General Retrial Times Research Articles

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.

Analysis of bulk retrial queue using maximum entropy principle

In the present investigation, we study the maximum entropy principle (MEP) for the Mx/G/1 retrial queue with general retrial times and unreliable server. The customers arrive in bulk in Poisson fashion and are served if they find the server idle otherwise join the virtual pool of customers called orbit. By using maximum entropy principle, approximate results for the queue length as well as waiting time are obtained. We construct the maximum entropy function by using several known constraints so as to determine the probability distributions. Moreover, a comparative study has been done between the exact analytic expressions obtained by the supplementary variable technique and approximate results obtained by MEP.

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 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.

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 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.

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.

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.

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 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.

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 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.

The Geo(superscript [X])/G/1 Retrial Queue with General Retrial Times, Admission Control and Starting Failures

This paper analyzes a discrete-time Geo(superscript [X])/G/1 retrial queue with control of admission where the retrial time has a general distribution and the server is subject to starting failures. Batches of customers arrive at the system according to a geometrical process and each individual customer in a batch is subject to a control admission policy upon arrival. If the server is found idle, one of the customers admitted to the system may start his service and the rest joins the retrial group, whereas all the admitted customers go to the retrial group when the server is unavailable upon arrival. An arriving customer (primary or retrial) must turn- on the server, which takes negligible time. If the server is started successfully, the customer gets service immediately. Otherwise, the repair for the server commences immediately and the customer must leave for the orbit and make a retrial at a later time. It is assumed that the customers who find the server unavailable are queued in the orbit in accordance with an FCFS discipline and only the customer at the head of the queue is allowed for access to the server. Both external arrivals and first customer in orbit may arrive at the same epoch. Under the early arrival system (EAS) assumption, the Markov chain underlying the considered queueing system is studied and the generating functions of the orbit size and the system size distributions are obtained along with the marginal distributions of the orbit size when the server is idle, busy or down. Finally, two stochastic decomposition laws for the system size distribution are presented.

A batch arrival retrial queue with general retrial times under Bernoulli vacation schedule for unreliable server and delaying repair

This paper deals with the steady state behaviour of an M x / G/1 queue with general retrial time and Bernoulli vacation schedule for an unreliable server, which consists of a breakdown period and delay period. Here we assume that customers arrive according to compound Poisson processes. 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 are queued in the orbit in accordance with FCFS (first come first served) retrial policy. After the completion of a service, the server either goes for a vacation of random length with probability p or may continue to serve for the next customer, if any with probability (1 − p). We carry out an extensive analysis of this model. Finally, we obtain some important performance measures and reliability indices of this model.

A discrete-time Geo[X]/G/1 retrial queue with general retrial time and M-additional options for service

This paper concerns a discrete time Geo[X] /G/1 retrial queue with general retrial time in which all the arriving customers require first essential service with probability while only some of them demand one of other optional services: type − r (r = 1, 2, 3,...M ) service with probability . The system state distribution, the orbit size and the system size distributions are obtained in terms of generating functions. The stochastic decomposition law holds for the proposed model. Performance measures of the system in steady state are obtained. Finally, some numerical illustrations are presented to justify the influence of parameters on several performance characteristics.

A general retrial system with additional vacation

This paper studies a retrial system with batch arrival and general retrial time, where the server may take an additional vacation after the essential vacation. In this system, primary customers arrive according to Poisson processes and receive corresponding service immediately if the server is available upon arrival. When the server is busy or on vacation, arriving customers enter a retrial orbit. If the orbit is empty, the server immediately leaves for the essential vacation. At the end of essential vacation, he may take one additional vacation. It is assumed that the retrial time and service time are all arbitrary distributed. This retrial system has potential applications in the packet-switched networks and the proxy of WWW server. By applying the supplementary variable technique, some important performance measures are derived, which may be useful for network system designers and software system engineers.

A repairable discrete-time retrial queue with recurrent customers, Bernoulli feedback and general retrial times

This paper relates to a repairable Geo/G/1 retrial queue with general retrial times, Bernoulli feedback, the server subjected to starting failures and two types of customers: transit and a fixed number of recurrent customers. After service completion, recurrent customers always return to the orbit and transit customers either immediately return to the orbit for another service with probability \(\theta (0\leq\theta<1)\) or leave the system forever with probability 1 − θ. We construct the mathematical model and present some performance measures of the model in steady-state. We provide a stochastic decomposition law and analyze the relationship between our discrete-time system and its continuous-time counterpart. Finally, some numerical examples show the influence of the parameters on the performance characteristics of the system.

A discrete-time Geo/G/1 retrial queue with preemptive resume and collisions

We consider a discrete-time Geo/G/1 retrial queue with preemptive resume, collisions of customers and general retrial times. We analyze the Markov chain underlying the considered queueing system and derive its ergodicity condition. Using generating function technique, the system state distribution as well as the orbit size and the system size distributions are studied. Some interesting and important performance measures are obtained. Besides, the stochastic decomposition property is investigated. Finally, some numerical examples are provided.

A Single Server Retrial Queue with Bernoulli Feedback and Collisions

A non-Markovian feedback single-server retrial queue with collisions and general retrial times is investigated. A necessary and sufficient condition for the system to be stable is studied. Using the supplementary variable technique, the joint distribution of the server state and the orbit length under steady-state is obtained. Some interesting and important performance measures of the system are provided. Finally, numerical illustrations are presented.