Lecture Notes in Economics and Mathematical Systems

(2003). Comparison Methods for Stochastic Models and Risks. Technometrics: Vol. 45, No. 4, pp. 370-371.

Comparison Methods for Stochastic Models and Risks

Studies stochastic models of queueing, reliability, inventory, and sequencing in which random influences are considered. One stochastic mode--rl is approximated by another that is simpler in structure or about which simpler assumptions can be made. After general results on comparison properties of random variables and stochastic processes are given, the properties are illustrated by application to various queueing models and questions in experimental design, renewal and reliability theory, PERT networks and branching processes.

https://link.springer.com/content/pdf/10.1057%2Fjors.1984.224.pdf

Comparison Methods for Queues and Other Stochastic Models

The engine knock analysis – An overview

Potentials of cooled EGR and water injection for knock resistance and fuel consumption improvements of gasoline engines

https://hal.archives-ouvertes.fr/hal-00841028/document

Exact computation of emergy based on a mathematical reinterpretation of the rules of emergy algebra

Safe operating conditions determination for stationary SI gas engines

The computational problem of emergy within a general system of interconnected processes at steady state is a subject of interest in literature today. When there is no co-product the proposed method coincides with the Track Summing Method of Tennenbaum which was developed precisely for interconnected networks with feedbacks and splits of emergy. As the underlying algebraic structure of the Tennenbaum's method is the linear algebra, it is not well-suited to account for the co-product problem which induces the idempotent operator max. Thus, authors have chosen another underlying algebraic structure which is the idempotent semiring structure (i.e. a semiring equipped with an idempotent addition). This method is divided into two parts. The first part is the emergy flow enumeration, where paths from an emergy source to the input of a given process are enumerated avoiding double counting of emergy assignation. This part is a path-finding problem which is a slight modification of gerbier of null square approach to find elementary/simple paths in a graph. The second part evaluates the emergy flowing between two components of the system. It is a quantitative part in which the problem of avoiding double counting split and co-product flows are dealt with by introducing a way to mark splits and co-products flows. The method is partially parallelizable. However, the method enumerates paths on a graph thus, in worst cases, its complexity is not polynomial. This paper provides a rigorous framework based on an axiomatic basis to conduct the emergy evaluation of an emergy graph. © 2012 ISEIS All rights reserved.

/pdf/a-rigourous-mathematical-framework-for-computing-a-1n5jplj7eo.pdf

A rigourous mathematical framework for computing a sustainability ratio: The emergy

Linear systems over naturally ordered dioids are other kinds of positive systems than the usual ones over semiring (R + , +,.). In this short paper we study some monotonicity concepts of linear systems over dioids inspired by results on monotonicity of Markov chains which are also particular cases of positive systems. We derive a necessary and sufficient condition for monotonicity in a simple case which requires strong assumptions on dioid (lattice distributivity and invertibility of the multiplication law). The result suggests links between monotonicity and positive invariance which plays an important role in control theory and also with aggregation problems of linear systems (i.e. conditions for the existence of aggregated variables and their linear dynamic from which the complete behavior can be retraced).

Monotone linear dynamical systems over dioids

Haar’s Lemma (1918) deals with the algebraic characterization of the inclusion of polyhedral sets. This Lemma has been involved many times in automatic control of linear dynamical systems via positive invariance of polyhedrons. More recently, it has been used to characterize stochastic comparison w.r.t. linear/integral ordering of Markov (reward) chains. In this paper we develop a state space oriented approach to the control of Discrete Event Systems (DES) based on the remark that most of control constraints of practical interest are naturally expressed as the inclusion of two systems of linear (w.r.t. idempotent semiring or semifield operations) inequalities. Thus, we establish tropical version of Haar’s Lemma to obtain the algebraic characterization of such inclusion. As in the linear case this Lemma exhibits the links between two apparently different problems: comparison of DES and control via positive invariance. Our approach to the control differs from the ones based on formal series and is a kind of dual approach of the geometric one recently developed. Control oriented applications of the main results of the paper are given. One of these applications concerns the study of transportation networks which evolve according to a time table. Although complexity of calculus is discussed the algorithmic implementation needs further work and is beyond the scope of this paper.

Idempotent versions of Haar's Lemma: links between comparison of discrete event systems with different state spaces and control.

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