paper considers an M/M/1 queueing inventory system with (s, S) policy. The server may break down during the working vacation and each customer takes one product away after being served. When there is no product in the system, the system starts a working vacation, and the service continues at a lower rate at that time. The server may breakdown only during a working vacation period. When a vacation breakdown occurs, the system immediately stops service and starts repairing. Firstly, we utilize Markov process theory to construct a three-dimensional Markov chain to analyze the stationary process of the system. Using Gaussian iteration and matrix geometry solution method, the steady-state performance measures about queueing and inventory are obtained. Then, through numerical experiments, we analyze the system parameters’ influence on performance indexes. Finally, we define a cost function and use the genetic algorithm to obtain the optimal inventory level and the lowest cost at a certain set of parameters. [ABSTRACT FROM AUTHOR]

In this article, a working vacation policy-based on bulk arrival feedback retrial queueing system with variable server capacity has been analyzed. The server can serve a minimum of one customer and a maximum of B customers in a batch in accordance with the variable server capacity bulk service rule. As soon as the orbit becomes empty at the time of service completion, the server goes for a working vacation. The server works at a lower speed during a working vacation period. In addition, the steady state probability generating function for system size and orbit size is generated by incorporating the supplementary variables technique (SVT). Further, the conditional decomposition law is shown for this retrial queueing system. Moreover, system performance metrics, and significant special instances are discussed. Finally, the effects of various parameters on the system performance are analyzed numerically. [ABSTRACT FROM AUTHOR]

A modified differentiated vacation queueing system with a close-down period and impatient customers is investigated in this research paper. The proposed model can be applied to study the energy management mechanisms of WiMAX and tethered high altitude platform (HAP) systems. In this system, the WiMAX mobile station (MS) switches to a close-down period for a random duration at the end of the busy period. At the end of the close-down period, if the system is non-empty, the MS switches to a functional state. Otherwise, the MS takes an ordinary vacation. The MS switches to a working vacation (WV) if the system is empty at the end of the ordinary vacation period. Jobs arriving during the WV are served at a slower service rate. Jobs may become impatient in any state of the system except during the regular busy period. Explicit expressions for the transient and steady-state probabilities of this system are obtained using continued fraction and confluent hypergeometric functions. Furthermore, performance indices such as mean, variance, average abandonment rate, and system throughput are computed. Finally, a cost-profit analysis and graphical illustrations are provided. [ABSTRACT FROM AUTHOR]

A stochastic clearing system is studied from a game-theoretic perspective in the paper where the server is subject to a Poisson-generated catastrophe and a follow-up repair process. Whenever a fatal shock (catastrophe) occurs, all customers are cleared from the system and the server fails. A repair is rendered immediately to fix the server with an exponential repair time. During the repair process, no customers are allowed to enter the system. Customers are strategic and they have the right to decide whether to join the system or balk based on a linear reward-cost structure with two types of rewards: A service reward for those customers that receive service and a compensation for those customers that are forced to abandon the system due to a catastrophe. During the service process, the server takes a working vacation after serving all the customers in the system. Our study is the first attempt to provide models to jointly characterize and analyse the queueing system with working vacation and catastrophes, with an emphasis on game-theoretic modeling of such a service system. The customer's equilibrium strategy and social benefit of the system under four different information scenarios are obtained. In particular, we find that customers obey the follow-the-crowd (FTC) property in almost observable condition, which provides managerial insight on the operations management perspective. Numerical experiments are presented to show the effects of system parameters and information levels on the equilibrium joining behavior of customers. [ABSTRACT FROM AUTHOR]

This study investigated the steady-state characteristics of a non-Markovian feedback retrial queue with reneging, delayed repair, and working vacation. In this scenario, we assumed that consumers arrive through Poisson processes and the server provides service to consumers during both regular and working vacation periods. However, it is subject to breakdowns at any moment, resulting in a service interruption for a random duration. Additionally, the concept of delay time was also presented. The consumer that is dissatisfied with the service may re-enter the orbit to receive another service; this individual is considered a feedback consumer. The server will go on a working vacation if the orbit is empty after successfully serving a satisfied consumer. By utilizing the supplementary variable technique (SVT), we examined the steady-state probability generating function of the system and orbit sizes. Finally, numerical outcomes and a sensitivity analysis were given to verify the analytical findings of important performance indicators.

The cloud storage service, known for its flexible and expandable nature, often has difficulties managing operating costs while ensuring dependable service and quick response times. This investigation presents a novel approach to optimizing cost efficiency in cloud storage systems by applying particle swarm optimization of the Markovian retrial queueing model in a generic setup by incorporating the working vacation and users' discouragement behavior. Some users may opt not to enter the system or join the retry pool to wait for their turn if the server is occupied. After returning from working vacation, if there is one user available for service, the server can interrupt the vacation period. The server is subject to breakdown and can be recovered after getting the repair. In the proposed model, the server is prone to catastrophes and can fail at any time, leading to the entire system breaking down, and no users being able to access it during this period. Chapman–Kolmogorov (CK) steady-state equations associated with the quasi-birth-death (QBD) process are constructed to make a mathematical design. The governing equations framed to derive the queue length distributions and various performance indices are solved using the recursive method and difference equation theory. The fuzzified parameters are used to develop the FM/FM/1/WV model, which is analyzed using a parametric nonlinear programming approach. To determine the optimal design parameters, the cost minimization problem has been done using the quasi-Newton method and particle swarm optimization. This model incorporates features such as server failures, retrials, and catastrophes, thereby reflecting the complex nature of cloud storage operations. A suitable illustration of cloud storage is taken for both classical and fuzzified models to facilitate the numerical results of performance indices and optimal decision descriptors. [ABSTRACT FROM AUTHOR]

This paper is concerned with the optimal control of a Markovian queueing system subjected to multiple adaptive vacation and working vacation policies. This system is applicable in diverse modern technologies, in particular in call centers. We establish the steady-state solution as well as important system characteristics by means of probability generating functions technique. We also construct the expected total cost for this model and develop a procedure to determine the optimal service rate that yields the minimum cost. Further, we carried out a comparative analysis to obtain the minimum cost using the Newton--Raphson method and particle swarm optimization (PSO) algorithm. [ABSTRACT FROM AUTHOR]

In this article, we consider an M/M/C queue in which the arrival rate and service rate depends on the state of the system. In addition, the servers takes working vacation and the system may breakdown. Whenever breakdown takes place, the repair process immediately commences. During the repair period the customers are given service in a reduced service rate. Based on the vacation termination point, two models have been defined. The steady state probability vector of the number of customers in the queue and the stability condition are obtained using Matrix-Geometric method. The stationary waiting time distributions have been obtained. Some illustrative examples are also provided. [ABSTRACT FROM AUTHOR]

An M/M/1 retrial queue with working vacation and interruption in Bernoulli schedule under Ncontrol pattern is investigated in this article. To describe the system, we employ a QBD analogy. The model’s stability condition is deduced. The stationary probability distribution is generated by utilizing the matrix-analytic technique. The performance measures and special cases are designed. The model’s firmness is demonstrated numerically. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the single server queueing model where arrival of units is based on Markovian arrival process and service provided by server is in phase type distribution (MAP/PH/1) with working vacation and working breakdown. The study of working vacation and working breakdown is very important and applicable in real life. The absence of a server during vacation or server breakdown due to any reason may cause service interruption in queueing systems. We consider the repair process in two phases to recover the server from breakdown to working state with improved service rate phase-wise. Arrival process is based on Neuts' versatile point process that follows MAP and service times follow phase type distributions. The arrival rate, the service rate, working vacation rate and working breakdown are mutually independent. Matrix geometric solution method is used to obtain stationary probability vectors. Long run probabilities for all the states are derived in the results. Also with the different combination of arrival and service process the flow of average expected length with several parameters have been presented graphically. [ABSTRACT FROM AUTHOR]

The retrial queueing inventory system with working vacation, flush out, balking, breakdown, and repair, as well as a constant retrial rate and orbital client collision are all examined in this study. We made the assumption that customers arrive through a Markovian arrival process and that they would get phase-type services from the server. The inventory is replenished using a (s, S) and (s, Q) strategy, and it is expected that the replenishment time will follow an exponential distribution. If there are zero inventory items, no customers in the orbit, or both, the server will go into working vacation mode. When a customer retries an orbit while the server is serving arriving customers, the orbital customer may collide with the arriving customer during that retry, in which case both of them will be shifted back into orbit; otherwise, the orbital customer may avoid colliding with the arriving customer and may rejoin the orbit for another retry. The number of customers in the orbit and the inventory level may be found in the steady state. A cost analysis is produced along with the establishment of various important performance measures. Moreover, some numerical examples are provided to clarify our mathematical notion. [ABSTRACT FROM AUTHOR]

The maximum entropy principle has grown progressively more pertinent to queueing systems. The principle of maximum entropy (PME) presents an impartial framework as a promising method to examine complex queuing processes. This principle can be employed to assess the most appropriate probability distributions for queueing scenarios in a variety of widespread industrial issues. The aspects of general service bulk arrival retrial G-queue including working vacation, state-dependent arrival, priority users, and working breakdown are all explored in this article. Real-world applications for this kind of waiting line include computer systems, industrial companies, packet-switching networks, and communication facilities, etc. The adverse users (or negative arrivals) can make an appearance when the server (operator) is preoccupied with a positive user. Consumer’s arrival patterns follow the Poisson distribution. Priority consumers and regular (ordinary) consumers are the two groups of consumers that are considered in this investigation. Priority consumers do not have to wait in line and are granted a special right of prevention that allows them to receive services before ordinary consumers. Initially, we have estimated performance metrics including orbit size and long-run probabilities in this research work. The maximum entropy approach is then used to give a comparative perusal between the system’s exact and estimated waiting times. Apart from that a bi-objective optimization model is developed to diminish both consumers waiting times and estimated costs simultaneously. It is manageable to establish an effective balance between the standard of service and operating expenses using the analytical strategy that has been provided.

The idea behind the proposed queueing model is its adequacy in daily existent workplaces like hospitals, banks, and multinational companies where the service quality of computer networks and telecommunication systems are to be improved. The study consists of an MX/G/1 retrial G-queue with preemptive priority arrivals where the service provider takes a working vacation and the vacation may get interrupted. The novelty of the proposed model lies as it anticipates state-dependent arrivals wherein unsatisfied clients may come up with immediate Bernoulli feedback. Also, the concept of preemptive priority is included which can be seen in various service systems of day-to-day life. The supplementary variable technique (SVT) is used to unfold the model and obtain the system performance indices and reliability measures. In continuation, the Grey Wolf Optimization (GWO) algorithm is used to obtain the optimal working vacation service rate and retrial rate under two scalar optimization problems where the system works under minimal cost and minimal awaiting time respectively. Lastly, the Pareto optimal solution of a bi-objective problem is also presented.

The maximum entropy principle has grown progressively more pertinent to queueing systems. The principle of maximum entropy (PME) presents an impartial framework as a promising method to examine complex queuing processes. This principle can be employed to assess the most appropriate probability distributions for queueing scenarios in a variety of widespread industrial issues. The aspects of general service bulk arrival retrial G-queue including working vacation, state-dependent arrival, priority users, and working breakdown are all explored in this article. Real-world applications for this kind of waiting line include computer systems, industrial companies, packet-switching networks, and communication facilities, etc. The adverse users (or negative arrivals) can make an appearance when the server (operator) is preoccupied with a positive user. Consumer's arrival patterns follow the Poisson distribution. Priority consumers and regular (ordinary) consumers are the two groups of consumers that are considered in this investigation. Priority consumers do not have to wait in line and are granted a special right of prevention that allows them to receive services before ordinary consumers. Initially, we have estimated performance metrics including orbit size and long-run probabilities in this research work. The maximum entropy approach is then used to give a comparative perusal between the system's exact and estimated waiting times. Apart from that a bi-objective optimization model is developed to diminish both consumers waiting times and estimated costs simultaneously. It is manageable to establish an effective balance between the standard of service and operating expenses using the analytical strategy that has been provided. [ABSTRACT FROM AUTHOR]

A single server queueing system with working vacation operating in a multi-level environment controlled by a random switch is considered. The environment has N levels and the random switch puts the server in one of the N levels to perform a service. The server resides in a level and reports to the random switch immediately after completing the service of the last customer in the queue. The random switch initiates an assignment process immediately at the arrival of a customer. The assignment process takes a random time and immediately at the end of the assignment interval, the server is put to operate in any one of the N levels with a positive probability. In each level r of the environment, the queueing system behaves like an M(λr,1)/M(μr,1)/1 queue and switches to a working vacation at a random time and serves with a lesser rate μr,2. During the working vacation, all newly arriving customers are lost. When the server is with the control switch, customers are permitted to join in the queue. For this model, time-dependent state probabilities are explicitly found and the corresponding steady-state probabilities are deduced. Some key performance measures are also obtained. A numerical study is also made. [ABSTRACT FROM AUTHOR]

We study the M/M/1 working vacation queueing system with N-policy and different arrival rates in this paper. The arrival rate of customers in a different period is different. Using the method of matrix geometry solution, We have given the steady-state performance indicators of this system. In addition, we also get the conditional stochastic decomposition structures of the queue length as well as the waiting time. The last part of the article show the numerical analyses and the trend of each performance indicators. [ABSTRACT FROM AUTHOR]

A Markovian retrial queue with reneging and working vacation under N-control pattern is investigated in this article. To describe the system, we employ a QBD analogy. The model's stability condition is deduced. The stationary probability distribution is generated by utilizing the matrix-analytic technique. The conditional stochastic decomposition of the line length in the orbit is calculated. The performance measures and special cases are designed. The model's firmness is demonstrated numerically. [ABSTRACT FROM AUTHOR]

This paper investigates an unreliable $ M/G(P_{1}, P_{2})/1 $ retrial queueing system with a woking vacation. An arriving customer successfully starts the first phase service with the probability $ \alpha $ or the server fails with the probability $ \bar{\alpha} $. Once failure happens, the serving customer is taken to the orbit. The failed server is taken for repair with some delay. Once the repair is comleted, the server is ready to provide service once again. In this background, we implemented the working vacation scenario. During working vacation, the service will be provided at a slower rate, rather than entirely stopping the service. The supplementary variable method was adopted to find the orbit and system lengths. Additionally, some unique results and numerical evaluations have been presented.

Non-preemptive priority queue system is a type of priority queue where customers with higher priorities cannot interrupt low priority one while being served. High priority consumers will still be at the head of the queue. This article discusses the non-preemptive priority queue system with multiple working vacations, where the vacation can be interrupted. Customers are classified into two classes, namely class I (non-preemptive priority customers) and class II, with exponentially distributed service rates. Customers will still receive services at a slower rate than during normal busy periods when they enter the system while it is on vacation. Suppose other customers are waiting in the queue after completing service on vacation. In that case, the vacation will be interrupted, and the service rate will switch to the busy period service rate. The model's performance measurements are obtained using the complementary variable method and analyzing the state change equation following the birth and death processes to find probability generating function for both classes. The results of the numerical solution show that the expected value number of customers and waiting time of customers in the queue for both class customers will be reduced when the vacation times rate (θ) and the vacation service rate (μ_0 ) increase.

This paper investigates an unreliable M=G(P1; P2)=1 retrial queueing system with a woking vacation. An arriving customer successfully starts the first phase service with the probability α or the server fails with the probability α. Once failure happens, the serving customer is taken to the orbit. The failed server is taken for repair with some delay. Once the repair is comleted, the server is ready to provide service once again. In this background, we implemented the working vacation scenario. During working vacation, the service will be provided at a slower rate, rather than entirely stopping the service. The supplementary variable method was adopted to find the orbit and system lengths. Additionally, some unique results and numerical evaluations have been presented. [ABSTRACT FROM AUTHOR]

In this article, an M/M/1 queueing model with bi-level network service provider, balking process and bi-level vacation policy that comprises of working vacation and complete vacation after fixed service, is developed. Matrix form expressions have been derived for the distributions of the queued customers with some performance metrics with the help of matrix geometric method. The maximum entropy principle is also used to derive the distributions of the steady state probabilities of queue size. The cost function has been formed to optimize the decision variables of the system. We perform the cost optimization by employing the steepest descent search method. Numerical illustrations along with the sensitivity analysis have been drawn to validate the model. Finally, the conclusions of the investigation done are drawn by mentioning the novel features and future scope. [ABSTRACT FROM AUTHOR]

In this paper, we consider a single server queueing model with working vacation and multiple working breakdowns. When on vacation, the server works at a different pace. Disturbances occur in the server due to multiple breakdowns. In working breakdowns server works at a different rate. During the time of interruption caused by working vacation and working breakdowns, the main server can find many implementations in operation. Both the server's lifespan and the time it takes to repair it are considered to be exponentially dispersed. Also, we have considered balking and reneging behaviours of customers. The stationary queue length distribution is computed using a matrix-analytic technique. Using Genetic Algorithm (GA) we optimize the cost function. The predicted length of a busy period, the expected length of a working vacation period, the expected length of a working breakdown period, the mean waiting time, and the average delay are all established. We compute numerical results to verify the analytical point of view. The effect of individual parameters is investigated using sensitivity analysis.

In this paper, we consider two queuing models. Model 1 considers a single-server working vacation queuing system with interdependent arrival and service processes. The arrival and service processes evolve by transitions on the product space of two Markovian chains. The transitions in the two Markov chains in the product space are governed by a semi-Markov rule, with sojourn times in states governed by the exponential distribution. In contrast, in the second model, we consider independent arrival and service processes following phase-type distributions with representation (α , T) of order m and (β , S) of order n, respectively. The service time during normal working is the above indicated phase-type distribution whereas that during working vacation is a phase-type distribution with representation (β , θ S) , 0 < θ < 1 . The duration of the latter is exponentially distributed. The latter model is already present in the literature and will be briefly described. The main objective is to make a theoretical comparison between the two. Numerical illustrations for the first model are provided. [ABSTRACT FROM AUTHOR]

This work considers a preemptive priority queueing system with vacation, where the single server may break down with imperfect coverage. Various combinations of server vacation priority queueing models have been studied by many scholars. A common assumption in these models is that the server will only resume its normal service rate after the vacation is over. However, such speculation is more limited in real-world situations. Hence, in this study, the vacation will be interrupted if a customer waits for service in the system at the moment of completion of service during vacation. The stationary probability distribution is derived by using the probability generating function approach. We also develop varieties of performance measures and provide a simple numerical example to illustrate these measures. Optimization analysis is finally carried out, including cost optimization and tri-object optimization. [ABSTRACT FROM AUTHOR]

Any arriving customer who arrives and finds that the server is free, enters the service station and the remaining customers connect into the orbit. When the normal busy server is running, the system may at any time become defective due to a disaster. All users are forced to quit the system due to a disaster, which also brings about the failure of the main server. When a primary server breaks, it is shipped out for repair, and the repair process starts instantly. The server stops running as soon as the orbit is empty at a typical service finish instant. During the working breakdown or working vacation, the replacement server offers arriving customers a lower level of service. The arriving customer receives service instantly if the server is idle. If not, he will choose whether to leave the system without service or returning to receive service. Using the supplementary variable technique, we calculate the steady state PGF for system and orbit sizes. We generate performance measures and particular cases. With the use of specific numerical examples, we analyse the model. [ABSTRACT FROM AUTHOR]

Maintaining a high-level reliability and efficiency without interruption are the key concerns for many real-time machining systems. Using the redundancy and repair facility features, we develop a double retrial orbit queueing model for the fault-tolerant machining system (FTMS) operating under the restriction of admission of repair jobs based on threshold policy and working vacation. The provision of primary and secondary orbits is made so that the failed units can wait there based on the facility available in case the repairman is occupied. From the orbits, the failed units retry to get the repairman free so that the repair job can be accomplished. Chapman–Kolmogorov equations for the system states of FTMS have been constructed to evaluate the transient reliability and queueing indices using the spectral technique. The sensitivity along with the relative sensitivity analysis of crucial system parameters, have facilitated. The impacts of parameter variability on the system metrics and total expected cost are examined for illustrative examples. • Reliability Modeling of Fault-tolerant system with working vacation and double retrial orbits have been studied. • Adopt the spectral expansion method to examine the transient behavior of the developed model. • Address the sensitivity analysis of the system reliability and mean time to failure. • The numerical simulations are presented to validate the computational tractability. • A real-life based example of a Digital Subscriber Line Access Multiplexer is illustrated. [ABSTRACT FROM AUTHOR]

An M/G/1 feedback retrial queue with working vacation and a waiting server is taken into consideration in this study. Both retrial times and service times are assumed to follow general distribution and the waiting server follows an exponential distribution. During the working vacation period customers are served at a lesser rate of service. Before going for a vacation the server waits for some arbitrary amount of time and so is called a waiting server. We obtain the probability generating function (PGF) for the number of customers and the mean number of customers in the invisible waiting area by utilizing the supplementary variable technique. We compute the mean waiting time. Out of interest a few special cases are conferred. Numerical outcomes are exhibited. [ABSTRACT FROM AUTHOR]

The paper deals with an M/M/2 Queueing Model with working vacations and reneging of customers due to impatience. The matrix geometric method is used to find the distribution of the number of customers in the system. A cost function is constructed to obtain the optimal value of the service rate to optimize (minimize) the cost function using the Quadratic Fit Search Method (QFSM). Further, the effects on the system's performance measures using numerical analysis and graphical representation are studied. [ABSTRACT FROM AUTHOR]

This manuscript analyses a retrial queueing system with working vacation, interruption, feedback, and setup time with the perfect repair. In the proposed model, the server takes vacation whenever the system gets empty but it still serves the customers at a relatively lower speed. The concept of power saving is included in the model. To save the power the server is turned off immediately on being empty at vacation completion instant. The customer who arrives when the system is turned off activates the server and he has to wait for his turn till the server is turned on. The unreliable server may sometimes fail to activate during setup. It is then sent for repair and repaired server immediately starts serving the waiting customers. Using probability generating function, explicit expressions for system size and different states of server for the model are obtained and results are analyzed graphically using MATLAB software.

In this investigation, we establish a steady-state solution of an infinite-space single-server Markovian queueing system with working vacation (WV), Bernoulli schedule vacation interruption, and impatient customers. Once the system becomes empty, the server leaves the system and takes a vacation with probability p or a working vacation with probability 1 − p, where 0 ≤ p ≤ 1. The working vacation period is interrupted if the system is non empty at a service completion epoch and the server resumes its regular service period with probability 1 − q or carries on with the working vacation with probability q. During vacation and working vacation periods, the customers may be impatient and leave the system. We use a probability generating function technique to obtain the expected number of customers and other system characteristics. Stochastic decomposition of the queueing model is given. Then, a cost function is constructed by considering different cost elements of the system states, in order to determine the optimal values of the service rate during regular busy period, simultaneously, to minimize the total expected cost per unit time by using a quadratic fit search method (QFSM). Further, by taking illustration, numerical experiment is performed to validate the analytical results and to examine the impact of different parameters on the system characteristics.

Working vacation queues with breakdowns and customers impatience have many applications in different real-life situations. The development of these models to determine their performance is extremely important. In this paper, we deal with a finite population Markovian multi-server machine system with breakdowns, repairs, Bernoulli feedback, balking, reneging, and retention of reneged customers, under multiple synchronous working vacations. The investigated model has a potential application in the real-world machine systems, such as the automated manufacturing systems with limited space. For the analysis purpose, we employ Q-matrix method in order to obtain the steady-state probabilities as well as closed-form expressions for several system characteristics. Particular cases of the current study are presented. We construct the expected cost function and develop an optimization problem to determine the optimum cost. The direct search method and the Quasi–Newton method are applied to find the optimum system capacity, the minimum number of servers, and the optimum service rates during both working vacation and regular busy periods at minimum cost. Lastly, a sensitivity analysis for the numerical simulation is carried out. [ABSTRACT FROM AUTHOR]

With the increasing reliance on cloud computing as the foundational manufacturing systems with intricate dynamics, featuring multiple service areas, varying job arrival rates, diverse service requirements, and the interplay of failures and impatience, significant analytical challenges arise. Queueing networks offer a powerful stochastic modeling framework to capture such complex dynamics. This paper develops a novel, exhaustive queueing model for a finite-capacity redundant multi-server system operating in a multi-phase random environment. The proposed model uniquely integrates real-world factors, including server breakdowns and repairs, waiting servers, synchronous working vacations, and state dependent balking and reneging, into a single queueing model, representing a significant advancement in the field. Using the matrix-analytic method, we establish the steady-state solution and derive key performance metrics. Numerical experiments and sensitivity analyses elucidate the impact of system parameters on performance measures. Additionally, a cost model is formulated, enabling cost optimization analysis using direct search method and Particle Swarm Optimization (PSO) to identify efficient operating configurations. [Display omitted] • Analytical framework for finite capacity redundant multi-server queue in multi-phase environment. • Matrix analytic method calculates steady-state probabilities to assess system performance. measures. • Through sensitivity analysis, key cost variables are identified, aiding decision-making. • Optimizing the cost function by merging Direct Search Method (DSM) and Particle Swarm Optimization (PSO). [ABSTRACT FROM AUTHOR]

• We discuss reliability analysis of a fault tolerant system with imperfect coverage. • We explore system reliability, MTTF and other queuing measures. • The cost-effective ratio is evaluated to upgrade and improve availability. • We obtain optimal control parameters via a direct search approach and PSO. Due to widespread usage in many real time systems, reliability modeling and cost optimization of fault tolerance system have drawn attention of the practitioners. The fault tolerance in these systems can be provided by the support of maintenance and redundant components that help in smooth operation of the system in spite of failure of some active components. This investigation deals with the performance modeling of a fault-tolerant system consisting of a finite number of active (online) and standby components. During the switching from active to standby, the recovery procedure is performed, which may be imperfect. In case of imperfect recovery, the system reboot takes place. The maintenance of all the components is managed by a repairman (server) which is subject to failure. When the server is interrupted for rendering the service, functioning does not get stopped due to the system switch-over from perfect working to working breakdown mode. The system works even when the server is on working vacation and performs repair jobs of the failed components. The machine repair model based on Markovian process is developed to derive the transient probabilities and other performance indices of the fault tolerant system using Laplace transforms and matrix analytical method. Using the direct search strategy and particle swarm optimization, the cost-benefit analysis is done. The optimal design of the control parameters for the fault-tolerant system are presented by framing a cost-effective ratio function. The model is examined computationally by performing the numerical simulation and cost optimization. [ABSTRACT FROM AUTHOR]

This paper considers the customers' equilibrium and socially optimal joining–balking behavior in a single-server Markovian queue with a single working vacation and Bernoulli interruptions. The model is motivated by practical service systems where the service rate can be adjusted according to whether or not the system is empty. Specifically, we focus on a single-server queue in which the server's service rate is reduced from a regular to a lower one when the system becomes empty. This lower rate period is called a working vacation for the server which may represent that part of the service facility is under a maintenance process or works on other non-queueing job, or simply for saving the energy (for a machine server case). In this paper, we assume that the working vacation period is terminated after a random period or with probability p after serving a customer in a non-empty system. Such a system is called a queue with single working vacation and Bernoulli interruptions. Customers are strategic and can make choice of joining or balking based on different levels of system information. We consider four scenarios: fully observable, almost observable, almost unobservable, and fully unobservable queue cases. Under a reward-cost structure, we analyze the customer's equilibrium and social-optimal strategies. In addition, the effects of system parameters on optimal strategies are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

In this manuscript, a queueing system with two optional vacation policies, power-saving mode under reneging and retention of reneged customers in both vacations is analyzed. If the server is free, it chooses either of the vacations, classical vacation or working vacation. During vacations, the customers may get impatient due to delays and may leave the system, but they are retained in the system with some convincing mechanisms. On vacation completion, if the system is empty, the server is turned off to facilitate better utilization of the resources. Some of the operating system characteristics are derived using the probability generating functions technique. The numerical results are graphically represented by using MATLAB software. [ABSTRACT FROM AUTHOR]

This paper addresses machine repair problem (MRP) with M identical operating machines and control arrival policy. The server is unreliable, and can break down during service which is further repaired soon so as to avoid interruption with the service process. The server may go for working vacation in case all the customers are served. The transient analysis of machine repair problem has been done using numerical technique. Various performance measures have been derived. With the help of tables and graphs the numerical results have been shown. The present investigation find applications in various industrial and workshop situations like Automobile repair shop (ARS).

Markovian queue with working vacation, retrial and impatient customers has been investigated by including the feature of imperfect service during working vacation. If the server is occupied, an incoming customer may either decide to balk from the system or join the retrial orbit following the classical retrial policy. When there is no customer in the system, the server switches over to a working vacation (WV) mode. The customers who enter into the system during working vacation are also served at a slower speed. The probability generating functions (PGFs) are derived for evaluating the distributions of the number of customers in the system and orbit, which are further utilized to derive several performance metrics in closed form. The sensitivity analysis has been done to explore the sensitiveness of various descriptors on the system performance measures. Finally, a cost function is constructed and minimized using quasi-Newton method (QNM) and genetic algorithm (GA). [ABSTRACT FROM AUTHOR]

This work considers a preemptive priority queueing system with vacation, where the single server may break down with imperfect coverage. Various combinations of server vacation priority queueing models have been studied by many scholars. A common assumption in these models is that the server will only resume its normal service rate after the vacation is over. However, such speculation is more limited in real-world situations. Hence, in this study, the vacation will be interrupted if a customer waits for service in the system at the moment of completion of service during vacation. The stationary probability distribution is derived by using the probability generating function approach. We also develop varieties of performance measures and provide a simple numerical example to illustrate these measures. Optimization analysis is finally carried out, including cost optimization and tri-object optimization.