Showing papers on "Retrial queue published in 2018"

Unreliable M/M/1/1 retrial queues with set-up time

Abstract: We consider a single-server retrial queue with set-up time and server breakdown. There is no waiting space in front of the server, and customers who cannot occupy the server upon arrival will go into a retrial orbit and they retry independently of each other after an exponentially distributed time. To save power, the server is turned off immediately after a service if there is no customer in the orbit. If the orbit is not empty, the server will stay idle to wait a customer coming from the external or from the orbit. A newly coming customer can reactivate the off server, and the server needs some set-up time to work. The server may fail to restart. Once failed, it will be repaired immediately and the repair time is exponentially distributed. Two models are considered according to whether the server can be perfectly repaired or not. The first model is concerned with imperfect repair and the server may still fail to activate after repair. In the second model, the server is perfectly repaired. In both models,...

The state-dependent M / G / 1 queue with orbit

Abstract: We consider a state-dependent single-server queue with orbit. This is a versatile model for the study of service systems, where the server needs a non-negligible time to retrieve waiting customers every time he completes a service. This situation arises typically when the customers are not physically present at a system, but they have a remote access to it, as in a call center station, a communication node, etc. We introduce a probabilistic approach for the performance evaluation of this queueing system, that we refer to as the queueing and Markov chain decomposition approach. Moreover, we discuss the applicability of this approach for the performance evaluation of other non-Markovian service systems with state dependencies.

Sensitivity analysis of an M/G/1 retrial queueing system with disaster under working vacations and working breakdowns

Abstract: This paper deals with the new type of retrial queueing system with working vacations and working breakdowns. The system may become defective by disasters at any point of time when the regular busy server is in operation. The occurrence of disasters forces all customers to leave the system and causes the main server to fail. At a failure instant, the main server is sent to the repair and the repair period immediately begins. As soon as the orbit becomes empty at regular service completion instant or disaster occurs in the regular busy server, the server goes for a working vacation and working breakdown (called lower speed service period). During this period, the server works at a lower service rate to arriving customers. Using the supplementary variable technique, we analyze the steady state probability generating function of system size. Some important system performance measures are obtained. Finally, some numerical examples and cost optimization analysis are presented.

Regenerative Analysis of Two-Way Communication Orbit-Queue with General Service Time

Abstract: In this paper, we apply a regenerative approach to reprove some recent steady-state results [1, 8, 9] for an orbit-queue (also known as retrial queue with a constant retrial rate) with outgoing calls. Stability conditions are discussed as well. Moreover, some generalizations of the model are also considered.

A Priority Retrial Queue with Constant Retrial Policy

Abstract: We analyse a priority queueing system with a normal queue (high priority) and an orbit (low priority). Only the first customer in orbit can retry during times that the queue and server are empty (constant retrial policy). In contrast with existing literature, we assume different service time distributions for the high- and low-priority customers. We obtain closed-form expressions for the probability generating function of the number of customers in queue and orbit, in steady state, and for the Laplace Stieltjes transforms of the stationary waiting times of both type of customers.

Retrial Queue M/M/N with Impatient Customer in the Orbit

Abstract: In the paper, the retrial queueing system of M/M/N type with Poisson flow of events and impatient calls is considered. The delay time of calls in the orbit, the calls service time and the impatience time of calls in the system have exponential distribution. Asymptotic analysis method is proposed for the solving problem of finding distribution of the number of calls in the orbit under a system heavy load and long time patience of calls in the orbit condition. The theorem about the Gauss form of the asymptotic probability distribution of the number of calls in the orbit is formulated and proved. Numerical illustrations, results are also given.

Performance analysis of retrial queue with server subject to two types of breakdowns and repairs

Abstract: This paper analyses a Markovian retrial queue where the server is subject to breakdowns and repairs. It is assumed that the breakdowns/repairs behaviour when the server is idle is different from the one when it is busy. Under the steady-state condition, explicit expressions for the partial probability generating functions of the server status and the number of customers in the orbit are obtained along with some key performance measures of the system. In addition, we study two new orbit characteristics, namely, the orbit idle period and the orbit busy period by using the first principle arguments. An approximate method of analysis for the system with losses is also suggested. The stochastic decomposition property is shown to hold good for the underlying retrial queueing system. Besides, we study the asymptotic behaviour of the system size under extreme conditions. Finally, some numerical results are illustrated.

Retrial queues with starting failure and service interruption

Abstract: The purpose of this study is to investigate an M/M/R retrial queue with geometric loss and Bernoulli feedback, in which all servers are subject to breakdowns and starting failures. After the completion of service, unsatisfied customers can join the retrial group with probability p or depart from the system with probability 1 − p. All servers may breakdown at any time, and the failed server undergoes repair immediately when a breakdown occurs. An arriving customer finding all servers unavailable (busy or down), will either join the orbit with probability b or leave the system with probability 1 − b. For such a queuing model, the authors apply the matrix-geometric method to compute the stationary probabilities and develop system performance measures in the steady state. Moreover, they construct a cost model and formulate an optimisation problem of minimising the expected cost per unit time. Finally, numerical results are given for illustrative purposes.

A Retrial Queueing System with Orbital Search of Customers Lost from an Offer Zone

Abstract: A tandem retrial queueing system with orbital search in which two self-service stations namely, the main station and the offer zone and an orbit for passive customers lost from the offer zone without joining the main station is considered. The main service station is of infinite capacity while the offer zone which works in a random environment and the orbit for passive customers are of finite capacities. Two types of customers arrive to the service stations according to a Marked Markovian Arrival Process (MMAP) with representation \((D_0,D_1,D_2).\) The service times in both stations are exponentially distributed. A virtual search mechanism associated with the main station will be working when the number of customers in the main station is below a pre-assigned level L. The duration of search is exponentially distributed. The condition for system stability is established. The system state distribution in the steady state is obtained. Several system performance characteristics are derived. An associated optimization problem is investigated.

Performance analysis of a discrete-time Geo/G/1 retrial queue under J-vacation policy

Abstract: We have investigated a discrete-time Geo/G/1 retrial queue under J vacation policy and general retrial times, which can be applicable to many digital systems, viz. broadband integrated services digital network (B-ISDN), asynchronous transfer mode (ATM) and circuit-switched time-division multiple access (TDMA) systems. In discrete-time systems, we consider time to be a discrete random variable and measure it in fixed size data units, such as machine cycle time, bits, bytes, packets, etc. The server follows J vacation policy according to which as soon as the retrial orbit becomes empty and no new customer arrives then the server may take at most J number of repeated vacations. The server immediately returns from the vacation if at least one customer arrives in the system. The inter arrival time is assumed to be geometrically distributed whereas retrial time, service time and vacation time are assumed to be generally distributed in the discrete environment. The underlying Markov process is analysed by using generating function method whereby we obtain various performance measures of interest. We provide stochastic decomposition laws for the system size of the developed model.

Refined Tail Asymptotic Properties for the $M^X/G/1$ Retrial Queue

Abstract: In the literature, retrial queues with batch arrivals and heavy service times have been studied and the so-called equivalence theorem has been established under the condition that the service time is heavier than the batch size. The equivalence theorem provides the distribution (or tail) equivalence between the total number of customers in the system for the retrial queue and the total number of customers in the corresponding standard (non-retrial) queue. In this paper, under the assumption of regularly varying tails, we eliminate this condition by allowing that the service time can be either heavier or lighter than the batch size. The main contribution made in this paper is an asymptotic characterization of the difference between two tail probabilities: the probability of the total number of customers in the system for the $M^X/G/1$ retrial queue and the probability of the total number of customers in the corresponding standard (non-retrial) queue. The equivalence theorem by allowing a heavier batch size is another contribution in this paper.

A Retrial Queuing Model with Unreliable Server in K Policy

Abstract: The retrial queue with unreliable server with provision of temporary server has been studied. A temporary server is installed when the primary server is over loaded. It means that a fixed queue length of K-policy customers including the customer with the primary server has been build up. The primary server may breakdown while rendering service to the customers; it is sent for the repair. This type of queuing system has been investigated using matrix geometric method and obtains the probabilities of the system steady state. From the probabilities, we found some performance measures.

Unreliable Single-Server Queue with Two-Way Communication and Retrials of Blocked and Interrupted Calls for Cognitive Radio Networks

Abstract: In this paper, we consider an M/GI/GI/1/1 retrial queue where incoming fresh calls arrive at the server according to a Poisson process. Upon arrival, an incoming call either occupies the server if it is idle or joins an orbit if the server is busy. From the orbit, an incoming call retries to occupy the server and behaves the same as a fresh incoming call. After some idle time, the server makes an outgoing call to outside. We consider the system with an unreliable server. In a free state and while servicing outgoing calls the server is reliable and unable to crash. If while servicing an incoming call the server crashes, the incoming call goes into the orbit. The service time of such an interrupted call follows the same distribution as that of an incoming call. For that system we obtained probability distribution of the states of the server, the condition for the existence of a stationary mode and probability distribution of a number of calls in the system.

Tail Asymptotics for a Retrial Queue with Bernoulli Schedule

Abstract: In this paper, we study the asymptotic behavior of the tail probability of the number of customers in the steady-state $M/G/1$ retrial queue with Bernoulli schedule, under the assumption that the service time distribution has a regularly varying tail. Detailed tail asymptotic properties are obtained for the (conditional and unconditional) probability of the number of customers in the (priority) queue, orbit and system, respectively.

A study on M/G/1 retrial queueing system with three different types of customers under working vacation policy

Abstract: This paper deals with a single server retrial queueing system with working vacations and vacation interruption. There are three different types of customers are considered, which are priority customers, ordinary customers and negative customers. The priority customers do not form any queue and have an exclusive preemptive priority to receive their services over ordinary customers. The negative customer is arriving during the service time of any positive customer, will remove the positive customer from the service. If the interrupted customer is an ordinary customer, he will leave the system. 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. Using the supplementary variable technique, the steady-state probability generating a function of the system and its orbit are found. Some numerical examples are presented.

Stochastic comparison bounds for an M1, M2/G1, G2/1 retrial queue with two way communication

Abstract: The main goal in the present paper is to provide a technique that considers the stochastic comparison approach for investigating monotonicity and comparability of an $M_{1},M_{2}/G_{1},G_{2}/1$ retrial queues with two way communication. This approach is developed for comparing a non Markov process to Markov process with many possible stochastic orderings. Particularly, we show the monotonicity of the transition operator of the embedded Markov chain relative to the strong stochastic ordering and convex ordering, as well as the comparability of two transition operators. Bounds are also obtained for the stationary distribution of the number of customers at departure epochs. Additionally, the performance measures of the system considered can be estimated by those of an $M_1,M_2/M_1,M_2/1$ retrial queue with two way communication when the service time distribution is NBUE (respectively NWUE). Finally, we validate stochastic comparison results by presenting a numerical example illustrating the interest of the approach.

Cost optimisation analysis of retrial queue with K optional phases of service under multiple working vacations and random breakdowns

Abstract: In this paper, we consider a cost optimisation analysis of M/G/1 retrial queue with k optional phases of service under multiple working vacations and vacation interruption, where each phase consists of an optional re-service. An arriving customer may balk the system at some particular times. When the orbit becomes empty at regular service completion instant, the server goes for a working vacation. During a working vacation period, the server works in lower service rate. The normal busy server may get to breakdown and the service channel will fail for a short interval of time. We analyse the steady state probability generating function for the system size by using the supplementary variable method. Some important system performance measures are obtained. Finally, some numerical examples and cost optimisation analysis are presented.

Unreliable retrial queue with loss and feedback under threshold-based policy

Abstract: This paper considers an M/M/1 unreliable retrial queueing system with geometric loss and feedback under the threshold-based policy. After the customer is served completely, he may decide either to leave the system or to join the retrial orbit again for another service. If the server is found busy, the customer may either leave the system or join the retrial orbit. The server may be broken down at any time during the server is operating. When the server is broken down, it cannot be repaired immediately until the number of customers in the system reaches a specified threshold value. This system is modelled by a quasi-birth-and-death process, and some system performance measures are derived. The formulae for computing the rate matrix and stationary probabilities are derived by means of matrix-analytical approach. We obtain the optimal threshold value and the optimal service rate simultaneously to minimise the cost function of the system.

Second Order Asymptotic Properties for the Tail Probability of the Number of Customers in the M/G/1 Retrial Queue

Abstract: When an explicit expression for a probability distribution function $F(x)$ can not be found, asymptotic properties of the tail probability function $\bar{F}(x)=1-F(x)$ are very valuable, since they provide approximations or bounds for system performance, and approaches for computing probability distribution. In this paper, we study tail asymptotic properties for the number of the customers in the $M/G/1$ retrial queueing system. For queueing systems, studies on asymptotic tails have mainly concentrated on the first order asymptotic behavior. To best our knowledge, there is no second order tail asymptotic analysis for retrial queueing models with heavy-tailed service time. Second order asymptotic expansions provide the refined asymptotic results from the first order approximation, and are often more difficult to obtain as expected. The main contribution of this paper is the second order asymptotic analysis for the $M/G/1$ retrial queue.

Optimizing buffer size for the retrial queue: two state space collapse results in heavy traffic

Abstract: We study a single server queueing model with admission control and retrials. In the heavy traffic limit, the main queue and retrial queue lengths jointly converge to a degenerate two-dimensional diffusion process. When this model is considered with holding and rejection costs, formal limits lead to a free boundary curve that determines a threshold on the main queue length as a function of the retrial queue length, above which arrivals must be rejected. However, it is known to be a notoriously difficult problem to characterize this curve. We aim instead at optimizing the threshold on the main queue length independently of the retrial queue length. Our main result shows that in the small and large retrial rate limits, this problem is governed by the Harrison–Taksar free boundary problem, which is a Bellman equation in which the free boundary consists of a single point. We derive the asymptotically optimal buffer size in these two extreme cases, as the scaling parameter and the retrial rate approach their limits.

Optimum alternatives of tandem G/G/K queues with disaster customers and retrial phenomenon: interactive voice response systems

Abstract: In this paper, we study a tandem queue with retrials where the queue experiences disasters. The probability of system failure depends on the strength of equipment, which makes servers idle and causes the removal of all customers in queues and service areas at once. The customers in the queue are forced to orbit in a retrial queue during the system failure where they decide whether or not to come back to the system. Reducing the disaster arrival rate (the probability of system failure) by employing more servers and reducing the number of lost customers is very costly. Moreover, it is important to service the customers with no interruption and reduce the time in system. The developed scenarios are compared in five dimensions including time in system, cost of lost customer, operator cost, the number of uninterrupted service customers and cost of reducing disaster arrival rate (or empowering system cost). The scenarios are modeled by computer simulation. Then, the optimal scenario is chosen using data envelopment analysis. The optimal scenario maximizes system efficiency in terms of disaster arrival rate, cost of lost customers and the number of satisfied customers. In the main problem, the disasters arrive at the system according to Poisson process; the effect of changing the distribution function of disaster arrival has been investigated finally. We are among the first ones to study and optimize G/G/K tandem queuing systems with system failures and retrial phenomena in interactive voice response systems.

Retrial Queue with Search of Interrupted Customers from the Finite Orbit

Abstract: In this paper we consider a single server retrial queue with two orbits of which the first orbit is occupied by primary customers who on arrival find the server busy or interrupted. The other orbit has finite capacity and consists of customers whose service get interrupted due to server breakdown. Interrupted customers are picked up with probability p by the server at the epoch at which he/she becomes free either by the successful completion of a service or by completion of repair. Also there arises a competition between primary customers, retrial customers from the first and the second orbit to access the server. Failed retrials result in the customers returning to the respective orbits. The primary customers arrive according to a Markovian arrival process (MAP), the interruption occur according to a Poisson process. Fixing of interruption takes a random duration having phase type distribution. The service time follows phase type distribution. Stability condition of the system is established. Steady-state system size distribution is obtained. Performance characteristics of the system are evaluated.

System State Distribution of a Finite-Source Retrial Queue with Subscribed Customers

Abstract: The paper deals with a single server, finite-source retrial queue where the server serves two types of customers, called regular customers and subscribed customers. The service times of both types customers follow two distinct arbitrary probability distributions. In addition, the subscribed customers do not join the orbit of repeated regular customers if the server is busy at the time of their arrival. Instead, such an unsuccessful subscribed customer waits till the current regular service is over, and then is accepted for service. Using the supplementary variable approach and the discrete transformations technique we derive formulas for computing the stationary joint distribution of the server state and the orbit size.

On a retrial queue with abandoned customers and multi-optional vacations

Abstract: This paper treats an M/G/1 retrial queue with abandoned customers and multi-optional vacations. If the server is found busy upon a new arrival, the customer being served could be abandoned by the s...

Analysis of mixed priority retrial queueing system with two way communication and working breakdown

Abstract: Incoming calls are arrive at the service system according to compound Poisson process. During the idle time, the server making an outgoing call with an exponentially distributed time. If the incoming call that finds the server busy will join an orbit. Here we use mixed priority services i.e., an arriving call may interrupt the service of an outgoing call or join the retrial queue (orbit). The server takes Bernoulli vacation. The server may become inactive due to normal as well as abnormal breakdown. After the completion of service, vacation and repair the server is in idle state. We consider reneging to occur at the orbit. Using supplementary variable technique, the stability condition is derived.

A Retrial Queueing System with Multiple Hierarchial Orbits and Orbital Search

Abstract: We consider a MAP/PH/1 retrial queueing model with orbital search, consisting of a finite number of orbits which are hierarchially ordered. The model consists of an initial orbit of infinite capacity and a finite number, say M finite capacity orbits, each of which is hierarchially numbered according to the number of unsuccessful retrials made by the customers present in them. Each of the M orbits can hold both individually and collectively a maximum of N customers where \( N \ge M \) and this results in the loss of customers from the system after each unsuccessful retrial. The server searches for those customers who failed to get service even after making a maximum number, say N retrials. At the end of each service completion epoch, the server searches for customers in orbit\( _ M \) with probability p where \( 0 \le p \le 1 \) and with its complementary probability \((1-p)\) the server remains idle. Search time is assumed to be negligible. Steady state analysis of the system is performed. Some performance measures of the system are evaluated.

MAP/PH/1 Retrial Queue with Abandonment, Flush Out and Search of Customers

Abstract: This paper considers a single server retrial queueing system with search, abandonment and flush out of customers from the system (system clearance) periodically with exponentially distributed duration. A customer on arrival, enters for service, if the server is found to be idle and enter into an orbit of infinite capacity if the server is busy. Orbital customers receive service either by successful retrials or by an orbital search. At the epoch of completion of a service, sever goes for search with probability p as long as the orbit size is atmost L-1. Search stops the moment there are L or more customers in the orbit. Further orbital customers are assumed to renege with certain probability on an unsuccessful retrial. In addition, clearance of system takes place each time a random duration following exponential distribution, expires. The customers arrive to the system according to Markovian arrival Process, inter-retrial times are exponentially distributed and service time follows phase type distribution. We analyze the resulting GI/M/1 Type queue. Steady-state analysis of the model is performed. Some performance measures are evaluated.