Showing papers on "Retrial queue published in 2007"

A discrete-time Geo/G/1 retrial queue with starting failures and second optional service

Abstract: We consider a discrete-time Geo/G/1 retrial queue with starting failures in which all the arriving customers require a first essential service while only some of them ask for a second optional service. We study the Markov chain underlying the considered queueing system and its ergodicity condition. Explicit formulae for the stationary distribution and some performance measures of the system in steady state are obtained. We also obtain two stochastic decomposition laws regarding the probability generating function of the system size. Finally, some numerical examples are presented to illustrate the influence of the parameters on several performance characteristics.

An M[X]/G/1 retrial queue with server breakdowns and constant rate of repeated attempts

Abstract: We consider an M[X]/G/1 retrial queue subject to breakdowns where the retrial time is exponential and independent of the number of customers applying for service. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest joins a retrial group (called orbit) to repeat his request later; otherwise, if the server is busy or down, all customers of the coming batch enter the orbit. It is assumed that the server has a constant failure rate and arbitrary repair time distribution. We study the ergodicity of the embedded Markov chain, its stationary distribution and the joint distribution of the server state and the orbit size in steady-state. The orbit and system size distributions are obtained as well as some performance measures of the system. The stochastic decomposition property and the asymptotic behavior under high rate of retrials are discussed. We also analyse some reliability problems, the k-busy period and the ordinary busy period of our retrial queue. Besides, we give a recursive scheme to compute the distribution of the number of served customers during the k-busy period and the ordinary busy period. The effects of several parameters on the system are analysed numerically.

A multiserver retrial queue: regenerative stability analysis

Abstract: We consider a multiserver retrial GI/G/m queue with renewal input of primary customers, interarrival time ? with rate $\lambda=1/\mathsf{E}\tau$ , service time S, and exponential retrial times of customers blocked in the orbit. In the model, an arriving primary customer enters the system and gets a service immediately if there is an empty server, otherwise (if all m servers are busy) he joins the orbit and attempts to enter the system after an exponentially distributed time. Exploiting the regenerative structure of the (non-Markovian) stochastic process representing the total number of customers in the system (in service and in orbit), we determine stability conditions of the system and some of its variations. More precisely, we consider a discrete-time process embedded at the input instants and prove that if $\rho=:\lambda\mathsf{E}S and $\mathsf{P}(\tau>S)>0$ , then the regeneration period is aperiodic with a finite mean. Consequently, this queue has a stationary distribution under the same conditions as a standard multiserver queue GI/G/m with infinite buffer. To establish this result, we apply a renewal technique and a characterization of the limiting behavior of the forward renewal time in the (renewal) process of regenerations. The key step in the proof is to show that the service discipline is asymptotically work-conserving as the orbit size increases. Included are extensions of this stability analysis to continuous-time processes, a retrial system with impatient customers, a system with a general retrial rate, and a system with finite buffer for waiting primary customers. We also consider the regenerative structure of a multi-dimensional Markov process describing the system.

Multi-server retrial queue with negative customers and disasters

Abstract: We consider a multi-server retrial queue with waiting places in service area and four types of arrivals, positive customers, disasters and two types of negative customers, one for deleting customers in orbit and the other for deleting customers in service area. The four types of arrivals occur according to a Markovian arrival process with marked transitions (MMAP) which may induce the dependence among the arrival processes of the four types. We derive a necessary and sufficient condition for the system to be positive recurrent by comparing sample paths of auxiliary systems whose stability conditions can be obtained. We use a generalized truncated system that is obtained by modifying the retrial rates for an approximation of stationary queue length distribution and show the convergence of approximation to the original model. An algorithmic solution for the stationary queue length distribution and some numerical results are presented.

Tail asymptotics for the queue size distribution in AN M/G/1 retrial queue

Abstract: We consider an M/G/1 retrial queue, where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. The result is obtained by investigating analytic properties of probability generating functions for the queue size and the server state.

Algorithmic Analysis of the Maximum Queue Length in a Busy Period for the M/M/c Retrial Queue

Abstract: This paper deals with the maximum number of customers in orbit (and in the system) during a busy period for the M/M/c retrial queue Determining the distribution for the maximum number of customers in orbit is reduced to computation of certain absorption probabilities By reducing to the single-server case we arrive at a closed analytic formula For the multi-server case we develop an efficient algorithmic procedure for computation of this distribution by exploiting the special block-tridiagonal structure of the system Numerical results illustrate the efficiency of the method and reveal interesting facts concerning the behavior of the M/M/c retrial queue

On retrial queueing model with fuzzy parameters

Abstract: This work constructs the membership functions of the system characteristics of a retrial queueing model with fuzzy customer arrival, retrial and service rates. The α-cut approach is used to transform a fuzzy retrial-queue into a family of conventional crisp retrial queues in this context. By means of the membership functions of the system characteristics, a set of parametric non-linear programs is developed to describe the family of crisp retrial queues. A numerical example is solved successfully to illustrate the validity of the proposed approach. Because the system characteristics are expressed and governed by the membership functions, more information is provided for use by management. By extending this model to the fuzzy environment, fuzzy retrial-queue is represented more accurately and analytic results are more useful for system designers and practitioners.

The busy period and the waiting time analysis of a MAP/M/c queue with finite retrial group

Abstract: We concentrate on the analysis of the busy period and the waiting time distribution of a multi-server retrial queue in which primary arrivals occur according to a Markovian arrival process (MAP). Since the study of a model with an infinite retrial group seems intractable, we deal with a system having a finite buffer for the retrial group. The system is analyzed in steady state by deriving expressions for (a) the Laplace–Stieltjes transforms of the busy period and the waiting time; (b) the probabiliy generating functions for the number of customers served during a busy period and the number of retrials made by a customer; and (c) various moments of quantites of interest. Some illustrative numerical examples are discussed.

A Feedback Retrial Queuing System with Starting Failures and Single Vacation

Abstract: The M/G/1 retrial queue with Bernoulli feedback and single vacation is studied in this paper, where the server is subjected to starting failure. The retrial time is assumed to follow an arbitrary distribution and the customers in the orbit access the server under FCFS discipline. The server leaves for a vacation as soon as the system becomes empty. When the server returns from the vacation and finds no customers, he waits free for the first customer to arrive from outside the system. The system size distribution at random points and various performance measures are derived. The general decomposition law is shown to hold good for this model also. Some of the existing results in [7] are deduced as special cases from our results.

Waiting time analysis of the M/G/1 queue with finite retrial group

Abstract: We consider an M/G/1 retrial queue with finite capacity of the retrial group. First, we obtain equations governing the dynamic of the waiting time. Then, we focus on the numerical inversion of the density function and the computation of moments. These results are used to approximate the waiting time of the M/G/1 queue with infinite retrial group for which direct analysis seems intractable.

A discrete-time retrial queueing system with starting failures, Bernoulli feedback and general retrial times

Abstract: It is well known that discrete-time queues are more appropiate than their continuous-time counterparts for modelling computer and telecommunication systems. We present a discrete-time Geo/G/1 retrial queue with general retrial times, Bernoulli feedback and the server subjected to starting failures. We gene- ralize the previous works in discrete{time retrial queue with unrealiable server due to starting failures in the sense that we consider general service and ge- neral retrial times. Also we consider the realistic phenomenon of feedback. We analyse the Markov chain underlying the considered queueing system and pre- sent some performance measures of the system in steady-state. We provide two stochastic decomposition laws. Besides, we prove that the M/G/1 retrial queue with feedback and starting failures can be approximated by our corresponding discrete-time system. Some numerical results are given to illustrate the impact of the unreability and the feedback on the performance of the syste

Unreliable Retrial Queues in a Random Environment

Abstract: : This dissertation investigates stability conditions and approximate steady-state performance measures for unreliable, single-server retrial queues operating in a randomly evolving environment. In such systems, arriving customers that find the server busy or failed join a retrial queue from which they attempt to regain access to the server at random intervals. Such models are useful for the performance evaluation of communications and computer networks which are characterized by time-varying arrival, service and failure rates. To model this time-varying behavior, we study systems whose parameters are modulated by a finite Markov process. Two distinct cases are analyzed. The first considers systems with Markov-modulated arrival, service, retrial, failure and repair rates assuming all interevent and service times are exponentially distributed. The joint process of the orbit size, environment state, and server status is shown to be a tri-layered, level-dependent quasi-birth-and-death (LDQBD) process, and we provide a necessary and sufficient condition for the positive recurrence of LDQBDs using classical techniques. Moreover, we apply efficient numerical algorithms, designed to exploit the matrix-geometric structure of the model, to compute the approximate steady-state orbit size distribution and mean congestion and delay measures. The second case assumes that customers bring generally distributed service requirements while all other processes are identical to the first case. We show that the joint process of orbit size, environment state and server status is a level-dependent, M/G/1-type stochastic process. By employing regenerative theory, and exploiting the M/G/1-type structure, we derive a necessary and sufficient condition for stability of the system. Finally, for the exponential model, we illustrate how the main results may be used to simultaneously select mean time customers spend in orbit, subject to bound and stability constraints.

Multi-Server Loss System with T-Limited Service for Traffic Control in Information Networks

Abstract: To handle more phone and personal computer users in such a natural disaster as terribly strong earthquake, a traffic control has been previously proposed by limiting the individual call holding time. This traffic control mechanism leads to our T-limited service. By T-limited service, we mean that the service time is limited to a threshold T. The call whose service time reaches T is assumed to be lost. For evaluating the traffic control performance, we present multi-server loss systems with T-limited service. Without any retrial queues, we analyze a Poisson input and general service time loss system to derive the steady-state distribution of the number of calls in the system. With a retrial queue, assuming further that the call sojourn time at the retrial queue is exponentially distributed and that the T-limited service time is also exponentially distributed, we propose an approximation for the steady-state distribution of the number of calls in the system. Our approximation accuracy is validated by a simulation result.

An Eigenvalue of M/M/1 Retrial Queueing Model with Special Retrial Times

Abstract: The results of this paper prove that 0 is an eigenvalue of the operator corresponding to M/M/1 retrial queueing model with special retrial times with geometric multiplicity one and 0 is an eigenvalue of its adjoint operator

A Note On The Busy Period Of The M / G /1 Queue With Finite Retrial Group

Abstract: We consider an M/G/1 retrial queue with finite capacity of the retrial group. We derive the Laplace transform of the busy period using the catastrophe method. This is the key point for the numerical inversion of the density function and the computation of moments. Our results can be used to approach the corresponding descriptors of the M/G/1 queue with infinite retrial group, for which direct analysis seems intractable.

OPTIMAL UTILIZATION OF SERVICE FACILITY FOR A k-OUT-OF-n SYSTEM WITH REPAIR BY EXTENDING SERVICE TO EXTERNAL CUSTOMERS IN A RETRIAL QUEUE

Abstract: In this paper, we study ak-out-of-n system with single server who provides service to external customers also. The system consists of two parts: (i) a main queue consisting of customers (failed components of thek-out-of-n system) and (ii) a pool (of finite capacityM) of external customers together with an orbit for external customers who find the pool full. An external customer who finds the pool full on arrival, joins the orbit with probability γ and with probability 1- γ leaves the system forever. An orbital customer, who finds the pool full, at an epoch of repeated attempt, returns to orbit with probability δ (< 1) and with probability 1- δ leaves the system forever. We compute the steady state system size probability. Several performance measures are computed, numerical illustrations are provided.

An M/G/1 Retrial Queue with Second Multi-Optional Service and Server Breakdowns

Abstract: We consider an M/G/1 retrial queue with two-phase service and unreliable server.All customers demand the first essential service,whereas only some of them demand the second "multi-optional" service.It is assumed that the service time and repair time of two phases are all generally distributed.By applying the supplementary variables method,we obtain the steady-state solutions for both queuing measures and reliability quantities of the server.

A Discrete-Time Retrial Queue with Starting Failure and Feedback

Abstract: We consider a discrete-time Geo/G/1 retrial queue with starting failure where all the arriving customers finish their service while only some of them returns to the orbit for another service.We study the Markov chain underlying the considered queueing system and its ergodicity condition.We present some performance measures of the system in steady-state.Then,we give two stochastic decomposition laws.Finally,some numerical examples show the influence of the parameters on several performance characteristics.

Tail asymptotics for the waiting time in an m/g/1 retrial queue

Abstract: We consider an M/G/1 retrial queue, where the service time distribution has a regularly varying tail with index -s, 1 ＜ s ＜ 2. It is shown that the waiting time distribution has a regularly varying tail with index 1 - s, and the pre-factor is determined explicitly. The result is obtained by comparing the waiting time in the M/G/1 retrial queue with the waiting time in the ordinary M/G/1 queue with random order service policy.