The theory of matrix-analytic methods, pioneered by Professor Marcel Neuts (1935–2014), the founding editor of Stochastic Models, is an area of applied probability that focuses on constructing methods of analysis that can be implemented using efficient algorithms and allow for practical numerical analysis of many real-life systems. In recent years, this active research area has expanded into a vast array of theoretical advancements and applications to a diverse range of real-world problems. Researchers in the field have gradually broadened the number of models studied and the theoretical tools at their disposal, and they have found effective numerical methods to provide solutions to many of these models. Meanwhile, the computational techniques have been successfully applied to various areas of practical modeling, from telecommunications problems to health care and biology.
The Tenth International Conference on Matrix-Analytic Methods in Stochastic Models (MAM10) was held at the University of Tasmania in Hobart from the 13th to the 15th of February 2019, continuing the established tradition of previous fruitful MAM conferences in Flint (1995), Winnipeg (1998), Leuven (2000), Adelaide (2002), Pisa (2005), Beijing (2008), New York (2011), Calicut (2014), and Budapest (2016). This issue of Stochastic Models, which is dedicated to Matrix-Analytic Methods, provides a sample of the research presented at MAM10.
The seven papers contained in this issue cover a broad range of research in the areas of quasi-birth-and-death processes, phase-type and matrix exponential distributions, continuous-state Markov processes, and applications. An eighth paper (still under review at the time this special issue was finalised) may appear in a successive issue.
Soohan Ahn and Beatrice Meini in “Matrix equations in Markov modulated Brownian motion: theoretical properties and numerical solution” consider the study of a Markov modulated Brownian motion, a generalization of the classical Brownian Motion obtained by allowing the Brownian parameters to be modulated by an underlying Markov chain of environments. The distributions of the first passage time between the levels in this model can be obtained by solving a suitable quadratic matrix equation (QME) or a nonsymmetric algebraic Riccati equation (NARE). The authors establish an explicit algebraic relation between the QME and the NARE and present numerical experiments that are closely related to the applications.
Kayla Javier and Brian Fralix in “A further study of some Markovian Bitcoin models from Göbel et al.” consider a continuous-time Markov chain model and a discrete-event simulation model which were used by Göbel et al. to evaluate the effect of a pool of selfish Bitcoin miners on a community of honest miners. The authors give an alternative derivation of the stationary distribution of the process when the pool and the community are honest, and when the pool is dishonest (selfish) and the community is honest. Also, they present a closed form expression for the normalizing constant and derive expressions for the Laplace transforms of the transition functions in these models.
Zbigniew Palmowski and Eleni Vatamidou in “Phase-type approximations perturbed by a heavy-tailed component for the Gerber-Shiu function of risk processes with two-sided jumps” study a Cramér-Lundeberg model with two sided jumps, where the upward jumps have a phase-type distribution while downward jumps have a mixture of phase-type and heavy-tailed distribution. The authors approach the problem by embedding their model into a suitable Markov additive process and analyze the seminal Gerber-Shiu function coding the joint distribution of the time to ruin, the surplus immediately before ruin, and the deficit at ruin.
Jiahao Diao, Tristan L. Stark, David A. Liberles, Małgorzata M. O’Reilly and Barbara R. Holland in “Level-dependent QBD models for the evolution of a family of gene duplicates” introduce multi-dimensional continuous time Markov models for the evolution of families of gene duplicates. They show that a two-dimensional model does not capture enough information for the gene evolution process. Using matrix-analytic methodology, they further investigate a three-dimensional Markov model, which corresponds to a level-dependent QBD process, and they show that this model is able to capture the qualitative behavior of the gene evolution process.
Aregawi K. Abera, Małgorzata M. O’Reilly, Mark Fackrell, Barbara R. Holland and Mojtaba Heydar in “On the decision support model for the patient admission scheduling problem with random arrivals and departures: A solution approach” present a stochastic decision model for the patient admission scheduling problem with random arrivals and departures, modeled using phase-type distributions and integer programing, and present heuristics for the solution of the problem. The authors show that the optimal solutions of their stochastic model, in terms of the total expected costs accumulated over a finite planning horizon, are more adequate than the solutions obtained from deterministic models.
Aviva Samuelson, Małgorzata M. O’Reilly and Nigel Bean in “Construction of algorithms for discrete-time quasi-birth-and-death processes through physical interpretation” consider one of the classical problems in matrix-analytic methods, the one of computing the Laplace-Stieltjes transform of the so-called R- and G-matrix in a QBD process. The authors present a framework that allows one to construct 144 different algorithms and analyze them both algebraically and through physical interpretations. They identify among them two quadratically convergent algorithms: a known one (Newton’s method), and a novel one.
Gábor Horváth, Illés Horváth and Miklós Telek in “High order concentrated matrix-exponential distributions” construct explicitly a family of matrix-exponential distributions that are highly concentrated, i.e., their coefficient of variation is as small as possible. They take an ansatz based on a trigonometric representation and use optimization methods to minimize the coefficient of variation numerically up to dimension 1000. Their computational results are made available publicly and can be used to implement an inversion formula for the Laplace-Stieltjes transform, which is an essential building block for various algorithms.
This issue of Stochastic Models highlights only a subset of many exciting research topics in the theory of matrix-analytic methods. We would like to take this opportunity to invite the authors in the wider applied probability community to contribute to the advances of this fascinating and fruitful area of mathematics.
We thank our sponsors, the Australian Mathematical Sciences Institute, the Australian Mathematical Society, the Australian Research Council Center of Excellence for Mathematical and Statistical Frontiers, Australian Laureate Fellow Professor Peter Taylor from the University of Melbourne, and Professor Andrew Bassom from the University of Tasmania. The very generous financial support of our sponsors has contributed to the success of this workshop.
We gratefully acknowledge the excellent work of the referees and thank the authors for their involvement. Special thanks to Editor-in-Chief of Stochastic Models Mark S. Squillante and to the publisher Taylor & Francis who made this publication possible.
Finally, we would like to express our deep gratitude to the organizing committee, the program committee and the steering committee, and to all researchers who attended the workshop for their support and participation in this event.
The University of Melbourne, Australia The Australian Research Council (ARC) Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), Australia
Małgorzata M. O'Reilly
University of Tasmania, The Australian Research Council (ARC) Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), Australia
[email protected]
Federico Poloni
University of Pisa, Italy