Retrial Orbit Research Articles

Exact Tail Asymptotics for a Queueing System with a Retrial Orbit and Batch Service

Exact Tail Asymptotics for a Queueing System with a Retrial Orbit and Batch Service

Cloud or In-House Service? Strategic Joining and Social Optimality in Hybrid Service Systems With Retrial Orbit

Cloud computing provides people with fast services, web service companies that need computing resources often submit some of their incoming jobs to the cloud and the rest to their in-house subsystem to seek services. This forms a hybrid service system with cloud service and an in-house server. Taking into account customers' strategic behavior, this article studies the optimal decision making problems in the hybrid service system. A queue-length-based admission control mechanism is adopted in this article to regulate whether the cloud server is open to customers. When the cloud server cannot be accessed immediately, some customers send their jobs to the in-house subsystem, while others (called opportunists) try to send their jobs to the cloud server again. We derive the conditional equilibrium joining probability of entering the in-house subsystem when the cloud is not open. Based on the cooperative and noncooperative game-theoretic analyses, we determine the cooperatively optimal retrial rate and the noncooperatively optimal retrial rate for a given queue-length information. Numerical experiments show that the two optimal retrial rates are equivalent. Finally, it is found that the existence of the opportunists significantly harms social interests, regardless of whether these opportunists are cooperative or noncooperative.

Performance Study of an M/M/1 Retrial Queueing System with Balking, Dissatisfied Customers, and Server Vacations

This study considers a single server retrial queueing system with dissatisfied customers, server vacations, and balking customers. When new arrivals clients see an idle server available, they immediately enter service. If a new arrival sees the server busy, that arrival may balk and leave the system. Otherwise, the new arrival will enter a retrial orbit and try again later. Customers who are dissatisfied with the service will be served again until the service is completed successfully. The server begins a vacation when the current customer completes service. Birth and death state transitions are given. Steady-state distributions for queue length are found. Furthermore, a recursive technique is used to evaluate probabilities. Several performance measures of the system are identified and effects of changes to parameters are observed.

Reliability analysis and optimization design of a repairable k-out-of-n retrial system with two failure modes and preventive maintenance

This article deals with a repairable k-out-of-n: G system with retrial feature and two maintenance policies. All components in the system have two failure modes, mode a failure and mode b failure. When a component fails, the failed component will be repaired instantly if the repair equipment is idle. Otherwise, it enters into a retrial orbit and tries again after some time until it is repaired. The single repair equipment may be subject to breakdown in a busy period, and in order to improve system reliability, the idle repair equipment will be performed preventive maintenance. The Markov process theory and Laplace transform method are adopted to compute the reliability indexes of the system. Also, four optimization design models are constructed to determine the optimal repair rates of the components with two failure modes, and the minimization problems of the total expected cost function per unit time and the cost-effectiveness ratio are solved. To verify the correctness of the model, numerical analysis is executed to demonstrate the impact of each parameter on system reliability indexes, the optimal solutions of different optimization design models are obtained by using differential evolution algorithm and genetic algorithm.

Strategic Analysis of Retrial Queues with Setup Times, Breakdown and Repairs

This paper considers a repairable M/M/1 retrial queueing model with setup times. Once the system is empty, the server will be closed down to reduce operating costs. And the system will be activated only when a new customer arrives. The customer who activates the server will enter the retrial orbit waiting to reapply for service. The server may break down during the busy period. First, the steady-state probability of the model is obtained by using the probability generating function method. And we derive performance measures of the system such as the queue length of the orbit, the numerical examples are given to show the sensitivity of the performance measures. Second, the cost function is established to find the minimum cost of the system, and we study the effects of some parameters on the cost by numerical examples. Finally, from the perspective of the customer and social planner, we construct the individual utility function and the social welfare function in the almost and fully unobservable cases, and then the optimal strategy of the customers is analyzed.

ANFIS and metaheuristic optimization for strategic joining policy with re-attempt and vacation

This research work deals with the queueing analysis and strategic behavior of primitive users (PUs) and ordinary users (OUs) in cognitive radio system (CRS). Whenever fusion center (FC) is occupied, the ordinary users must decide whether to join a retrial orbit and make re-attempts or depart from the system. The OUs residing in the retrial orbit try to access the spectrum through FC. The FC becomes inactive and considered on vacation in case if there are no users in the CRS. Returning back from the vacation, the FC renders service to the primitive users. The service of ordinary users (OUs) may be interrupted due to breakdown of FC. The probability generating function approach is used to derive the performance indices related to the queueing characterization. The joining strategic behavior of the primitive as well as ordinary users has been investigated by considering a linear reward cost function. The social welfare by formulating the cooperative strategy is studied. The hybrid soft computing technique viz. adaptive network fuzzy inference system (ANFIS) is employed for the numerical simulation and sensitivity analysis of various system metrics. The optimal service rates are determined by minimizing the total cost involved in various activities using the quasi-Newton method (QNM) and metaheuristic algorithms.

Queueing and Reliability Analysis of Unreliable Multi-server Retrial queue with Bernoulli feedback

The study consists of an unreliable multiserver retrial queue with feedback where clients may resist to join the orbit and the service provider is erratic. In this model we have considered c service providers at a time and if all the c servers are busy, the incoming clients wait in the retrial orbit. The model is unfolded using Matrix Geometric Method (MGM) to obtain the rate matrix and stationary probabilities. The system with retrials is modelled and the performance measures and reliability indices are procured. These derived quotients are then visualised and validated with the help of tables and graphs. Further, the cost analysis of the model is carried out and the optimal cost for the system is obtained using Particle Swarm Optimization (PSO) algorithm for matrix method. The optimal repair and retrial rates are obtained using PSO technique which helps the computer and manufacturing industry to work towards better decision policies.

Transient analysis and reliability modeling of fault-tolerant system operating under admission control policy with double retrial features and working vacation

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 and analysis for linear consecutive-[formula omitted]-out-of-[formula omitted]: F retrial systems with two maintenance activities

This paper focuses on a linear consecutive-k-out-of-n: F repairable retrial system with an unreliable repair facility, in which preventive maintenance (PM) and repair for working breakdowns are considered for the repair facility. When a component fails, it can be repaired immediately if the repair facility is idle. Otherwise, the failed one would enter the retrial orbit and try again according to the first in first out. Based on repair priority of key components, the steady-state probabilities of the system are derived by using Markov process theory and Crammer’s rule. We analytically calculate steady-state indexes such as steady-state availability, steady-state failure frequency, and mean cycle time of the system. Furthermore, the Runge–Kutta method and the Laplace transform method are employed to obtain the reliability function and the mean time to first failure (MTTFF). Moreover, cost-effectiveness ratio (CER) analysis of the system is provided. Numerical experiments are conducted to illustrate the developed system model and perform influence analysis of PM on steady-state availability and CER for the models with and without PM.

Optimization of retrial queue with unreliable servers subject to imperfect coverage and reboot delay

ABSTRACT A retrial queue with constant retrial policy, unreliable servers and the failed servers may not be detected is investigated in this paper. Upon arrival, a customer either is attended to one of the available servers or enters a retrial orbit and retries in a random time as all servers are not available. The unreliable server can be breakdown when he is working. The breakdown server may not be detected due to some fault matters. Under such a situation, the repair facility will issue a reboot operation to locate the non-detected breakdown server. Once the breakdown server is detected and located, he will be sent for repair. After utilizing the general theory of matrix-geometric approach to determine the stationary distribution, some performance measures are proposed. A single-objective model aiming at minimizing the average cost is constructed. We apply particle swarm optimization algorithm and genetic algorithm to find the optimum combination of parameters. We also formulate a bi-objective model aiming at minimizing the expect cost and the mean waiting time of customers in orbit. Two multi-objective algorithms are developed to solve it.

Analysis of a retrial queueing system with priority service and modified multiple vacations

This paper studies an M/G/1 retrial queueing system with modified multiple vacations, in which a new external arrival may expel the customer being served out of the system and directly starts to be served or join the retrial orbit. Once the server finds the orbit is empty at the end of a service, it will immediately take a random length vacation. If the orbit is still empty when a vacation is finished, the server takes another same vacation. This pattern continues until the orbit is not empty when a vacation is completed or the server has already taken M vacations. By constructing an embedded Markov chain, we provide the sufficient and necessary condition of system stability. The distributions of the orbit size and the system size in steady-state are derived through the supplementary variable method. Then, some system performance measures and the Laplace-Stieltjes transform of sojourn time distribution are obtained. Besides, we perform the cost analysis of the system. Finally, several numerical illustrations are given to investigate the effects of system parameters on the essential system characteristics.

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

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.

Handoff calls’ joining behavior and incentive mechanism in wireless cellular networks with retrial orbit

Handoff calls’ joining behavior and incentive mechanism in wireless cellular networks with retrial orbit

Exact tail asymptotics for the Israeli queue with retrials and non-persistent customers

In this paper, we consider the Israeli queue which consists of a main queue with at most N groups and an infinite capacity retrial orbit. The retrial customers may become non-persistent before receiving service. This model was considered before and the decay rate function of the stationary distribution was obtained. To strengthen the result, we characterize the exact tail asymptotics by calculating the coefficient before the decay rate function.

Markovian working vacation queue with imperfect service, balking and retrial

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).

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

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.

On the retrial queue with imperfect coverage and delay reboot

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.

On retrial queue with customer balking and feedback subject to server breakdowns

Motivated by some real stochastic service systems, we study a feedback retrial queue with customer balking and unreliable servers (servers subject to breakdowns). A customer will either join a retrial orbit or balk if all servers are busy upon his arrival. After a service completion, the customer either leave the system or enter the retrial group again for additional service. The customer who is interrupted by the server failure will be served by another server if available; otherwise, he will be contacted later and treated as leaving the system. We use a quasi-birth-and-death process to analyze this system with almost all possible uncertain factors and derive major performance measures of the system. We also develop a cost function determining the optimal parameter settings of the system under the stability condition. Probabilistic Global Search Lausanne approach is employed in solving the optimization issue. The effects of parameters on performance measures of the system are examined numerically. Finally, we discuss the application of the model in the telephone medical consulting service system.

Reliability and availability analysis of a retrial system with mixed standbys and an unreliable repair facility

We study a K-out-of-M+W+C: G mixed standby system with an unreliable repair facility, in which there are M primary operating components, W warm standby components and C cold standby components. The life times of primary components and warm components are assumed to be exponentially distributed random variables. The repair facility is performed preventive maintenance (PM) when it is idle, and it is subject to breakdowns when it repairs a failed component (called as active breakdowns). Upon arrival, a failed component is repaired immediately if the repair facility is found free, and the failed component enters a retrial orbit for seeking repair service after some time later if the repair facility is found busy, under PM, or under repair. By adopting a Markov process approach and solving equations of statistical equilibrium in matrix form, we obtain the system steady-state availability of the machine system, the steady-state availability and the expected busy cycle of the repair facility. Furthermore, we obtain the reliability function and the mean time to the failure (MTTF) of the system by using Laplace transform method. Numerical examples are presented to show the effects of system parameters on the reliability function, steady-state availability and MTTF and to analyze the optimization problem of the cost-effectiveness ratio.

Reliability and sensitivity analysis of a repairable k-out-of-n:G system with two failure modes and retrial feature

This paper investigates the reliability and sensitivity analysis for a repairable k-out-of-n:G system with retrial of failed components. Such a model has important practical applications in fully automatic systems, and the most typical one is fully automatic manufacturing system. Markov models for availability and reliability of the system whose components are all subject to two failure modes are presented. There is no waiting space for failed components in the system. If a failed component finds the repairman busy and it can not be repaired at once, it will enter into the retrial orbit and try again for repair after some random period of time. Some reliability indexes, including steady-state availability, reliability function and mean time to system first failure, are derived by using vector Markov process and Laplace transform theory. Sensitivity analysis and relative sensitivity analysis are provided as well. Finally, some numerical experiments are conducted to show the effects of system parameters on the system reliability indexes.