Showing papers on "Retrial queue published in 2002"

The M/G/1 retrial queue with Bernoulli schedules and general retrial times

Abstract: This paper is concerned with the analysis of a single-server queue with Bernoulli vacation schedules and general retrial times. We assume that the customers who find the server busy are queued in the orbit in accordance with an FCFS (first-come-first-served) discipline and only the customer at the head of the queue is allowed access to the server. We first present the necessary and sufficient condition for the system to be stable and derive analytical results for the queue length distribution, as well as some performance measures of the system under steady-state condition. We show that the general stochastic decomposition law for M/G/1 vacation models holds for the present system also. Some special cases are also studied.

Numerical Calculation of the Stationary Distribution of the Main Multiserver Retrial Queue

Abstract: We are concerned with the main multiserver retrial queue of M/M/c type with exponential repeated attempts. It is known that an analytical solution of this queueing model is difficult and does not lead to numerical implementation. Based on appropriate understanding of the physical behavior, an efficient and numerically stable algorithm for computing the stationary distribution of the system state is developed. Numerical calculations are done to compare our approach with the existing approximations.

A Multi-Server Retrial Queue with BMAP Arrivals and Group Services

Abstract: In this paper, we consider a c-server queuing model in which customers arrive according to a batch Markovian arrival process (BMAP). These customers are served in groups of varying sizes ranging from a predetermined value L through a maximum size, K. The service times are exponentially distributed. Any customer not entering into service immediately orbit in an infinite space. These orbiting customers compete for service by sending out signals that are exponentially distributed with parameter t. Under a full access policy freed servers offer services to orbiting customers in groups of varying sizes. This multi-server retrial queue under the full access policy is a QBD process and the steady state analysis of the model is performed by exploiting the structure of the coefficient matrices. Some interesting numerical examples are discussed.

Analysis of a single-server retrial queue with quasi-random input and nonpreemptive priority☆

Abstract: In this paper, we model a single-server retrial queue with quasi-random input and two priority classes. In the case of blocking, a high priority unit is queued, whereas a low priority unit joins the orbit to start generating a Poisson flow of repeated attempts until it finds the server free. Since units in orbit will be served only when the high priority queue is empty, high priority units have nonpreemptive priority over low priority units. We present a simple analysis for the outside observer distribution of the system state as well as for the arriving unit distribution in steady state. Besides, we give numerical examples to illustrate the effect of the parameters on several performance characteristics.

A second order analysis of the waiting time in the M/G/1 retrial queue

Abstract: We are concerned with the M/G/1 queue with repeated attempts where a customer who finds the server busy leaves the service area and repeats his request after a random amount of time. We concentrate on the study of the waiting time process. Its analysis in terms of Laplace transforms has been discussed in the literature. However, this solution has important limitations in practice. For instance, we cannot calculate the first moments of the waiting time, W, by direct differentiation. This paper supplements the existing work and provides a direct method of computation for the second moment of W. Then the maximum entropy approach is used to estimate the true waiting time distribution.

On the M/G/1 retrial queue subjected to breakdowns

Abstract: Retrial queueing systems are characterized by the requirement that customers finding the service area busy must join the retrial group and reapply for service at random intervals. This paper deals with the M/G/1 retrial queue subjected to breakdowns. We use its stochastic decomposition property to approximate the model performance in the case of general retrial times.

Served in an m/g/1 retrial queue

Abstract: We present a recursive method of computation for the probability that at most k customers were served during the busy period of an M/G/1 retrial queue. performance can be described in terms of its main characteristics, such us limit distribution, waiting time, busy period, number of customers served, etc. In this paper, an M/G/1 queue with exponential retrial times is considered. In particular, we focus on I, the number of customers served during the busy period, L; the system busy period is defined to be the period of time between an epoch when an arriving customer finds an empty system and the first departure epoch at which the system is empty again. The analysis of I and L on retrial systems has been the subject of several papers. The joint Laplace transform was obtained by Falin in (8), the structure of the busy period was studied by Artalejo and Falin in (3), and in (4) Artalejo and Lopez-Herrero determined a closed expression for E(L2). When exponential service times are considered, Choo and Conolly (6) developed a method for the calculation of the moments of L. Related to the number of customers served, the theoretical solution in terms of its Laplace transform presents important practical limitations.

Approximation of multiserver retrial queues by means of generalized truncated models

Abstract: It is well-known that an analytical solution of multiserver retrial queues is difficult and does not lead to numerical implementation. Thus, many papers approximate the original intractable system by the so-called generalized truncated systems which are simpler and converge to the original model. Most papers assume heuristically the convergence but do not provide a rigorous mathematical proof. In this paper, we present a proof based on a synchronization procedure. To this end, we concentrate on theM/M/c retrial queue and the approximation developed by Neuts and Rao (1990). However, the methodology can be employed to establish the convergence of several generalized truncated systems and a variety of Markovian multiserver retrial queues.

Distribution of the number of customers served in an M/G/1 retrial queue

Abstract: We present a recursive method of computation for the probability that at most k customers were served during the busy period of an M/G/1 retrial queue.

GI/M/1/1 queue with finite retrials and finite orbits

Abstract: In this paper we consider the GI/M/1/1 retrial queue with finite number of retrials to each orbital customer and a finite number of orbits. The interarrival times from outside the system have a general distribution. The sojourn time of a unit in an orbit until its retrial for service and its service time are exponentially distributed with parameters depending on the orbit number. The maximum number of retrials any unit is permitted to take is restricted to k. There are a finite number, say m, of orbits with at most one customer in each orbit. At the time of an arrival if the server is busy and all orbits are occupied, then the customer is lost to the system. If the server is idle at an arrival epoch, the unit directly goes for service. If the server is busy and at least one of the orbits is empty then the arriving customer occupies the first empty orbit. A unit in the orbit retries for service which returns to the same orbit (if the server is busy) with probability P and leaves the system with probability...

Operator-Geometric Solutions for the M/G/k Queue and its Variants

Abstract: One of the classical, but yet unsolved queueing systems is the M/G/k queue with Poisson input, general service time distribution and serving facilities. In the present paper, this queue is analyzed as a piecewise–deterministic Markov process. First, an iteration formula for the transient distributions is derived. This formula is a new result for piecewise–deterministic Markov processes, too, since it yields their transition probability kernel. In the main part, recent results on stability of piecewise–deterministic Markov processes are used in order to show that the computation of the stationary distribution for the M/G/k queue can be reduced to the determination of the stationary distribution for a birth and death process with kernel entries. This results in operator–geometric solutions for the M/G/k queue. For the M/G/k retrial queue, a similar analysis is presented. Here, closed formulae for the transient distribution are derived. These expressions hold for the class of piecewise–deterministic processes without jumps from a border set, too. Additionally, it is sketched how these results can be generalized for BMAP/G/k queues.

A finite capacity single server retrial queue with two types of calls

Abstract: A single server retrial queueing system with two types of calls type I and type II are considered. Arrivals are according to Poisson processes with rates λ 1 and λ 2 and service time distribution is exponential for both the calls. In case that arriving calls are blocked due to the server being busy, type I calls are queued in a priority queue of finite capacity K whereas type II calls enter the retrial grou in order to get service again after a random amount of time. For this system the joint distribution function of the number of calls in the priority queue and the retrial group in closed from is obtained. The results for particular cases are presented. The optimal value for K is numerically obtained.