The max-Łukasiewicz semiring is defined as the unit interval [0, 1] equipped with the arithmetics “a + b” = max(a, b) and “ab” = max(0, a + b− 1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Łukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, w...

This paper discusses the usefulness of (max, +) algebra as a mathematical modelling framework for discrete event systems (DESs). A Max-Plus Algebra Toolbox developed at Lehrstuhl für Systemtheorie technischer Prozesse is presented. This software package is a set of functions to take advantage of the (max, +) algebra in the Matlab environment for rapid prototyping, design, and analysis of DESs. ...

In this paper we prove a new characterization of the max-plus singular values of a maxplus matrix, as the max-plus eigenvalues of an associated max-plus matrix pencil. This new characterization allows us to compute max-plus singular values quickly and accurately. As well as capturing the asymptotic behavior of the singular values of classical matrices whose entries are exponentially parameteriz...

In this paper we propose a modelling technique for the control of computational processes. The Petri Net model, particularly a Timed Event Graph (TEG) can be used for analyzing. The proposed model enables the determination of state equations. The max-plus algebra represents linear algebraic form of discrete systems and supplies new tools to their modelling. We develop a linear mathematical mode...

This paper deals with control of transportation systems. We propose an approach based on dioid theory to compute timetables of a transportation network. We generalize the problem by considering additional constraints for the control objective. Copyright c ©2006 IFAC

We consider (max,+)-algebra products of random matrices, which arise from performance evaluation of acyclic fork-join queueing networks. A new algebraic technique to examine properties of the product and investigate its limiting behaviour is proposed based on an extension of the standard matrix (max,+)-algebra by endowing it with the ordinary matrix addition as an external operation. As an appl...

Proceedings of the 29th Conference on Decision and Control Honolulu, Dec. 1990 Abstract In this paper, we study the general system of linear equations in the algebra. We introduce a symmetrization of this algebra and a new notion called balance which generalizes classical equations. This construction results in the linear closure of the algebra in the sense that every nondegenerate system of li...

In this paper we discuss the characteristic equation of a matrix in the maxplus algebra. In their Linear Algebra and Its Applications paper (vol. 101, pp. 87– 108, 1988) Olsder and Roos have used a transformation between the max-plus algebra and linear algebra to show that the Cayley-Hamilton theorem also holds in the max-plus algebra. We show that the derivation of Olsder and Roos is not entir...