An Algebraic Approach to Timed Petri Nets with Applications to Communication Networks

In this report, we define a formalism for a time-extension to algebraic place/transition (P/T) nets. This allows time durations to be assigned to the transitions of a P/T net, representing delays present in the systems that are being modelled, which in turn influence (restrict) the firing behaviour of the nets. This is especially useful when modelling time-dependent systems. The new contribution of this approach is the definition of categories for the timed net classes of timed P/T nets, timed P/T systems and timed P/T states. Moreover, we define functorial relations between these categories as well as functorial relations to categories of untimed P/T nets and systems. The first main result is the formalisation of morphisms for all three net classes that preserve firing behaviour. The second main result is the equivalence of the categories of timed P/T systems and states, establishing a relation between structurally identical nets with a time offset. As a third main result we formalise structuring techniques for timed P/T nets and show that timed P/T nets fit in the framework of M-adhesive categories

Algebraic Approach to Timed Petri Nets

One aspect often needed when modelling systems of any kind is time-based analysis, especially for real-time or in general time-critical systems. Algebraic place/transition (P/T) nets do not inherently provide a way to model the passing of time or to restrict the firing behaviour with regards to passing time. In this paper, we present an extension of algebraic P/T nets by adding time durations to transitions and timestamps to tokens. We define categories for different timed net classes and functorial relations between them. Our first result is the definition of morphisms preserving firing behaviour for all timed net classes. As second result, we define structuring techniques for timed P/T nets in a way that our category fulfills the properties of M-adhesive systems, a general categorical framework for structuring and transforming high-level algebraic structures. We demonstrate our approach by applying it to model a real-time communication network