# Fluid Net Models: From Behavioral Properties to Structural Objects
## 1 Introduction and Motivation
For the sake of simplicity, the results in this work are formulated for Mono-T-Semiflow nets; however, they can be easily extended to MTSR nets using the reduction rule already discussed in [10]. For the class of MTSR nets, the number of firings of transitions per time unit can be interpreted as the finished jobs per time unit in the flowshop, which can be used to measure the performance of the manufacturing system and is related to the equilibrium throughput of the TCPN system. In particular, qualitative properties of the equilibrium throughput such as _monotonicity_, _continuity_, and _deadlock-freeness_, as functions of the initial marking, firing transitions rates, and net structure, are herein systematically analyzed. The equilibrium throughput is:  
* Monotonic, when increasing the initial marking and/or the transition firing rates leads to a non-slower system (a desired behavior); that is, more or faster resources in the system should at least not reduce the number of finished jobs per time unit;
* Continuous, when no abrupt changes of the equilibrium throughput are possible in the system when firing rates and/or the initial marking vary (again a desired system property);* Deadlock-free, when no blocking situation can occur in the system (this property can be reinterpreted as \"persistence\" in chemical reaction networks [11]). As a motivation, here, a cell composed of one input and one output stores of six slots each, one conveyor belt, a computer numerical control (CNC) machine, and a robot is presented. The cell is devoted to the manufacture of gears. The raw material resides in the input store. The robot retrieves the raw material from the input store and places it onto the conveyor. The conveyor is a two-site belt, and it leads the raw material into the CNC machine, which fabricates gears. The robot unloads the machine when it finishes a gear, and places the finished gear into the output store. Figure 1a shows a photo of this manufacturing cell. A TCPN model of the manufacturing cell is presented in Figure 1b, and the description of places and transitions are presented in Tables 1 and 2, respectively. The firing of \(t_{1}\) indicates that a part arrives to the cell and it is stored in the input store. The firing of \(t_{2},t_{3}\) and \(t_{4}\) represent that a part is retrieved by the robot form the input store, deposited on the conveyor, and loaded into the CNC machine, respectively. The unloading of the CNC machine and the storing of a part in the output store are represented by the firing of \(t_{5}\) and \(t_{6}\), respectively. Finally, \(t_{7}\) indicates that a part leaves the cell. \begin{table}
\begin{tabular}{l l} \hline \hline
**Place** & **Description** \\ \hline \multicolumn{3}{c}{Robot} \\ \hline \(p_{8}\) & Robot is idle \\ \(p_{3}\) & Robot retrieved a part from input store \\ \(p_{6}\) & Robot unloaded a part from CNC machine \\ \hline \multicolumn{3}{c}{CNC machine} \\ \hline \(p_{11}\) & CNC is idle \\ \(p_{5}\) & CNC is processing a part \\ \hline \multicolumn{3}{c}{Input Store} \\ \hline \(p_{9}\) & Free slots of the Input Store \\ \(p_{2}\) & Stored parts in Input Store \\ \hline \multicolumn{3}{c}{Output Store} \\ \hline \(p_{12}\) & Free slots of the Output Store \\ \(p_{7}\) & Stored parts in Output Store \\ \hline \multicolumn{3}{c}{Conveyor} \\ \hline \(p_{10}\) & Free sites in the conveyor \\ \(p_{4}\) & Occupied sites on the conveyor \\ \hline \multicolumn{3}{c}{System capacity} \\ \hline \(p_{1}\) & Number of jobs that can be processed in the system \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Places and their description in the \(TCPN\) model of Figure 1b.