Retrial queue

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

Heterogeneous finite-source retrial queues

Abstract: In this paper we investigate a single server retrial queue with a finite number of heterogeneous sources of calls. It is assumed when a given source is idle it will generate a primary call after an exponentially distributed time. If the server is free at the time of the request’ s arrival then the call starts to be served. The service time is also exponentially distributed. During the service time the source cannot generate a new primary call. After service the source moves into free state and can generate a new call again. If the server is busy at the time of the arrival of a primary call, then the source starts generating so called repeated calls with exponentially distributed times until it finds the server free. As before, after service the source becomes free and can generate a new primary call again. We assume that the primary calls, repeated attempts and service times are mutually independent. This queueing system and its variants could be used to model magnetic disk memory systems, local area networks with CSMA/CD protocols and collision avoidance local area networks. The novelty of this model is the heterogeneity of the calls, which means that each call is characterized by its own arrival, repeated and service rates. The aim of the paper is to give the usual steady-state performance measures of the system. To do so, an efficient software tool, MOSEL ( Modeling, Specification and Evaluation Language ) developed at the University of Erlangen, Germany, is used to formulate and solve the problem. Several sample numerical results illustrate the power of the tool showing the effect of different parameters on the system measures.

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

Abstract: We consider a MAP/G/1 retrial queue where the service time distribution has a finite exponential moment. We derive matrix differential equations for the vector probability generating functions of the stationary queue size distributions. Using these equations, Perron---Frobenius theory, and the Karamata Tauberian theorem, we obtain the tail asymptotics of the queue size distribution. The main result on light-tailed asymptotics is an extension of the result in Kim et al. (J. Appl. Probab. 44:1111---1118, 2007) on the M/G/1 retrial queue.

Scaling limits for single server retrial queues with two-way communication

Abstract: This paper studies M/G/1 retrial queues in which there are two arrival flows, i.e., incoming calls made by regular customers and outgoing calls made by the server in idle time. The stationary analysis of this system has been carried out in a recent paper by Artalejo and Phung-Duc (Appl Math Model 37(4):1811–1822, 2013). In this paper, we obtain a decomposition property where we prove that the queue length is decomposed into the sum of three independent random variables with clear physical meaning. We then derive scaling limits for the queue length distribution under some extreme conditions (i) heavy traffic, (ii) slow retrials and (iii) instantaneous connection to outgoing calls. Furthermore, we also investigate the convergence of our model to that without outgoing calls.

An explicit solution for a tandem queue with retrials and losses

Abstract: We consider a retrial tandem queueing system with two servers whose service times follow two exponential distributions. There are two types of customers: type one and type two. Customers of type one arrive at the first server according to a Poisson process. An arriving customer of type one that finds the first server busy joins an orbit and retries to enter the server after some time. We assume that the arrival rate of customers from the orbit is a linear function of the number of retrial customers. After being served at the first server, a customer of type one moves to the second server. Customers of type two directly arrive at the second server according to another Poisson process. Customers of both types one and two are lost if the second server is busy upon arrival. For this model, we derive explicit expressions of the joint stationary distribution between the number of customers in the orbit and the states of the servers. We prove that the stationary distribution is computed by a numerically stable algorithm. Numerical examples are provided to show the influence of parameters on the performance of the system.