# Fluid Net Models: From Behavioral Properties to Structural Objects
## 1 Introduction and Motivation  
The more basic analysis of Discrete Event Systems (DES) focuses on properties such as _liveness_, _boundedness_, and, when timed, system _performance_. Considering Flexible Manufacturing Systems (FMSs), as an example, the performance may deal with the number of goods fabricated per unit time (i.e., the throughput) or the number of items in a storage (i.e., its marking). If the throughput should be increased, then it may be an initial idea to increase the speed of the machines or the number of resources. Unfortunately, it may not only decrease the system throughput, but even worse, reduce it to zero (deadlock). An overview of these paradoxical throughput behaviors are studied in this work when the systems are modeled with Timed Continuous Petri nets (TCPN). Petri nets (PN) are a fine and well-known formalism to model _Discrete Event Systems_, since they have a sound mathematical background, capturing in a compact manner basic system characteristics, such as causal relationships, synchronization, and concurrence. Moreover, their nice graphical representation provides a communication framework between engineers and practitioners. This work assumes that the reader is familiar with some basic concepts of PN, such as (structural) liveness, (structural) boundness, or reachability graph [1; 2]. Unfortunately, as any other expressive DES formalism, they suffer from the so-called _state explosion_ problem, particularly when they are heavily populated (heavily marked in PN terms). This problem is overcome with a classical relaxation: the _fluidization_ of the system, i.e., both, the marking and the firing vectors are now considered non-negative real numbers. This leads to the concept of _Fluid_ or _Continuous_ PNs (CPNs). CPNs make some computational problems _decidable_ or much tractable in practice [3]. When a quantitative notion of time is introduced in the paradigm of the CPN, then the evaluation of the system performance indices is also possible (throughput, response time, average marking), leading to the notion of _Timed_ CPN (TCPN). We use the _infinite server semantics_ (ISS), also known as _variable speed_, to express the firing rate of transitions. Under this consideration, the fluid systems are continuous piecewise linear with polyhedral regions. For certain net subclasses, which can model several types of manufacturing systems, ISS provides better approximations of the system throughput [4]. Nonetheless, depending on the system, other timing interpretations may be more adequate, for example, _finite server semantics_ (FSS) in some hydraulic systems (see [5]) or _product semantics_ (PS) in biologic systems (see [6]). In some sense, this work complements the overview [7] (for a more technical and detailed presentation, see [6]). Performance evaluation is a major concern when analyzing DES. For instance, a study of the performance of manufacturing systems can be found in [8], where _generalized stochastic Petri nets_ are used to model the systems. The analysis is based mainly on the simulation of the model to obtain the performance indices. Under the assumption of modeling with TCPN-ISS, ref. [9] presents the performance evaluation of manufacturing systems, where an insight of the relation between the structure of the model and the performance indices is discussed. That relation with the structure represents how the resources and the processes interact. The present work makes an effort to clarify how the performance of a system is affected when the resources or the velocity of processes are modified, relating them with the structure of the system. This work focuses on the class of _Mono-T-Semiflow-Reducible_ MTSR nets [10], since they allow to model a certain degree of _cooperative_ and _competitive_ relationships that arise in many practical systems, for instance flowshop or mass production systems.