Retrial Times Research Articles

A Discrete-TimeGeo/G/1 Retrial Queue with Two Different Types of Vacations

We analyze a discrete-timeGeo/G/1 retrial queue with two different types of vacations and general retrial times. Two different types of vacation policies are investigated in this model, one of which is nonexhaustive urgent vacation during serving and the other is normal exhaustive vacation. For this model, we give the steady-state analysis for the considered queueing system. Firstly, we obtain the generating functions of the number of customers in our model. Then, we obtain the closed-form expressions of some performance measures and also give a stochastic decomposition result for the system size. Moreover, the relationship between this discrete-time model and the corresponding continuous-time model is also investigated. Finally, some numerical results are provided to illustrate the effect of nonexhaustive urgent vacation on some performance characteristics of the system.

An Unreliable Server Retrial Queue with Two Phases of Service and General Retrial Times Under Bernoulli Vacation Schedule

This paper deals with the steady state behavior of an M/G/1 retrial queue with two successive phases of service and general retrial times under Bernoulli vacation schedule for an unreliable server. While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. The primary customers finding the server busy, down, or on vacation are queued in the orbit in accordance with the FCFS (first come, first served) retrial policy. After the completion of the second phase of service, the server either goes for a vacation of random length with probability p or serves the next unit, if any, with probability (1 – p). For this model, we first obtain the condition under which the system is stable. Then, we derive the system size distribution at a departure epoch and the probability generating function of the joint distributions of the server state and orbit size, and prove the decomposition property. We also provide a reliability analysis of this model, which is one of the chief objectives of the paper. Finally, numerical illustrations show how the performance measures obtained are used to manage the system.

AN M/G/1 RETRIAL QUEUE WITH GENERAL RETRIAL TIMES, WORKING VACATIONS AND VACATION INTERRUPTION

We consider an M/G/1 retrial queue with general retrial times, and introduce working vacations and vacation interruption policy into the retrial queue. During the working vacation period, customers can be served at a lower rate. If there are customers in the system at a service completion instant, the vacation will be interrupted and the server will come back to the normal working level. Using supplementary variable method, we obtain the stationary probability distribution and some performance measures. Furthermore, we carry out the waiting time distribution and prove the conditional stochastic decomposition for the queue length in orbit. Finally, some numerical examples are presented.

STABILITY ANALYSIS AND SIMULATION OF N-CLASS RETRIAL SYSTEM WITH CONSTANT RETRIAL RATES AND POISSON INPUTS

In this paper, we study a new retrial queueing system with N classes of customers, where a class-i blocked customer joins orbit i. Orbit i works like a single-server queueing system with (exponential) constant retrial time (with rate [Formula: see text]) regardless of the orbit size. Such a system is motivated by multiple telecommunication applications, for instance wireless multi-access systems, and transmission control protocols. First, we present a review of some corresponding recent results related to a single-orbit retrial system. Then, using a regenerative approach, we deduce a set of necessary stability conditions for such a system. We will show that these conditions have a very clear probabilistic interpretation. We also performed a number of simulations to show that the obtained conditions delimit the stability domain with a remarkable accuracy, being in fact the (necessary and sufficient) stability criteria, at the very least for the 2-orbit M/M/1/1-type and M/Pareto/1/1-type retrial systems that we focus on.

APPROXIMATION OF PH/PH/c RETRIAL QUEUE WITH PH-RETRIAL TIME

We consider the PH/PH/c retrial queues with PH-retrial time. Approximation formulae for the distribution of the number of customers in service facility, sojourn time distribution and the mean number of customers in orbit are presented. We provide an approximation for GI/G/c retrial queue with general retrial time by approximating the general distribution with phase type distribution. Some numerical results are presented.

An improved truncation technique to analyze a Geo/PH/1 retrial queue with impatient customers

This paper considers a discrete-time retrial queue with impatient customers. We establish the global balance equations of the Markov chain describing the system evolution and prove that this queueing system is stable as long as the customers are strict impatient and the mean retrial time is finite. Direct truncation with matrix decomposition is used to approximate the steady-state distribution of the system state and hence derive a set of performance measures. The proposed matrix decomposition scheme is presented in a general form which is applicable to any finite Markov chain of the GI/M/1-type. It represents a generalization of the Gaver–Jacobs–Latouche's algorithm that deals with QBD process. Different sets of numerical results are presented to test the efficiency of this technique compared to the generalized truncation one. Moreover, an emphasis is put on the effect of impatience on the main performance measures.

An unreliable retrial queue with delaying repair and general retrial times under Bernoulli vacation schedule

This paper deals with the steady state behavior of an M/G/1 queue with general retrial time and Bernoulli vacation schedule for an unreliable server, which consist of a breakdown period and delay period. While the server is working with primary customers, it may breakdown at any instant and server will be down for short interval of time. Further concept of the delay time is also introduced. The primary customer finding the server busy, down or vacation enters a group of unsatisfied customer called orbit. After the completion of a service, the server either goes for a vacation of random length with probability p or may continue serve for the next unit, if any with probability (1−p). For this model, we first derive the system size distribution at a departure epoch. Secondly, we derive the probability generating function of joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Finally, we obtain some important performance measures and reliability indices of this model.

Performance analysis of a batch arrival retrial queue with Bernoulli feedback

This paper facilitates the performance prediction of an M X /G/1 retrial queue wherein the single server follows a modified vacation policy . We assume that the job who finds the server busy joins the retrial group (orbit) to get its service in random order and only the job at the head of the queue is allowed to receive service from the server first. As soon as the system becomes empty, the server leaves for at most J vacations of random length V each. When the server returns from the vacation and finds at least one job in the orbit, it renders service to these jobs otherwise it goes for next vacation. We assume that the service time, retrial time and vacation times are general distributed. The balking behaviour of the jobs is also taken into consideration. By using supplementary variable technique, we derive the formulas for system size distribution and mean system size at departure points and other performance measures. Numerical illustrations have been provided to validate the analytical results. In addition, by setting appropriate parameters, some special cases are deduced which tally with existing results. Tables and figures have been facilitated to examine the system behaviour with regard to different parameters.

Equilibrium Customer Strategies in the Single-Server Constant Retrial Queue with Breakdowns and Repairs

We consider a single-server constant retrial queueing system with a Poisson arrival process and exponential service and retrial times, in which the server may break down when it is working. The lifetime of the server is assumed to be exponentially distributed and once the server breaks down, it will be sent for repair immediately and the repair time is also exponentially distributed. There is no waiting space in front of the server and arriving customers decide whether to enter the retrial orbit or to balk depending on the available information they get upon arrival. In the paper, Nash equilibrium analysis for customers’ joining strategies as well as the related social and profit maximization problems is investigated. We consider separately the partially observable case where an arriving customer knows the state of the server but does not observe the exact number of customers waiting for service and the fully observable case where customer gets informed not only about the state of the server but also about the exact number of customers in the orbit. Some numerical examples are presented to illustrate the effect of the information levels and several parameters on the customers’ equilibrium and optimal strategies.

Steady State Analysis of Repairable M/G/1 Retrial Queuing Model with Modified Server Vacation Balking and Optional Reservice

A retrial queue with general retrial times, modified vacation policy and Bernoulli feedback is analyzed in this paper. If the server is busy or on vacation, an arriving customer either enters on orbit with probability b, or balks with probability 1-b. Otherwise the service of the arriving customer commences immediately. At the service completion epoch, the test customer may either enter the orbit for another service with probability p or leave the system with probability q (=1-p). If the orbit is empty, the server takes at most J vacations until at least one customer is recorded in the orbit when the server returns from a vacation. By applying supplementary variable technique some analytical results for the system are derived. The effects of various parameters on the system performance are analyzed numerically.

Availability of a repairable retrial system with warm standby components

In this paper, we study a repairable K-out-of-(M+W) retrial system with M identical primary components, W standby components and one repair facility. The time-to-failure and time-to-repair of the primary and standby components are assumed to be exponential and general distributions, respectively. The failed components are immediately for repair if the server is idle, otherwise the failed machines would enter an orbit. It is assumed that the retrial times are exponentially distributed. We present a recursive method using the supplementary variable technique and treating the supplementary variable as the remaining repair time to obtain the steady-state probabilities of down components at arbitrary epoch. Then, a unified and efficient algorithm is developed to compute the steady-state availability. The method is illustrated analytically for the exponential repair time distribution. Sensitivity analysis of the steady-state availability with respect to system parameters for a variety of repair time distributions is also investigated.

A discrete-time Geo/G/1 retrial queue with J-vacation policy and general retrial times

The authors discuss a discrete-time Geo/G/1 retrial queue with J-vacation policy and general retrial times. As soon as the orbit is empty, the server takes a vacation. However, the server is allowed to take a maximum number J of vacations, if the system remains empty after the end of a vacation. If there is at least one customer in the orbit at the end of a vacation, the server begins to serve the new arrivals or the arriving customers from the orbit. For this model, the authors focus on the steady-state analysis for the considered queueing system. Firstly, the authors obtain the generating functions of the number of customers in the orbit and in the system. Then, the authors obtain the closed-form expressions of some performance measures of the system and also give a stochastic decomposition result for the system size. Besides, the relationship between this discrete-time model and the corresponding continuous-time model is also investigated. Finally, some numerical results are provided.

FURTHER RESEARCH ON THE M/M/1 RETRIAL QUEUEING MODEL WITH SPECIAL RETRIAL TIMES

We study the spectrum of the underlying operator which cor- respondents to the M/M/1 retrial queueing model with special retrial times

Delay Analysis of Orderly Reattempts in Retrial Queueing System with Phase Type Retrial Time

Retrial queueing systems play a major role in many communication systems. The delay analysis of such systems poses an interesting problem in communication engineering. This has motivated us to analyze the waiting time distribution of a multi-server retrial queueing system with phase type retrial time, impatience and Bernoulli abandonments. The impact of retrial time on waiting time and other system characteristics is analyzed by determining the waiting time distribution. Numerical examples are presented to reveal the impact of the parameters.

Unreliable M<SUP align="right">x</SUP>/G/1 retrial queue with multi-optional services and impatient customers

The present investigation deals with Mx/G/1 retrial queue with unreliable server and general retrial times. The server is subject to breakdowns and takes some setup time before starting the repair. The server renders first essential phase of service (FES) to all the arriving customers whereas second optional phase services (SOS) are provided after FES to only those customers who opt for it. The impatient customers are allowed to balk depending upon server’s status; they may also renege after waiting sometime in the queue. By incorporating the supplementary variables corresponding to service time, repair time, retrial time and setup time and then using generating function method, the queueing analysis has been done to obtain the queue size and orbit size distributions, and some other queueing as well as reliability measures. The effects of several parameters on the system performance are examined numerically by taking an illustration.

A Discrete-Time UnreliableGeo/G/1Retrial Queue with Balking Customers, Second Optional Service, and General Retrial Times

This paper deals with the steady-state behavior of a discrete-time unreliableGeo/G/1retrial queueing system with balking customers and second optional service. The server may break down randomly while serving the customers. If the server breaks down, the server is sent to be repaired immediately. We analyze the Markov chain underlying the considered system and its ergodicity condition. Then, we obtain some performance measures based on the generating functions. Moreover, a stochastic decomposition result of the system size is investigated. Finally, some numerical examples are provided to illustrate the effect of some parameters on main performance measures of the system.

Discrete-Time Geox/G/1 Retrial Queue with General Retrial Times, Working Vacations and Vacation Interruption

We consider a discrete-time GeoX/G/1 retrial queue with general retrial times, and introduce working vacations and vacation interruption policy into the retrial queue. Firstly, we analyze the stationary condition for the embedded Markov chain at the departure epochs. Secondly, using supplementary variable method, we obtain the stationary probability distribution and some performance measures. Furthermore, we prove the conditional stochastic decomposition for the queue length in the orbit. Finally, some numerical examples are presented.

A discrete-time Geo/G/1 retrial queue with preferred and impatient customers

This paper is concerned with a discrete-time Geo/G/1 retrial queue with preferred, impatient customers and general retrial times. We analyze the Markov chain underlying the considered queueing system and derive its ergodicity condition. The system state distribution as well as the orbit size and the system size distributions are obtained in terms of their generating functions. These generating functions yield exact expressions for different performance measures. Besides, the stochastic decomposition property and the corresponding continuous-time queueing system are investigated. Finally, some numerical examples are provided to illustrate the effect of priority and impatience on several performance characteristics of the system.

Discrete-time Geo/G/1 retrial queues with general retrial time and Bernoulli vacation

This paper considers a discrete-time Geo/G/1 retrial queue where the retrial time has a general distribution and the server is subject to Bernoulli vacation policy. It is assumed that the server, after each service completion, begins a process of search in order to find the following customer to be served with a certain probability, or begins a single vacation process with complementary probability. This paper analyzes the Markov chain underlying the queueing system and obtain its ergodicity condition. The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle, busy or on vacation. Finally, the author gives two stochastic decomposition laws, and as an application the author gives bounds for the proximity between the system size distributions of the model and the corresponding model without retrials.

Retrial Queuing System with Starting Failure, Single Vacation and Orbital Search

This paper is concerned with the analysis of a single server retrial queue with vacation and orbital search. The server is subject to starting failure and repair. At the completion epoch of each service, the server may take a single vacation. After vacation completion, the server searches for the customers in the orbit or remains idle. Retrial times, service times and vacation times are assumed to be arbitrarily distributed. Various performance measures are derived and numerical results are presented.