Previous article Next article Full AccessSIGESThttps://doi.org/10.1137/SIREAD000044000003000415000001BibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract Engineered systems such as communications networks, digital circuits, distributed operating systems, manufacturing plants, and airport traffic control are typically modeled as discrete-event systems. By far the most common discrete-event systems are based on operations and rules designed by humans and consist of asynchronous repetition of finitely many events. (The word "discrete" refers to the definitive role of these events, which are characterized by starting and ending times.) For example, an event in a communications network begins when an information packet arrives at a given node; in manufacturing, an event corresponds to the performance of a task by a specific machine. The dynamics of discrete-event systems include synchronization, competition for resources, and prioritization, none of which is easily expressible through differential equations or their discretizations. The "max-plus" algebra, whose basic operations are maximization and addition, provides a convenient and powerful way to express certain linear discrete-event systems, where linearity is understood with respect to the associated nonstandard algebraic structures. Max-plus (sometimes written as (max, +)) algebra has been applied to performance evaluation, graph and Hamilton--Jacobi theory, and asymptotics in statistical physics. Beyond this expressive power, the max-plus algebra allows direct analogies to be drawn between its operations and those of conventional algebras. As a consequence, properties and concepts such as the Cayley--Hamilton theorem, eigenvalues, and eigenvectors can be translated directly into the max-plus regime. Similarly, certain constructs from linear system theory carry over to linear discrete-event systems. Even so, the theory of max-plus linear algebra is far from complete. This issue's SIGEST paper, "The QR Decomposition and the Singular Value Decomposition in the Symmetrized Max-Plus Algebra Revisited," by Bart De Schutter and Bart De Moor, which first appeared in 1998 in the SIAM Journal on Matrix Analysis and Applications, makes a significant contribution to that theory. The "QR decomposition" and "singular value decomposition" of its title refer not to the familiar factorizations from conventional linear algebra but to their analogues in the max-plus algebra. The authors present a clear introduction to the max-plus and symmetrized max-plus algebras; they show how to prove the existence of the max-plus QR and singular value decompositions, in the process developing techniques that can be adapted to prove existence of other max-plus factorizations; and, finally, they show how max-plus decompositions can be computed. For its appearance in SIGEST, the authors have updated and extended the original paper by adding background, examples, and references, along with the latest results. We are grateful for this tour de force of exposition, which gives SIAM Review readers a clear, useful overview of discrete-event systems and the max-plus algebra. Previous article Next article FiguresRelatedReferencesCited byDetails The relationship between axillary temperature and lifestyle in elementary school childrenJapan Journal of Human Growth and Development Research, Vol. 2011, No. 51 Cross Ref Volume 44, Issue 3| 2002SIAM Review History Published online:04 August 2006 InformationCopyright © 2002 Society for Industrial and Applied Mathematics Article & Publication DataArticle DOI:10.1137/SIREAD000044000003000415000001Article page range:pp. 415-415ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics