Retrial Orbit Research Articles

Fluid limits of many-server retrial queues with nonpersistent customers

This work considers a many-server retrial queueing system in which nonpersistent (impatient) customers with i.i.d., generally distributed service times and independent sequences of i.i.d., generally distributed inter-attempt times. A newly arrived customer attempts to obtain service immediately upon arrival and joins a retrial orbit with probability $$p\in [0,1]$$p?[0,1] if all servers are busy, and re-attempts to obtain service after a random amount of time until it gets service. The dynamics of the system is represented in terms of a family of measure-valued processes that keep track of the amounts of time that each customer being served has been in service and the waiting times of customers in the retrial orbit since their previous attempts to obtain service. Under some mild assumptions, as both the arrival rate and the number of servers go to infinity, a law of large numbers (or fluid) limit is established for this family of processes with the aid of the one-dimensional Skorokhod map and a contraction map. The limit is shown to be the unique solution to the so-called (extended) fluid model equations. In addition, the set of invariant states for the (extended) fluid model equations is established and is used to yield some steady state performance measures, such as the steady state blocking probability and the steady state number of customers in the retrial orbit.

The multi-server retrial system with Bernoulli feedback and starting failures

In this paper, we present a detailed analysis of a multi-server retrial queue with Bernoulli feedback, where the servers are subject to starting failures. Upon completion of a service, a customer would decide either to leave the system with probability p or to join the retrial orbit again for another service with complementary probability 1−p. We analyse this queueing system as a quasi-birth–death process. Specifically, the equilibrium condition of the system is given for the existence of the steady-state analysis. Applying the matrix-geometric method, the formulae for computing the rate matrix and stationary probabilities are obtained. We further develop the matrix-form expressions for various system performance measures. A cost model is constructed to determine the optimal number of servers, the optimal mean service rate and the optimal mean repair rate subject to the stability condition. Finally, we give a practical example to illustrate the potential applicability of this model.

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.

An M[X]/G/1 retrial g-queue with single vacation subject to the server breakdown and repair

An M[X]/G/1 retrial G-queue with single vacation and unreliable server is investigated in this paper. Arrivals of positive customers form a compound Poisson process, and positive customers receive service immediately if the server is free upon their arrivals; Otherwise, they may enter a retrial orbit and try their luck after a random time interval. The arrivals of negative customers form a Poisson process. Negative customers not only remove the customer being in service, but also make the server under repair. The server leaves for a single vacation as soon as the system empties. In this paper, we analyze the ergodical condition of this model. By applying the supplementary variables method, we obtain the steady-state solutions for both queueing measures and reliability quantities.

A single-server retrial G-queue with priority and unreliable server under Bernoulli vacation schedule

We consider an M/G/1 retrial queue with negative customers and priority under Bernoulli vacation schedule subject to the server breakdowns and repairs. Arrivals of both positive customers and negative customers are two independent Poisson processes. Positive customers receive service immediately if the server is idle upon their arrivals. Otherwise, they may either with probability p join the priority queue or with complementary probability p¯ enter a retrial orbit. A breakdown at the busy server is represented by the arrival of a negative customer which causes the the customer being in service to be lost. The server takes Bernoulli vacation after a service or a repair completion. It is assumed that the server has arbitrary repair time and vacation time distributions. With the help of Lyapunov functions we have obtained the necessary and sufficient condition for ergodicity of embedded Markov chain. By applying the supplementary variables method, we obtain the steady-state solutions for both queueing measures and reliability quantities. Moreover, we investigate the stochastic decomposition law. Besides, some special cases of interest are discussed. Finally, the effects of various parameters on the system performance are analyzed numerically.

Performance analysis of the retrial queues with finite number of sources and service interruptions

This paper aims at presenting an analytic approach for investigating a single-server retrial queue with finite population of customers where the server is subject to interruptions. A free source may generate a primary call to request service. If the server is free upon arrival, the call starts to be served and the service times are independent, generally distributed random variables. During the service time the source cannot generate a new primary call. After service the source moves into the free state and can generate a new primary call. There is no waiting space in front of the server, and a call who finds the server unavailable upon arrival joins an orbit of unsatisfied customers. The server is subject to interruptions during the service processes. When the server is interrupted, the call being served just before server interruption goes to the retrial orbit and will retry its luck after a random amount of time until it finds the server available. The recovery times of the interrupted server are assumed to be generally distributed. Our analysis extends previous work on this topic and includes the analysis of the arriving customer’s distribution, the busy period, and the waiting time process.

On the multi-server retrial queue with geometric loss and feedback: computational algorithm and parameter optimization

We consider an M/M/c retrial queue with geometric loss and feedback. An arriving customer finding a free server enters into service immediately; otherwise the customer either enters into an orbit to try again after a random amount of time or leave the system without service. After the completion of service, he decides either to join the retrial orbit or to leave the system. The retrial system is modelled by a quasi-birth-and-death process, and some system performance measures are derived. The useful formulae for computing the rate matrix and stationary probabilities are derived by means of matrix-analytical approach. A cost model is derived to determine the optimal values of the number of servers and service rate simultaneously at the minimal total expected cost per unit time. Illustrative numerical examples demonstrate the optimization approach as well as the effect of various parameters on system performance measures.

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 RETRIAL QUEUEING MODEL WITH MAP ARRIVALS, CATASTROPHIC FAILURES WITH REPAIRS, AND CUSTOMER IMPATIENCE

We consider a single server retrial queueing system in which arrivals occur according to a Markovian arrival process. An arriving customer finding an idle server will get service immediately; otherwise the customer enters into a retrial orbit. The system is subject to catastrophic failures at which time all customers in the system (the one in service as well as the customers, if any, in the orbit) are lost. The system undergoes a repair and after completion of a repair the server in the system will be available for service. The customers in the orbit try to reach the server by sending a signal at random times and get service if the server is idle at those times. While customers are waiting in the orbit, they may become impatient and leave the system after a random amount of time. Any arrival finding the server unavailable (due to busy or under repair) and cannot enter the orbit due to the buffer being full (in the case of finite buffer only) is considered lost. Under exponential assumption for all except for the arrivals, the queueing model (for both finite and infinite orbit sizes) is studied using matrix-analytic methods and the qualitative nature of the model is brought out through some illustrative numerical examples.

An M/G/1 retrial G-queue with preemptive resume and feedback under N-policy subject to the server breakdowns and repairs

An M / G / 1 retrial G -queue with preemptive resume and feedback under N -policy vacation subject to the server breakdowns and repairs, is investigated in this paper. Arrivals of both positive customers and negative customers are two independent Poisson processes, and positive customers receive service immediately if the server is free upon their arrivals. Otherwise, they may enter a retrial orbit and try their luck after a random time interval. All positive customers must receive preliminary first phase of service and primary second phase of service. While at the preliminary service, the server may push out the customer undergoing such service to the orbit, to commence preliminary service of an arriving positive customer. Negative customers not only remove the customer being in service, but also make the server under repair. The server leaves for an N -policy vacation as soon as the system empties. By applying the supplementary variables method, we obtain the steady-state solutions for both queueing measures and reliability quantities. The effects of various parameters on the system performance are analyzed numerically.

AN M/G/1 RETRIAL QUEUE WITH UNRELIABLE SERVER FOR STREAMING MULTIMEDIA APPLICATIONS

As a model for streaming multimedia applications, we study an unreliable retrial queue with infinite-capacity orbit and normal queue for which the retrial rate and the server repair rate are controllable. Customers join the retrial orbit if and only if their service is interrupted by a server failure. Interrupted customers do not rejoin the normal queue but repeatedly attempt to access the server at independent and identically distributed intervals until it is found functioning and idle. We provide stability conditions, queue length distributions, stochastic decomposition results, and performance measures. The joint optimization of the retrial and server repair rates is also studied.

Transient analysis of an M/ G/1 retrial queue subject to disasters and server failures

An M/ G/1 retrial queueing system with disasters and unreliable server is investigated in this paper. Primary customers arrive in the system according to a Poisson process, and they receive service immediately if the server is available upon their arrivals. Otherwise, they will enter a retrial orbit and try their luck after a random time interval. We assume the catastrophes occur following a Poisson stream, and if a catastrophe occurs, all customers in the system are deleted immediately and it also causes the server’s breakdown. Besides, the server has an exponential lifetime in addition to the catastrophe process. Whenever the server breaks down, it is sent for repair immediately. It is assumed that the service time and two kinds of repair time of the server are all arbitrarily distributed. By applying the supplementary variables method, we obtain the Laplace transforms of the transient solutions and also the steady-state solutions for both queueing measures and reliability quantities of interest. Finally, numerical inversion of Laplace transforms is carried out for the blocking probability of the system, and the effects of several system parameters on the blocking probability are illustrated by numerical inversion results.

A discrete-time retrial queue with negative customers and unreliable server

This paper treats a discrete-time single-server retrial queue with geometrical arrivals of both positive and negative customers in which the server is subject to breakdowns and repairs. Positive customers who find sever busy or down are obliged to leave the service area and join the retrial orbit. They request service again after some random time. If the server is found idle or busy, the arrival of a negative customer will break the server down and simultaneously kill the positive customer under service if any. But the negative customer has no effect on the system if the server is down. The failed server is sent to repair immediately and after repair it is assumed as good as new. We analyze 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 down. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.

Generalized Truncated Methods for an Efficient Solution of Retrial Systems

We are concerned with the analytic solution of multiserver retrial queues including the impatience phenomenon. As there are not closed‐form solutions to these systems, approximate methods are required. We propose two different generalized truncated methods to effectively solve this type of systems. The methods proposed are based on the homogenization of the state space beyond a given number of users in the retrial orbit. We compare the proposed methods with the most well‐known methods appeared in the literature in a wide range of scenarios. We conclude that the proposed methods generally outperform previous proposals in terms of accuracy for the most common performance parameters used in retrial systems with a moderated growth in the computational cost.

Algorithmic analysis of theMAP/PH/1 retrial queue

In this paper the distribution of the maximum number of customers in a retrial orbit for a single server queue with Markovian arrival process and phase type services is studied. Efficient algorithm for computing the probability distribution and some interesting numerical examples are presented.

An M/G/1 retrial queue with second multi-optional service, feedback and unreliable server

An M/G/1 retrial queue with two-phase service and feedback is studied in this paper, where the server is subject to starting failures and breakdowns during service. Primary customers get in the system according to a Poisson process, and they will receive service immediately if the server is available upon arrival. Otherwise, they will enter a retrial orbit and are queued in the orbit in accordance with a first-come-first-served (FCFS) discipline. Customers are allowed to balk and renege at particular times. All customers demand the first “essential” service, whereas only some of them demand the second “multi-optional” service. It is assumed that the retrial time, service time and repair time of the server are all arbitrarily distributed. The necessary and sufficient condition for the system stability is derived. Using a supplementary variable method, the steady-state solutions for some queueing and reliability measures of the system are obtained.

Computational analysis of the maximal queue length in the MAP/ M/ c retrial queue

We consider a multi-server retrial queueing model in which arrivals occur according to a Markovian arrival process. Using continuous-time Markov chain with absorbing states, we determine the distribution of the maximum number of customers in a retrial orbit. Illustrative numerical examples that reveal some interesting results are presented.

Reliability analysis M/G/1 queues with general retrial times and server breakdowns*

Abstract This paper concerns the reliability issues as well as queueing analysis M/G/1 retrial queues with general retrial times and server subject to breakdowns and repairs. We assume that the server is unreliable and customers who find the server busy or down are queued in the retrial orbit in accordance with a firs-come-first-served discipline. Only the customer at the head of the orbit queue is allowed for access to the server. The necessary and sufficient condition for the system to be stable is given. Using a supplementary variable method, we obtain the Laplace-Stieltjes transform of the reliability function of the server and a steady state solution for both queueing and reliability measures of interest. Some main reliability indexes, such as the availability, failure frequency, and the reliability function of the server, are obtained. * Supported by Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, National Natural Science Foundation of China (Gran...

A Retrial Queue with a Constant Retrial Rate, Server Downs and Impatient Customers

ABSTRACT In this paper, we consider a retrial queueing system consisting of a waiting line of infinite capacity in front of a single server subject to breakdowns. A customer upon arrival may join the queue (waiting line) or go to the retrial orbit (another queue) to retry for service after a random time. Only the customer at the head of the retrial orbit is allowed to retry for service. Upon retrial, the customer enters the service if the server is idle; otherwise, it may go back to the retrial orbit or leave the system (become impatient). All the interarrival times, service times, server up times, server down times and retrial times are exponential, and all the necessary independence conditions in these variables are assumed. For this system, we provide sufficient conditions under which, for any given number of customers in the orbit, the stationary probability of the number of customers in the waiting line decays geometrically. We also provide explicitly an expression for the decay parameter.

Geo/G/1 discrete time retrial queue with Bernoulli schedule

This paper studies discrete time Geo/G/1 retrial queues with Bernoulli schedule in which the blocked customers either join the infinite waiting space with probability α or leave the server and enter the retrial orbit with probability α (=1−α ). The customers in the retrial orbit will retry their service after a random amount of time. First, the analytic formula for the generating function of the joint distribution of the numbers of customers in the waiting space and the retrial orbit in steady state is derived. It is shown that a stochastic decomposition law holds for the retrial queues under study. That is, the total number of customers in system is distributed as a sum of two independent random variables. Second, recursive formulas for the marginal steady state probabilities of the numbers of customers in the waiting space and in the retrial orbit was developed. Since a regular two-level priority Geo/G/1 queue (one without retrials) with Bernoulli schedule and head-of-line priority discipline is a special case of the studied retrial systems, the recursive formulas developed can be used to compute the marginal steady state probabilities of numbers of customers in the priority and non-priority groups for this case. Furthermore, a relationship between a continuous time M/G/1 retrial queue with Bernoulli schedule and its discrete time counterpart is established so that the recursive formulas can also be applied to a continuous time system. Last, several special cases are studied and some numerical examples are presented to demonstrate the use of the recursive formulas.