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A stochastic model for the spread of an SIS epidemic among a population consisting of N individuals, each having heterogeneous infectiousness and/or susceptibility, is considered and its behavior is analyzed under the practically relevant situation when N is small. The model is formulated as a finite timehomogeneous continuous-time Markov chain X . Based on an appropriate labeling of states, we first construct its infinitesimal rate matrix by using an iterative argument, and we then present an algorithmic procedure for computing steadystate measures, such as the number of infected individuals, the length of an outbreak, the maximum number of infectives, and the number of infections suffered by a marked individual during an outbreak. The time till the epidemic extinction is characterized as a phase-type random variable when there is no external source of infection, and its Laplace-Stieljtes transform and moments are derived in terms of a forward elimination backward substitution solution. The inverse iteration method is applied to the quasi-stationary distribution of X , which provides a good approximation of the process X at a certain time, conditional on non-extinction, after a suitable waiting time. The basic reproduction number R 0 is defined here as a random variable, rather than an expected value.

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.Keywords Markov chain of GI/M/1-type . Markov chain of M/G/1-type . Matrix-analytic method . Quasi-birth-and-death process . Retrial queue Preliminary commentsOver the last two decades a good deal of progress has been made in the analysis of retrial queues through matrix analytic techniques, and high quality research papers have appeared in diverse journals and conference proceedings in the field of applied probability, computer sciences, operations research and stochastic modelling.This paper claims to be a bibliographical guide to the use of matrix analytic techniques in retrial queues. Its main objective is to make the related bibliography accessible to researchers and, hopefully, to graduate students and practitioners. It includes an author index and a subject index of research papers written in English and published, or accepted for publication, in journals or collective publications. With permission of the corresponding authors, a few papers submitted for potential publication are also included.In the author index, we add a set of keywords per paper. Keywords are listed in alphabetical order and seek briefly to show the queueing model under study, the probabilistic descriptors of interest, and the tools closely related to those matrix analytic techniques used by the authors. However, we point out that they do not necessarily correspond to the list of keywords provided by the authors. Following standard terminology, we mainly group the methods for finding the descriptors under study into the three paradigms popularized by Neuts: Markov A. Gómez-Corral

Queueing systems with repeated requests have many useful applications in communications and computer systems modeling. In the majority of previous work the repeat requests are made individually by each unsatisfied customer. However, there is in the literature another type of queueing situation, in which the time between two successive repeated attempts is independent of the number of customers applying for service. This paper deals with the M/G/1 queue with repeated orders in its most general setting, allowing the simultaneous presence of both types of repeat requests. We first study the steady state distribution and the partial generating functions. When the service time distribution is exponential we show that the performance characteristics can be expressed in terms of hypergeometric functions.

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