More on linear and metric tree maps

We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree.

Coloured stochastic Petri nets modelling for the reliability and maintenance analysis of multi-state multi-unit systems

Purpose – The purpose of this paper is to introduce a method for modelling the multi-state repairable systems subject to stochastic degradation processes by using the coloured stochastic Petri nets (CSPN). The method is a compact and flexible Petri nets model for multi-state repairable systems and offers an alternative to the combinatory of Markov graphs. Design/methodology/approach – The method is grounded on specific theorems used to design an algorithm for systematic construction of multi-state repairable systems models, whatever is their size. Findings – Stop and constraint functions were derived from these theorems and allow to considering k-out-of-n structure systems and to identifying the minimal cut sets, useful to monitoring the states evolution of the system. Research limitations/implications – The properties of this model will be studied, and new investigations will help to demonstrate the feasibility of the approach in real world, and more complex structure will be considered. Practical implications – The simulation models based on CSPN can be used as a tool by maintenance decision makers, for prediction of the effectiveness of maintenance strategies. Originality/value – The proposed approach and model provide an efficient tool for advanced investigations on the development and implementation of maintenance policies and strategies in real life.