Retrial queue

About: Retrial queue is a research topic. Over the lifetime, 784 publications have been published within this topic receiving 12354 citations.

Transient distributions of level dependent quasi-birth-death processes with linear transition rates

Abstract: Many queueing systems such asM/M/s/K retrial queue with impatient customers, MAP/PH/1 retrial queue, retrial queue with two types of customers andMAP/M/∞ queue can be modeled by a level dependent quasi-birth-death (LDQBD) process with linear transition rates of the form λk = α+ βk at each levelk. The purpose of this paper is to propose an algorithm to find transient distributions for LDQBD processes with linear transition rates based on the adaptive uniformizaton technique introduced by van Moorsel and Sanders [11]. We apply the algorithm to some retrial queues and present numerical results.

Stochastic comparison bounds for an M1, M2/G1, G2/1 retrial queue with two way communication

Abstract: The main goal in the present paper is to provide a technique that considers the stochastic comparison approach for investigating monotonicity and comparability of an $M_{1},M_{2}/G_{1},G_{2}/1$ retrial queues with two way communication. This approach is developed for comparing a non Markov process to Markov process with many possible stochastic orderings. Particularly, we show the monotonicity of the transition operator of the embedded Markov chain relative to the strong stochastic ordering and convex ordering, as well as the comparability of two transition operators. Bounds are also obtained for the stationary distribution of the number of customers at departure epochs. Additionally, the performance measures of the system considered can be estimated by those of an $M_1,M_2/M_1,M_2/1$ retrial queue with two way communication when the service time distribution is NBUE (respectively NWUE). Finally, we validate stochastic comparison results by presenting a numerical example illustrating the interest of the approach.

Steady-state analysis of M/M/c/c-type retrial queueing systems with constant retrial rate

Abstract: The paper deals with a research of bivariate Markov process $$\{X(t), t\ge 0\}$$ whose state space is a lattice semistrip $$S(X)=\{0,1,{\ldots },c\} \times Z_{+}$$ . The process $$\{X(t), t\ge 0\}$$ describes the service policy of a multi-server retrial queue in which the rate of repeated flow does not depend on the number of sources of retrial calls. In this class of queues, a vector–matrix representation of steady-state distribution was obtained. This representation allows to write down the stationary probabilities through the model parameters in closed form and to propose the closed formulas of its main performance measures. The investigative techniques use an approximation of the initial model by means of the truncated one and the direct passage to the limit.

On a retrial queueing model with single/batch service and search of customers from the orbit

Abstract: A single server retrial queueing system, where customers arrive according to a batch Poisson process and are served either in single or as a batch, is considered here. An arriving batch, finding the server busy, enters an orbit. Otherwise, one customer, a few customers, or all customers from the arriving batch, depending on if the batch size exceeds a threshold value or not, enter service immediately, while the rest join the orbit. Customers from the orbit try to reach the server subsequently with the inter-retrial times, exponentially distributed. Additionally, at each service completion epoch, one of the two types of search mechanisms say, type I and type II search, to bring the orbital customers to service, is switched on—type I search when the orbit size is less than the threshold value and type II search otherwise. This means that, while the server is idle, a competition takes place among primary customers, customers who come by retrial and by one of the two types of search as the case may be. A type I search selects a single customer whereas a type II search takes a batch of customers from the orbit. In the case of primary customers and those who come by type II search, maximum size of the batch taken into service is restricted to a pre-assigned value. Both single and batch service are assumed to be arbitrarily distributed with different distributions, which are independent of each other. Steady state analysis is performed. Some important system descriptors are computed algorithmically and numerical illustrations are provided.

Analysis of a retrial queue with multiple vacations and state dependent arrivals

Abstract: This paper examines an M/G/1 retrial queueing system with multiple vacations and different arrival rates. Whenever the system is empty, the server immediately takes a vacation. At a vacation completion epoch, if the number of customers in the orbit is at least one the server remains in the system to activate service, otherwise the server avails multiple vacations until at least one customer is recorded in the orbit. The primary arrival rate is λ 1 when the server in idle and the primary arrival rate is λ 2 when the server is busy or on vacation (λ 1 > λ 2 ). The steady state queue size distribution of number of customers in the retrial group, expected number of customers in the retrial group and expected number of customers in the system are obtained. Some special cases are also discussed. Numerical illustrations are also provided.