Retrial Queueing System Research Articles

A two-class queueing system with constant retrial policy and general class dependent service times

A single server retrial queueing system with two-classes of orbiting customers, and general class dependent service times is considered. If an arriving customer finds the server unavailable, it enters a virtual queue, called the orbit, according to its type. The customers from the orbits retry independently to access the server according to the constant retrial policy. We derive the generating function of the stationary distribution of the number of orbiting customers at service completion epochs in terms of the solution of a Riemann boundary value problem. For the symmetrical system we also derived explicit expressions for the expected delay in an orbit without solving a boundary value problem. A simple numerical example is obtained to illustrate the system’s performance.

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

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.

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

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.

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

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.

A single server non-Markovian retrial queue with two types of service and Bernoulli vacation

A single server retrial queueing system with two types of service and Bernoulli vacation is analysed in this paper. It is assumed that the arriving job finds the server busy by providing either type of service is said to be in orbit with an FCFS discipline and repeat its request (demand) for service after some random time. The customer at the head of the orbit is allowed to access the server. For such queueing model, the system size probabilities are investigated in steady state by using supplementary variable technique. The effects of various parameters on the system performance are analysed numerically. Stochastic decomposition and some special cases of interest are also discussed.

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

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.

Asymptotic Analysis of an Retrial Queueing System M|M|1 with Collisions and Impatient Calls

Asymptotic Analysis of an Retrial Queueing System M|M|1 with Collisions and Impatient Calls

A study on M/G/1 retrial G - queue with two phases of service, immediate feedback and working vacations

In this paper, we discuss about the steady state behaviour of M/G/1 retrial queueing system with two phases of services and immediate feedbacks under working vacation policy where the regular busy server is affected due to the arrival of negative customers. Upon arrival if the customer finds the server busy, breakdown or on working vacation it enters an orbit; otherwise the customer enters into the service area immediately. After service completion, the customer is allowed to make finite number of immediate feedback. The feedback service also consists of two phases. At the service completion epoch of a positive customer, if the orbit is empty the server goes for a working vacation. The server works at a lower service rate during working vacation (WV) period. Using the supplementary variable technique, we found out the steady state probability generating function for the system and in orbit. System performance measures and reliability measures are discussed. Finally, some numerical examples are presented to validate the analyticalresults.

State dependent arrival in bulk retrial queueing system with immediate Bernoulli feedback, multiple vacations and threshold

The objective of this paper is to analyse state dependent arrival in bulk retrial queueing system with immediate Bernoulli feedback, multiple vacations, threshold and constant retrial policy. Primary customers are arriving into the system in bulk with different arrival rates λa and λb. If arriving customers find the server is busy then the entire batch will join to orbit. Customer from orbit request service one by one with constant retrial rate γ. On the other hand if an arrival of customers finds the server is idle then customers will be served in batches according to general bulk service rule. After service completion, customers may request service again with probability δ as feedback or leave from the system with probability 1 − δ. In the service completion epoch, if the orbit size is zero then the server leaves for multiple vacations. The server continues the vacation until the orbit size reaches the value ‘N’ (N > b). At the vacation completion, if the orbit size is ‘N’ then the server becomes ready to provide service for customers from the main pool or from the orbit. For the designed queueing model, probability generating function of the queue size at an arbitrary time will be obtained by using supplementary variable technique. Various performance measures will be derived with suitable numerical illustrations.

Multi stage unreliable retrial Queueing system with Bernoulli vacation

In this work we considered the Bernoulli vacation in group arrival retrial queues with unreliable server. Here, a server providing service in k stages. Any arriving group of units finds the server free, one from the group entering the first stage of service and the rest are joining into the orbit. After completion of the ith, (i=1,2,…k) stage of service, the customer may go to (i+1)th stage with probability θi, or leave the system with probability qi = 1 – θi, (i = 1,2,…k – 1) and qi = 1, (i = k). The server may enjoy vacation (orbit is empty or not) with probability v after finishing the service or continuing the service with probability 1-v. After finishing the vacation, the server search for the customer in the orbit with probability θ or remains idle for new arrival with probability 1-θ. We analyzed the system using the method of supplementary variable.

Strategic Joining and Optimal Pricing in the Cognitive Radio System With Delay-Sensitive Secondary Users

The transmission times with generally random distributed for both primary user (PU) and secondary user (SU) in a cognitive radio system with a single PU band is investigated. To characterize the random access protocols of wireless networks, a retrial queueing system is employed to model the decentralized and centralized behavior of SUs. Based on the tradeoff between the service reward and the delay cost, each SU must decide whether to join or balk the system upon arrival based on different levels of information provided by the system. Two situations in which SUs have no information or partially observable information are studied. In each situation, two kinds of strategies are considered: 1) individual equilibrium strategies (i.e., non-cooperative strategies) that maximize SUs’ own profit and 2) socially optimal strategies (i.e., cooperative strategies) that maximize the expected social welfare. Moreover, comparisons are carried out between these two kinds of strategies. The results indicate that the equilibrium joining probability in the non-cooperative case is not less than the optimal joining probability in the cooperative case. To regulate the SUs’ behavior, an equilibrium pricing scheme is proposed to make these strategies coincide. In addition, we observe that the expected delay and the mean number of SUs in the system only depend on the first two moments of the SU’s transmission time. Numerical examples also show that the partially observable queue is more profitable from the perspective of the social planner.

Equilibrium pricing in an M/G/1 retrial queue with reserved idle time and setup time

• Equilibrium strategic behavior and social optimization in the M/G/1 with reserved idle times and setup time are studied. • Equilibrium pricing issue is further discussed. • The admission fee and the optimal reserved time are derived to maximize the server’s profit. • The sensitivity of solutions of revenue optimization is presented. • The customers’ equilibrium behavior is illustrated through numerical experiments. Power consumption is a ubiquitous and challenging problem in modern society. To save energy, one should turn off an idle device which still consumes about 60% of its peak consumption and switch it on again when some jobs arrive. However, it is not tolerate for delay sensitive applications. Therefore, there is a trade-off between power consumption and delay performance. In this paper we study an M/G/1 retrial queueing system with setup times in which the server keeps idle for a reserved idle time after completion of a service. If there are arrivals during this reserved idle time, these customers can be served immediately. Otherwise, the server will be turned off for saving energy until a new customer comes to activate the server. The setup time follows an exponential distribution. Based on the reward-cost function and the expected payoff, all customers will make decisions on whether to join or balk the system upon arrival. Given these strategic behaviors we study the optimal pricing strategies from the perspective of the server and social planner, respectively. The optimization of the reserved idle time for maximizing the server’s profit is also studied. Finally, numerical experiments are presented to illustrate the impact of system parameters on the customers’ equilibrium behavior and profit maximization solutions.

Optimal design for a retrial queueing system with state-dependent service rate

This paper considers a single server retrial queue in which a state-dependent service policy is adopted to control the service rate. Customers arrive in the system according to a Poisson process and the service times and inter-retrial times are all exponentially distributed. If the number of customers in orbit is equal to or less than a certain threshold, the service rate is set in a low value and it also can be switched to a high value once this number exceeds the threshold. The stationary distribution and two performance measures are obtained through the partial generating functions. It is shown that this state-dependent service policy degenerates into a classic retrial queueing system without control policy under some conditions. In order to achieve the social optimal strategies, a new reward-cost function is established and the global numerical solutions, obtained by Canonical Particle Swarm Optimization algorithm, demonstrate that the managers can get more benefits if applying this state-dependent service policy compared with the classic model.

Retrial Queueing System M / M / 1 / 0 with Combined Service Discipline

The paper considers retrial queueing system M /M /1/ 0 with combined service discipline, namely, a customer from the orbit is serviced in its turn, but in case of a free channel an arrival from the original flow is serviced immediately. The author obtains the expressions for state probabilities as well as ergodicity conditions. The system is compared with the Lakatos-type system.

Stability analysis of a multiclass retrial system with classical retrial policy

We consider a multiserver retrial queueing system with a renewal input, K classes of customers, and a finite buffer. Service times are class-dependent, however, for each class, are independent, identically distributed (iid). A new class-i customer joins the primary system (servers and buffer), otherwise, if all servers and buffer are full, he joins the class-i (virtual) orbit, and attempts to enter the system after an exponentially distributed time with rate γi,i=1,…,K. The retrial discipline is classical because the attempts of different orbital (blocked) customers are independent. We exploit the regenerative structure of the (non-Markovian) queue-size process (total number of customers in the primary system and in orbits) to develop the stability analysis. First we establish the necessary stability conditions, and then show that these conditions are sufficient for stability as well. These conditions coincide with the known stability conditions of a conventional multiclass multiserver system with infinite buffer. Our analysis covers also the model in which, when a server is free, it makes an outgoing call which occupies the server for a server-dependent random time. Also we extend the stability analysis to the retrial system with a feedback when a served customer returns to the corresponding orbit with a positive probability.

A Geo[X]/G[X]/1 retrial queueing system with removal work and total renewal discipline

In this paper we consider a discrete-time retrial queueing system with batch arrivals of geometric type and general batch services. The arriving group of customers can decide to go directly to the server expelling out of the system the batch of customers that is currently being served, if any, or to join the orbit. After a successful retrial all the customers in the orbit get service simultaneously. An extensive analysis of the model is carried out, and using a generating functions approach some performance measures of the model, such as the first distribution’s moments of the number of customers in the orbit and in the system, are obtained. The generating functions of the sojourn time of a customer in the orbit and in the system are also given. Finally, in the section of conclusions and research results the main contributions of the paper are commented.

A study on M/G/1 feedback retrial queue with subject to server breakdown and repair under multiple working vacation policy

In this paper, we consider a single server feedback retrial queueing system with multiple working vacations and vacation interruption. An arriving customer may balk the system at some particular times. As soon as orbit becomes empty at regular service completion instant, the server goes for a working vacation. The server works at a lower service rate during working vacation (WV) period. After completion of regular service, the unsatisfied customer may rejoin into the orbit to get another service as feedback customer. The normal busy server may get to breakdown and the service channel will fail for a short interval of time. The steady state probability generating function for the system size is obtained by using the supplementary variable method. Some important system performance measures are obtained. Finally, some numerical examples and cost optimization analysis are presented.

On the MAP/PH/c retrial queueing system with two types of nonpersistent calls

In this paper, we establish a performance model of a cell in a cellular mobile wireless network with two types of calls arrivals: originating calls and handoff calls. Using the Markovian Arrival Processes to describe the calls arrival processes and PH type distributions to model the calls service times and the cell-sojourn times of any ongoing calls, we investigate the impact of retrial and nonpersistent phenomenon on the performance measures of the system. The algorithms for calculating the stationary state probabilities of the model and determining the optimal number of guard channels and total channels are elaborated. Main performance measures are obtained. Finally, we provide numerical results to illustrate the optimization problems and the impact of retrial and nonpersistent phenomena on performance measures. We find that the performance measures are mainly affected by the two types of calls’ arrivals and service patterns, but the retrial rate plays a little crucial role.

Transient Solution of M[x1],M[x2]/G1,G2/1 Retrial queueing system with Priority services, Modified Bernoulli Vacation, Bernoulli Feedback, Negative arrival, Breakdown, Delaying repair, Setup time and Balking

Transient Solution of M[x1],M[x2]/G1,G2/1 Retrial queueing system with Priority services, Modified Bernoulli Vacation, Bernoulli Feedback, Negative arrival, Breakdown, Delaying repair, Setup time and Balking

Retrial Queueing Systems in Series of Space Experiments “Kontur”

This article describes queueing systems model of data transmission system for remote control of robot from board of ISS using public networks. Model has two incoming flows, one service channel, limited buffer size and randomized push-out mechanism. Also model of a retrial queueing system and lost packets with specified probability are able to get back on service. Two parameters: retrial and pushing-out probabilities of randomized mechanism have extremely high influence on model parameters like loss probability and average queue length and allow changing them on tens of percents. Also shown that retrial queueing models are having areas of linear behavior and blocking too, as model without retrial.