General Retrial Times Research Articles

Unreliable M[X]/G(P1,P2)/1 feedback retrial queues with combined working vacation

The study examines M[X]/G(P1,P2)/1 feedback retrial queues coupled with starting failure, repair, delay to repair, working vacation, and general retrial times. Also, it explores how different batch sizes affect performance and how bulk arrival affects system behavior. When the server is not in use, a single customer initiates the system while the remaining customers transition to a state of orbit. A new customer must turn on the server to provide two phases of mandatory service at any time. The server could have starting issues. If the service is successfully started (with likelihood α), the customer receives service immediately. In the absence of that, the likelihood of starting failure happens (with likelihood 1−α=α¯). The server was taken for repair with some delay and that customer was transported to an orbital location. When the server was busy or unavailable, the arriving customers queued by FCFS in the orbit. We also discuss the idea of reworking with probability p, and restarting unsuccessful service attempts to improve customer happiness and service efficiency. We also introduce the concept of working vacation, which permits servers to temporarily stop providing services, affecting system performance and availability at both peak and off-peak times. A supplementary variable technique was adopted for the system's and orbit size's probability-generating function. Various performance measurements were provided with appropriate numerical examples.

Corrigendum: “Extended generator and associated martingales for M/G/1 retrial queue with classical retrial policy and general retrial times”

Corrigendum: “Extended generator and associated martingales for M/G/1 retrial queue with classical retrial policy and general retrial times”

Performance analysis of discrete-time GeoX/G/1 retrial queue with various vacation policies and impatient customers

This paper studies the behavior of a batch arrival single server retrial queueing model under three different vacation policies. Three types of vacation policies, single vacation, multiple vacations, and atmost J-vacations with impatient customers in general retrial times are considered. The probability generating function and marginal generating function of orbit size are obtained in a steady state. The stability condition for each vacation model is derived. Performance measures such as mean orbit size, mean system size, mean waiting time of a customer, and the probabilities of the server being in different states have also been determined. Based on performance characteristics, a comparative analysis is performed among the three vacations. Numerical illustrations are displayed to establish the consistency of the theory developed.

Extended generator and associated martingales for M/G/1 retrial queue with classical retrial policy and general retrial times

AbstractA retrial queue with classical retrial policy, where each blocked customer in the orbit retries for service, and general retrial times is modeled by a piecewise deterministic Markov process (PDMP). From the extended generator of the PDMP of the retrial queue, we derive the associated martingales. These results are used to derive the conditional expected number of customers in the orbit in the transient regime.

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

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.

Stochastic analysis of a single server unreliable queue with balking and general retrial time

In this investigation, we consider an M/G/1 queue with general retrial times allowing balking and server subject to breakdowns and repairs. In addition, the customer whose service is interrupted can stay at the server waiting for repair or leave and return while the server is being repaired. The server is not allowed to begin service on other customers until the current customer has completed service, even if current customer is temporarily absent. This model has a potential application in various fields, such as in the cognitive radio network and the manufacturing systems, etc. The methodology is strongly based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary distribution of the embedded Markov chain of the considered system.

An M/G/1 retrial queue with single working vacation under Bernoulli schedule

In this paper, an M/G/1 retrial queue with general retrial times and single working vacation is considered. We assume that the customers who find the server busy are queued in the orbit in accordance with a first-come-first-served (FCFS) discipline and only the customer at the head of the queue is allowed access to the server. During the normal period, if the orbit queue is not empty at a service completion instant, the server begins a working vacation with specified probabilityq(0 ≤ q ≤ 1), and with probability 1 − q, he waits for serving the next customer. During the working vacation period, customers can be served at a lower service rate. We first present the necessary and sufficient condition for the system to be stable. Using the supplementary variable method, we deal with the generating functions of the server state and the number of customers in the orbit. Various interesting performance measures are also derived. Finally, some numerical examples and cost optimization analysis are presented.

Cost optimization and ANFIS computing for admission control of M/M/1/K queue with general retrial times and discouragement

The present study deals with the admission control policy for the single server finite capacity queueing model with discouraged customers and general distributed retrial times. The arriving customer is forced to join the retrial orbit on finding the server busy. From the orbit, the customer can retry for the service again after sometime. Due to the long queue in front of the server, the customers may be discouraged and leave the system without receiving the service. The noble feature of proposed study is to control the arrivals in the system based on F-policy according to which as soon as the system capacity becomes full, no more customers are permitted to join the system until the system capacity again drops to threshold level ‘F’. The steady state queue size distribution of the system size is evaluated by introducing the supplementary variables corresponding to remaining retrial times and framing Chapman–Kolmogorov equations. The recursive and soft computing based artificial neuro fuzzy inference system (ANFIS) approaches are applied to solve governing equations and to establish various performance indices. Furthermore, a cost function is also constructed to evaluate the optimal service rate and the corresponding expected cost of the system. Genetic algorithm (GA) and quasi-Newton method (QNM) are used to minimize the expected cost of the system to decide the optimal decision parameter corresponding to service rate.

Admission control for finite capacity queueing model with general retrial times and state-dependent rates

The finite state dependent queueing model with \begin{document}$ F $\end{document} -policy is investigated by considering the general retrial attempts. On arrival in the system, if the job finds the server engaged, it is forced to enter into the retrial orbit. After a random period of time, the job from the retrial orbit re-attempts for the service. According to \begin{document}$ F $\end{document} -policy, as the system attains its full capacity, the arrivals are restricted to join the system until the number of jobs comes down to the prefixed threshold value ' \begin{document}$ F $\end{document} '. The supplementary variable corresponding to the remaining retrial time is used to frame the governing equations which are solved by using Laplace-Stieltjes transform and then applying the recursive method. Special models for machine repair and time-sharing queue are deduced by setting the state dependent rates. Several system indices are obtained explicitly which are further used to facilitate the sensitivity analysis by considering a numerical illustration. A cost function is constructed and minimized for evaluating the optimal threshold parameter and optimal service rate.

Asymptotic behavior of the solution of a queueing system modeled by infinitely many partial differential equations with integral boundary conditions

By studying the spectrum on the imaginary axis of the underlying operator, which corresponds to the M/G/1 retrial queueing model with general retrial times described by infinitely many partial differential equations with integral boundary conditions, we prove that the time-dependent solution of the model strongly converges to its steady-state solution. Next, when the conditional completion rates for repeated attempts and service are constants, we describe the point spectrum of the underlying operator and verify that all points in an interval in the left real line including 0 are eigenvalues of the underlying operator. Lastly, by using these results and the spectral mapping theorem we prove that the C_{0}-semigroup generated by the underlying operator is not compact, but not eventually compact and even not quasi-compact, and it is impossible that the time-dependent solution exponentially converges to its steady-state solution. In other words, our result on convergence is optimal.

Analysis of a single server batch arrival unreliable queue with balking and general retrial time

ABSTRACTThis paper deals with a batch arrivals queue with general retrial time, breakdowns, repairs and reserved time. Here we assume that customers arrive according to compound Poisson processes. Any arriving batch of primary customers finds the server free, one of the customers from the batch enters into the service area and the rest of them join into the orbit. The primary customers who find the server busy or failed are allowed to balk or are queued in the orbit in accordance with FCFS retrial policy. The customer whose service is interrupted can stay at the server waiting for repair or enter into service orbit. After the repair is completed, the server resumes service immediately if the customer in service has remained in the service position. This model has a potential applications in various fields, such as in the cognitive radio network and the manufacturing systems. By using supplementary variables technique, we carry out an extensive analysis of the considered model. Then, we obtain some important performance measures, stochastic decomposition property of the system size distribution and the reliability indices. Next, by setting the appropriate parameters, some special cases are discussed. Finally, some numerical examples and cost analysis are presented.

An M/G/1 retrial queue with balking customers and Bernoulli working vacation interruption

In this paper, an M/G/1 retrial queue with general retrial times and Bernoulli working vacation interruption is considered. If the server is busy, an arriving customer either enters an orbit with probability q () or balks (does not enter) with probability . During a working vacation period, if there are customers in the system at a service completion instant, the vacation either is interrupted with probability p () or continues with probability . By applying the supplementary variable technique, we obtain the steady state joint distribution of the server and the number of customers in the orbit. Various interesting performance measures are also derived. Finally, some numerical examples and cost optimization analysis are presented.

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

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.

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

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.

Analysis of adiscrete- time single server retrial queue with general retrial times, multiple vacations and statedependent arrivals

Analysis of adiscrete- time single server retrial queue with general retrial times, multiple vacations and statedependent arrivals

An M/G/1 Retrial G-Queue with General Retrial Times and Working Breakdowns

This paper considers an M/G/1 retrial G-queue with general retrial times, in which the server is subject to working breakdowns and repairs. If the system is not empty during a normal service period, the arrival of a negative customer can cause the server breakdown, and the failed server still works at a lower service rate rather than stopping the service completely. Applying the embedded Markov chain, we obtain the necessary and sufficient condition for the stability of the system. Using the supplementary variable method, we deal with the generating functions of the number of customers in the orbit. Various system performance measures are also developed. Finally, some numerical examples and a cost optimization analysis are presented.

Performance analysis of an M/G/1 retrial queue with general retrial time, modified M-vacations and collision

In this paper, a single server retrial queue with general retrial time and collisions of customers with modified M-vacations is studied. The primary calls arrive according to Poisson process with rate λ. If the server is free, the arriving customer/the customer from orbit gets served completely and leaves the system. If the server is busy, arriving customer collides with the customer in service resulting in both being shifted to the orbit. After the collision the server becomes idle. If the orbit is empty the server takes at most M vacations until at least one customer is recorded in the orbit when the server returns from a vacation. Whenever the orbit is empty the server leaves for a vacation of random length V. If no customers appear in the orbit when the server returns from vacation he again leaves for another vacation with the same length. This pattern continues until he returns from a vacation to find at least one customer recorded in the orbit or he has already taken M vacations. If the orbit is empty by the end of the Mth vacation, the server remains idle for customers in the system. The time between two successive retrials from the orbit is assumed to be general with arbitrary distribution R(t). By applying the supplementary variables method, the probability generating function of number of customers in the orbit is derived. Some special cases are also discussed. A numerical illustration is also presented.

APPROXIMATE ANALYSIS OF M/M/c RETRIAL QUEUE WITH SERVER VACATIONS

We consider the M/M/c/c queues in which the customers blocked to enter the service facility retry after a random amount of time and some of idle servers can leave the vacation. The vacation time and retrial time are assumed to be of phase type distribution. Approximation formulae for the distribution of the number of customers in service facility and the mean number of customers in orbit are presented. We provide an approximation for M/M/c/c queue with general retrial time and general vacation time by approximating the general distribution with phase type distribution. Some numerical results are presented.

A preemptive priority retrial queue with two classes of customers and general retrial times

In this paper, we consider a continuous-time retrial queue with two classes of customers: priority customers and ordinary customers, where priority customers don’t queue and have an exclusive preemptive priority to receive their services over ordinary customers. If an arriving ordinary customer finds the server busy, it enters a retrial group (called orbit) according to FCFS discipline. Only the ordinary customer at the head of the retrial queue is allowed to access the server. Firstly, we obtain the necessary and sufficient condition for the system to be stable by embedded Markov chain approach. Secondly, using supplementary variable method, we obtain the stationary probability distribution and some performance measures of interest. Thirdly, we give the analysis of the sojourn time in the system of an arbitrary ordinary customer. Lastly, numerical examples are given to show the effect of system parameters on several performance measures.

A batch arrival unreliable Bernoulli vacation model with two phases of service and general retrial times

This paper deals with the steady state behaviour of an Mx/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 first come, first served (FCFS) 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 may serve 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.