Antonio Gómez-Corral

A bibliographical guide to the analysis of retrial queues through matrix analytic techniques

This paper provides a bibliographical guide to researchers who are interested in the analysis of retrial queues through matrix analytic methods. It includes an author index and a subject index of research papers written in English and published in journals or collective publications, as well as some papers accepted for a forthcoming publication.

Stochastic analysis of a single server retrial queue with general retrial times

Retrial queueing systems are widely used in teletraffic theory and computer and communication networks. Although there has been a rapid growth in the literature on retrial queueing systems, the research on retrial queues with nonexponential retrial times is very limited. This paper is concerned with the analytical treatment of an M/G/1 retrial queue with general retrial times. Our queueing model is different from most single server retrial queueing models in several respectives. First, 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 for access to the server. Besides, a retrial time begins (if applicable) only when the server completes a service rather upon a sen ice attempt failure. We carry out an extensive analysis of the queue, including a necessary and sufficient condition for the system to be stable, the steady state distribution of the server state and the orbit length, the waiting time distribution, the busy period, and other related quantities. Finally, we study the joint distribution of the server state and the orbit length in non-stationary regime.

A Tandem Queue with Blocking and Markovian Arrival Process

Queueing networks with blocking have proved useful in modelling of data communications and production lines. We study such a network consisting of a sequence of two service stations with an infinite queue allowed before the first station and no intermediate queue allowed between them. This restriction results in the blocking of the first station whenever a unit having completed its service in that station cannot enter into the second one due to the presence of another unit there. The input of units to the network is the MAP (Markovian Arrival Process). At the first station, service requirements are of phase type whereas service times at the second station are arbitrarily distributed. The focus is on the embedded process at departures. The essential tool in our analysis is the general theory on Markov renewal processes of M/G/1-type.

Optimal balking strategies in single-server queues with general service and vacation times

Abstract In many service systems arising in OR/MS applications, the servers may be temporarily unavailable, a fact that affects the sojourn time of a customer and his willingness to join. Several studies that explore the balking behavior of customers in Markovian models with vacations have recently appeared in the literature. In the present paper, we study the balking behavior of customers in the single-server queue with generally distributed service and vacation times. Arriving customers decide whether to enter the system or balk, based on a linear reward–cost structure that incorporates their desire for service, as well as their unwillingness to wait. We identify equilibrium strategies and socially optimal strategies under two distinct information assumptions. Specifically, in a first case, the customers make individual decisions without knowing the system state. In a second case, they are informed about the server’s current status. We examine the influence of the information level on the customers’ strategic response and we compare the resulting equilibrium and socially optimal strategies.

Applications of maximum queue lengths to call center management

This paper deals with the distribution of the maximum queue length in two-dimensional Markov models. In this framework, two typical assumptions are: (1) the stationary regime, and (2) the system homogeneity (i.e., homogeneity of the underlying infinitesimal generator). In the absence of these assumptions, the computation of the stationary queue length distribution becomes extremely intricate or, even, intractable. The use of maximum queue lengths provides an alternative queueing measure overcoming these problems. We apply our results to some problems arising from call center management.

A matrix-geometric approximation for tandem queues with blocking and repeated attempts

Our interest is in the study of the MAP/PH/1/1->./PH/1/K+1 queue with blocking and repeated attempts. The main feature of its infinitesimal generator is the spatial heterogeneity caused by the transitions due to successful repeated attempts. We develop an algorithmic solution by making a simplifying approximation which yields an infinitesimal generator which is spatially homogeneous and has a modified matrix-geometric stationary vector. The essential tool in our analysis is the general theory on quasi-birth-and-death processes.

On a tandem G-network with blocking

An important class of queueing networks is characterized by the following feature: in contrast with ordinary units, a disaster may remove all work from the network. Applications of such networks include computer networks with virus infection, migration processes with mass exodus and serial production lines with catastrophes. In this paper, we deal with a two-stage tandem queue with blocking operating under the presence of a secondary flow of disasters. The arrival flows of units and disasters are general Markovian arrival processes. Using spectral analysis, we determine the stationary distribution at departure epochs. That distribution enables us to derive the distribution of the number of units which leave the network at a disaster epoch. We calculate the stationary distribution at an arbitrary time and, finally, we give numerical results and graphs for certain probabilistic descriptors of the network.

Analysis of a stochastic clearing system with repeated attempts

A stochastic clearing system is characterized by the existence of an output mechanism that instantaneously clears the system, i.e. removes all work currently present. In this paper we study the stochastic behavior of a single server clearing queue in which customers cannot be continuously in contact with the server, but can reinitiate the demand some time later. We develop a comprehensive analysis of the system including its limiting behavior, busy period, and waiting time

Algorithmic analysis of the Geo/Geo/c retrial queue

In this paper, we consider a discrete-time queue of Geo/Geo/c type with geometric repeated attempts. It is known that its continuous counterpart, namely the M/M/c queue with exponential retrials, is analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused from the retrial feature. In discrete-time, the occurrence of multiple events at each slot increases the complexity of the model and raises further computational difficulties. We propose several algorithmic procedures for the efficient computation of the main performance measures of this system. More specifically, we investigate the stationary distribution of the system state, the busy period and the waiting time. Several numerical examples illustrate the analysis.

Performance of two-stage tandem queues with blocking: the impact of several flows of signals

This paper considers a two-stage tandem G-queue with blocking, service requirements of phase type and arrivals of units and of signals - which cancel one unit waiting in line or in service - both assumed to be Markovian arrival processes. The main characteristic of the service process in this tandem queue is blocking, which occurs according to a BAS (blocking after service) mechanism. A finite queue of capacity K ≥ 0 is allowed between servers, so that a unit having completed processing on Station 1 attempts to join Station 2. If K + 1 units are accommodated into Station 2, then the completed unit is forced to wait at Station 1 occupying the server space until a space becomes available at Station 2. Thus the first server becomes blocked or not available for service on incoming units. Our purpose is to study the influence of several flows of signals on the performance evaluation of the queueing model through various probabilistic descriptors. By means of a numerical investigation, we study four two-stage tandem G-queues with blocking depending on the existence of flows of signals affecting Station 1 and/or Station 2. The essential tool in our analysis is the general theory of quasi-birth-and-death processes.