Showing papers on "Retrial queue published in 2021"

Equilibrium balking strategies in the single-server retrial queue with constant retrial rate and catastrophes

Abstract: We study an M/M/1 retrial queueing system with constant retrial rate and Poisson generated catastrophes. When a catastrophe arrives to the system that the server is working, it deletes all customer...

Equilibrium and pricing analysis for an unreliable retrial queue with limited idle period and single vacation

Abstract: This paper treats an M/M/1 retrial queue with an unreliable server, where the server is subject to random breakdowns and repairs when he is serving a customer. Whenever the server becomes idle, he will spare a limited period to wait possible new arrivals or repeated customers if any. If no customer arrives during this period, the server will take a single vacation. Under a given reward-cost structure, arriving customers have to decide to balk or to enter the orbit if they find the server unavailable at their arrival epochs. Firstly, we study some important performance measures. Secondly, we derive the equilibrium joining strategies for the customers, respectively, in noncooperative and cooperative cases. Thirdly, we study the pricing problem that the social planner faces to eliminate the gap between the individual optimal and socially optimal strategies. Finally, we present some numerical examples to illustrate the effect of some system parameters on optimal joining probabilities and Maximum of social welfare.

Strategic priority-purchasing and joining rules in a retrial queue

Abstract: This paper considers a retrial queueing system with a pay-for-priority option. A queueing-game-theoretic model that captures the interaction among the customers, the service provider (SP) and the social planner is developed. We obtain the equilibrium strategy of customers for any fixed priority premium and identify the unique Pareto-dominant strategy. The optimal pricing strategies for the SP and the social planner are derived and compared extensively. Interestingly, we find that the equilibrium outcome of customers is non-monotone in the service reward and the profit of the SP is bimodal in the priority premium. We reveal the fact that the SP’s optimization makes the system more congested than what is socially desirable. Finally, numerical examples indicate that the customer welfare can be improved by providing priorities when the market size is large.

On the Retrial-Queuing Model for Strategic Access and Equilibrium-Joining Strategies of Cognitive Users in Cognitive-Radio Networks with Energy Harvesting

Abstract: This article studies the strategic access of single-server retrial queue with two types of customers, where priority is given according to their category. On the basis of this concept, a cognitive-radio network was developed as retrial queue with energy harvesting. Cognitive radio allows for a secondary user to opportunistically access the idle spectrum of a primary user (PU). Upon arrival of a primary user, the service given to the secondary user by the cognitive radio is interrupted, and the PU band is available for the primary user. After completion of service for the primary user, the PU band is again available to secondary users. Performance metrics are derived to study the equilibrium strategies of secondary users. A Stackelberg game was formulated and Nash equilibrium was derived for the noncooperative strategy of the secondary user. Game-theory concepts are incorporated with queuing theory ideas to obtain the net benefit for the noncooperative strategy and social benefit for cooperative strategy. Lastly, analytical results are verified with numerical examples, and the effects of energy-harvesting rate are discussed.

Bernoulli vacation model for MX/G/1 unreliable server retrial queue with bernoulli feedback, balking and optional service

Abstract: The study of unreliable server retrial bulk queue with multiphase optional service is analyzed by incorporating the features of balking, Bernoulli vacation and Bernoulli feedback. On the occasion when the server is occupied with the service of the customers, an arriving customer finding the long queue, can join the retrial orbit and receives its service later on by making re-attempt. The system is reinforced with multi phase optional service along with essential service and joining customer can opt any one of optional services after getting essential service. Furthermore, the essential/optional service can be aborted due to abrupt failure of the server. There is an immediate support of multi phase repair facility to take care of the failed server, but sometimes repair may be put on hold by virtue of any unexpected cause. If the service is unsatisfactory, the customer can rejoin the queue as feedback customer. Bernoulli vacation is permitted to the server following the respective busy period. For evaluating the queue size distribution and other system performance metrics, supplementary variable technique (SVT) is used. The approximate solutions for the steady state probabilities and waiting time are suggested using maximum entropy principle (MEP). We perform a comparative study of the exact waiting time obtained by the supplementary variable technique and the approximate waiting time derived by using maximum entropy principle by taking the numerical illustration. Quasi Newton method is used to find optimal cost. To verify the outcomes of the model, numerical illustrations and senstivity analysis have been accomplished.

Joining strategies of noncooperative and cooperative in a single server retrial queue with N-policy and multiple server vacations

Abstract: Motivated by service price and cost control, in this work, we study a single-server queueing system with a constant retrial rate, N-policy, and multiple server vacations. There is no waiting space ...

Unreliable Single Server Double Orbit Retrial Queue with Balking

Abstract: The single unreliable server double orbit Markovian retrial queue with customers’ balking behavior has been investigated. The arriving customers may not wish to proceed for the service when the system seems overcrowded. On joining the system, if the server is busy then customers are forced to join ordinary orbit or executive orbit as per their demand. It is assumed that the server is unreliable and subject to breakdown and repair. For the steady-state analysis of the system, Chapman–Kolmogorov steady-state equations are framed and then solved by using probability generating function. Furthermore, various performance metrics are derived explicitly. An illustration is taken to analyze the effects of the system parameters on the performance metrics. ANFIS technique is implemented to authenticate the steady-state results. Further, a nonlinear cost function is also formulated which is then minimized by using the quasi-Newton method.

Joining strategies under two kinds of games for a multiple vacations retrial queue with $ N $-policy and breakdowns

Abstract: Motivated by cost control and information guidance, in this work, we study a multiple vacations retrial queue with $ N $-policy and breakdowns. This service system has the characteristics that there is no waiting space in front of the server and the waiting list is virtual. If the arriving customer finds that the system is available, he immediately receives the complete service. Otherwise, the customer leaves the system or joins the orbit (virtual waiting list). For cost control, the system is activated only when the current vacation is completed and at least $ N $ customers are waiting in the system, otherwise, the server continues to the next vacation until the number of customers in the system is not less than $ N $. Two types of customer joining cases apply to this paper, i.e., non-cooperative customers aim to optimize individual interests, and the social planner in the cooperative case considers the profit of the whole service system. The equilibrium joining strategy for the non-cooperative case and the socially optimal joining strategy for the cooperative case are determined. Since it is difficult to obtain analytical characterization, an improved particle swarm optimization (PSO) algorithm is used to explore the impact of system parameters on the profit of the service provider. At the same time, a large number of numerical experiments visualize the influence of parameters on the system.

Equilibrium balking strategies in the repairable m/m/1 g-retrial queue with complete removals

Abstract: We consider an M/M/1 retrial queue subject to negative customers (called as G-retrial queue). The arrival of a negative customer forces all positive customers to leave the system and causes the server to fail. At a failure instant, the server is sent to be repaired immediately. Based on a natural reward-cost structure, all arriving positive customers decide whether to join the orbit or balk when they find the server is busy. All positive customers are selfish and want to maximize their own net benefit. Therefore, this system can be modeled as a symmetric noncooperative game among positive customers and the fundamental problem is to identify the Nash equilibrium balking strategy, which is a stable strategy in the sense that if all positive customers agree to follow it no one can benefit by deviating from it, that is, it is a strategy that is the best response against itself. In this paper, by using queueing theory and game theory, the Nash equilibrium mixed strategy in unobservable case and the Nash equilibrium pure strategy in observable case are considered. We also present some numerical examples to demonstrate the effect of the information together with some parameters on the equilibrium behaviors.

Markovian Model of Unreliable Server Retrial Queue with Discouragement

Abstract: In this paper, we present the performance analysis of Markovian retrial queue with unreliable server by incorporating the features of balking and reneging. The customers may abandon the queue at the moment of busy server or at the departure epoch from the orbit zone. When the customers notice the server in busy state, they will join to the orbit zone and wait there to avail the service. After a random period, the customers from the orbit either may try for the re-attempts for the service or may renege from the system without getting the service. Markov model is developed by considering the inter-arrival, retrial and service times governed by the exponential distributions. The probabilities of the system states are determined by implementing recursive approach to solve the governing Chapman–Kolmogorov equations. The cost function is constructed which is further used to find the total expected cost and associated optimal service rate of the system. A numerical illustration is provided to explore the system metrics with respect to various system descriptors.

Stability Analysis of a Multi-class Retrial Queue with General Retrials and Classical Retrial Policy

Abstract: We deal with a single server multi-class retrial model, feed by Poisson input. The system is considered under classical retrial policy, while inter-retrial times are class dependent and generally distributed. Such systems have various applications like multi-access protocols or cellular mobile networks, where blocked messages are sent again after some waiting period. We rely on regenerative approach and results from renewal theory to obtain the stability criterion of the system under consideration and present some simulation results, to illustrate that obtained condition could be extended to the case with general input.

Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters

Abstract: We consider an \begin{document}$ M^X/G/1 $\end{document} retrial queue with impatient customers subject to disastrous failures at which times all customers in the system are lost. When the server finishes serving a customer and finds the orbit empty, the server becomes dormant until \begin{document}$ N $\end{document} or more customers accumulate. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest join the orbit to repeat their request later. Otherwise, if the server is dormant or busy or down, all customers of the coming batch enter the orbit. When the server is under repair, customers in the orbit can become impatient after waiting a random amount of time and leave the system. By using the characteristic method for partial differential equations, the steady-state distributions of the server state and the number of customers in the orbit are obtained along with various performance measures. In addition, the reliability of the system is analyzed detailed. Finally, an application to telecommunication networks is provided and the effects of various parameters on the system performance are demonstrated numerically.

Modeling of Stochastic Arrivals Depending on Base Stock Inventory System with a Retrial Queue

Abstract: This paper mainly deals with the customers whose arrivals depend upon the existing stock level and base stock ordering policy in a stochastic queuing environment. The time between any successive primary as well as retrial arrivals and lead time are following independent exponential distributions. By applying Neuts Matrix Geometric Method, we present the time invariant joint distribution of number of customers in the orbit as well as stock level. Adequate numerical results are explored according to different characteristics of the system performance measures. Further we discuss the advantages between stock depending and non-depending arrivals about the expected total cost and parameter analysis. Also we explore the model with a different class of stock dependent arrivals under the various cost structure and replenishment time. Managerial insights of this model helps to make the decision for an appropriate base stock ordering policy under certain random conditions with a stock dependent demand rate and a limited holding capacity for ordered items.

Analysis of $$MAP, PH_{2}^{OA}/PH_{1}^{I}, PH_{2}^{O}/1$$ M A P , P H 2 OA / P H 1 I , P H 2 O / 1 retrial queue with vacation, feedback, two-way communication and impatient customers

Abstract: This article concentrates on the steady-state analysis of a constant retrial queueing system with impatient customers, vacation, feedback, and two types of arrivals, namely the incoming calls which are made by the customers and the outgoing calls which are made by the server during the idle period. The incoming calls arrive at the system by following the Markovian Arrival Process(MAP) and service times of incoming/outgoing calls follow phase-type (PH) distribution, and the rest of the random variables are exponentially distributed. We have framed our model for analyzing some of the basic situations/problems in telecommunication systems. With the support of matrix analytic method, the invariant analysis of our system has been carried out. We have also discussed the busy period and have performed the cost analysis for our model. At last, we have validated our model through numerical and graphical exemplifications.

On the retrial queue with imperfect coverage and delay reboot

Abstract: Retrial systems have been used extensively to model many practical problems in call center, data center, cloud service computing center and computer network system. This paper deals with a multi-server retrial system with the features of imperfect coverage and delay reboot. In the investigated system, arrivals may not be detected because of some fault issues. When this situation happened, the system is cleared by a reboot operation. Once arrivals are detected and located, they are attended to when a server is available; otherwise, they join a retrial orbit and generate repeated attempts till a free server is found. We analyze the presented model as a quasi-birth-and-death process and develop various performance indices. The optimal number of servers and optimal service rate are searched by constructing an average cost function. A heuristic search technique is employed to obtain the optimization approximate solution at a minimum cost. Numerical illustrations are given to demonstrate the optimization procedure and the effects of varying parameters on performance indices. We also present an application example to demonstrate the applicability of investigated model.

Sojourn Times in a Queueing System with Breakdowns and General Retrial Times

Abstract: This paper centers on a discrete-time retrial queue where the server experiences breakdowns and repairs when arriving customers may opt to follow a discipline of a last-come, first-served (LCFS)-type or to join the orbit. We focused on the extensive analysis of the system, and we obtained the stationary distributions of the number of customers in the orbit and in the system by applying the generation function (GF). We provide the stochastic decomposition law and the application bounds for the proximity between the steady-state distributions for the queueing system under consideration and its corresponding standard system. We developed recursive formulae aimed at the calculation of the steady-state of the orbit and the system. We proved that our discrete-time system approximates M/G/1 with breakdowns and repairs. We analyzed the busy period of an auxiliary system, the objective of which was to study the customer’s delay. The stationary distribution of a customer’s sojourn in the orbit and in the system was the object of a thorough and complete study. Finally, we provide numerical examples that outline the effect of the parameters on several performance characteristics and a conclusions section resuming the main research contributions of the paper.

Analysis of classical retrial queue with differentiated vacation and state dependent arrival rate.

Abstract: In present paper we have introduced the concept of differentiated vacations in a retrial queueing model with state dependent arrival rates of customers. The arrival rate of customers is different in various states of the server. The vacation types are differentiated by means of their durations as well as the previous state of the server. In type I vacation, server goes just after providing service to at least one customer whereas in type II, it comes after remaining free for some time. In steady state, we have obtained the system size probabilities and other system performance measures. Finally, sensitivity and cost analysis of the proposed model is also performed. The probability generating function technique, parabolic method and MATLAB is used for the purpose.

Correction to: Equilibrium and pricing analysis for an unreliable retrial queue with limited idle period and single vacation

Abstract: In the original publication of the article, the below-mentioned equations had been incorrectly published.

Strategic joining in a single-server retrial queue with batch service

Abstract: We consider a single-server retrial queue with batch service where potential customers arrive according to a Poisson process. The service is divided into two periods: busy period and admission period, and they are corresponding to whether the server is in service or not respectively. These two periods constitute an alternate renewal process. When arrivals find busy period, they make join-or-balk decisions and the joining ones will stay at the orbit and try to get into server at constant rate; when arrivals find admission period, they get into the server directly. At the end of each admission period, all customers in the server will be served together regardless of the size of the batch. Therefore, we give the assumption of service reward that vary with the service size. Furthermore, customers in the orbit fail to get into the server before the end of each service cycle were forced to leave the system. We identify the (Nash) equilibrium joining strategies and the social- and profit- maximization problems of arrivals by assuming that they are informed about the service period upon arrivals. Finally, the optimal joining strategies are showed by numerical examples.

Reliability and Sensitivity Analysis of Retrial Queue with Optional k-Phases Services, Vacation and Feedback

Abstract: Queueing theory is implemented for modeling and analyzing actual conditions in industries and real-world problems. In many cases, the input is converted to the desired output after several successive steps. Lack of space, feedback and vacation are the main characters of these processes. This article deals with the modeling and analyzing the steady-state behavior of an $$M/G/1$$ retrial queueing system with first essential and $$k-1$$ optional phases of service. Also, the probabilistic feedback to orbit at each phase and Bernoulli vacation at the end of $$k$$ -th phase may occur in this system. If the customers find the server busy or on vacation, they join to the orbit. In this article, after finding the probability generating functions of the system and orbit sizes, some important performance measures are found. Also, the system reliability is defined. Eventually, to demonstrate the capability of the proposed model, the sensitivity analysis of cost indices and performance measures via some model parameters (arrival/retrial/vacation rate) in different reliability levels are investigated in two applicable examples. Additionally, for optimizing the performance of the system, some technical suggestions are presented.

Equilibrium customer and socially optimal balking strategies in a constant retrial queue with multiple vacations and $N$-policy

Abstract: In this paper, equilibrium strategies and optimal balking strategies of customers in a constant retrial queue with multiple vacations and the N-policy under two information levels, respectively, are investigated. We assume that there is no waiting area in front of the server and an arriving customer is served immediately if the server is idle; otherwise (the server is either busy or on a vacation) it has to leave the system to join a virtual retrial orbit waiting for retrials according to the FCFS rules. After a service completion, if the system is not empty, the server becomes idle, available for serving the next customer, either a new arrival or a retried customer from the virtual retrial orbit; otherwise (if the system is empty), the server starts a vacation. Upon the completion of a vacation, the server is reactivated only if it finds at least N customers in the virtual orbit; otherwise, the server continues another vacation. We study this model at two levels of information, respectively. For each level of information, we obtain both equilibrium and optimal balking strategies of customers, and make corresponding numerical comparisons. Through Particle Swarm Optimization (PSO) algorithm, we explore the impact of parameters on the equilibrium and social optimal thresholds, and obtain the trend in changes, as a function of system parameters, for the optimal social welfare, which provides guiding significance for social planners. Finally, by comparing the social welfare under two information levels, we find that whether the system information should be disclosed to customers depends on how to maintain the growth of social welfare.

Modified Bernoulli vacation batch arrival and retrial clients in a single server queuing model with server utilization

Abstract: This paper investigates a batch arrival feedback retrial queue with two types of service under modified Bernoulli vacation where each type consists of an optional re-service where the busy server is subjected to starting failures In Poisson form, the consumer comes to the system in batches but can also baulk at certain specific times. Customers may re-service the same type without joining the orbit after the completion of each type of service or may leave the system. The server either goes on vacation at the completion stage of each service or can wait for the next client to serve. The model is analysed during the supplementary variable technique and the probability generating function of system size, the server utilization and the probability that the system is empty are found. Stochastic decomposition law is shown to hold good for this model also when there is no bulking permitted along with other performance measures to predict the behaviour of the system are derived. Further, we carry out some special cases for the proposed model.

Genetic algorithm in retrial queueing system with server breakdown and caller intolerance with voluntary service

Abstract: The paper investigates the M/M/s retrial queueing system with servers prone to breakdown and with voluntary service. During a phone call, if the caller finds a free call line, then it connects immediately to the call agent, considered as server. Those who finds the call busy either join the orbit for retrial or disconnect the call. After receiving the essential service either caller leaves the system or requests for voluntary service. The miniature used in the queueing system employs a quasi-birth-and-death process. The stationary probabilities are derived using matrix-analytic approach and then performance measures are calculated using these stationary probabilities. The impact of system parameters on performance measures is illustrated using graphs and tables. Total system cost is calculated as a function of orbit capacity, repair rate of breakdown and two different service rates. The optimization methodology is demonstrated using various cases. Further the Genetic algorithm is used to find the optimum value of both essential and optional service rates, rate of breakdown and orbit capacity to minimize the overall cost. The study of consistency of retrial queue with server prone to breakdowns and repairs is of great significance because of inadequate capability of repairs and strong impact of the breakdowns on the performance measures.

Analysis of a Batch Arrival Retrial Queue with Two-Phase Services, Feedback and Admission

Abstract: Today, real-world problems modeling is the first step in controlling, analyzing, and optimizing them. One of the applied techniques for modeling some of these problems is the queueing theory. Usually, the conditions such as lack of space, feedback, admission limits, etc. are the inseparable parts of these problems. This paper deals with modeling and analyzing the steady-state behavior of an $$M^{X} /G/ 1$$ retrial queueing system with two phases of heterogeneous services and general retrial time. The arriving batches join the system with dependent admission due to the server state. If the customers find the server busy, they join the orbit to repeat their request. Although, the first phase of service is essential for all customers, any customer has three options after the completion of the $$i$$ -th phase $$\left( {i = 1,2} \right)$$ . They may take the $$\left( {i + 1} \right)$$ -th phase of service with probability $${\uptheta }_{{\text{i}}}$$ , otherwise, return the orbit with probability $$p_{i} \left( {1 - \theta_{i} } \right)$$ or leave the system with probability $${ }\left( {1 - p_{i} } \right)\left( {1 - \theta_{i} } \right)$$ . In this paper, after finding the steady-state distributions and the probability generating functions of the system and orbit size, some important performance measures are found. Then, the sensitivity analysis of performance measures and cost rate is done concerning arrival/retrial rates, feedback probabilities, and state-dependent admission in a telecommunication system.

On the $ BMAP_1, BMAP_2/PH/g, c $ retrial queueing system

Abstract: In this paper, we consider the BMAP/PH/c retrial queue with two types of customers where the rate of individual repeated attempts from the orbit is modulated according to a Markov Modulated Poisson Process Using the theory of multi-dimensional asymptotically quasi-Toeplitz Markov chain, we obtain the algorithm for calculating the stationary distribution of the system Main performance measures are presented Furthermore, we investigate some optimization problems The algorithm for determining the optimal number of guard servers and total servers is elaborated Finally, this queueing system is applied to the cellular wireless network Numerical results to illustrate the optimization problems and the impact of retrial on performance measures are provided We find that the performance measures are mainly affected by the two types of customers' arrivals and service patterns, but the retrial rate plays a less crucial role

A feed back m$^x$/g/1 retrial queue with impatient customers, active breakdown and orbital search

Abstract: ABSTRACT. The Present paper deals M/G/1 retrial queueing model with an unreliable server, where the server renders two types of essential services. Incoming customers may balk and renege at particular times. The repair of the failed server begins immediately. At the completion of repair, server starts the service for interrupted customer. After completing first essential service (FES), the customer has an option to join the orbit as a feedback customer or may go to second essential service (SES). After SES completion, the server searches for customers in the orbit with certain probability. The system is analysed using Supplementary Variable Technique and probability generating function (PGF) of system size, orbit size and various performance measures are derived.