# Fluid Net Models: From Behavioral Properties to Structural Objects
## 1 Introduction and Motivation
\begin{table}
\begin{tabular}{c c} \hline \hline
**Transition** & **Description** \\ \hline \(t_{1}\) & A part arrives to the cell and it is stored into the input store \\ \(t_{2}\) & The robot retrieves a part from the input store \\ \(t_{3}\) & The robot deposits a part onto the conveyor \\ \(t_{4}\) & A part leaves the conveyor and is loaded into the CNC machine \\ \(t_{5}\) & The robot unloads a finished part from the CNC machine \\ \(t_{6}\) & The robot stores a part into the output store \\ \(t_{7}\) & A part is retrieved from the output store and leaves the cell \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Transitions and their description in the \(TCPN\) model of Figure 1b. The net in Figure 1b is _structurally bounded_, i.e., for every initial marking, any reachable marking maintains a limited number of tokens in every place. In this model, tokens in place \(p_{1}\) represent the raw material, the firing of transition \(t_{2}\) represents the robot retrieving a part from the input store, and the firing of transition \(t_{5}\) represents that the robot unloads the machine. With regard to the _initial marking variation_ and how the qualitative properties of the continuous equilibrium throughput varies, the considered initial marking is \(m_{0}=[k_{1}\ 0\ 0\ 0\ 0\ 0\ 1\ 6\ 2\ 1\ 6]^{T}\) and the firing rate is fixed to \(\lambda=[1\ 2\ 1\ 1\ 1\ 1\ 1]^{T}\). In this case, \(m_{0}(p_{1})=k_{1}\) varies in the range \([0,\ 7]\), leading to a variation in the equilibrium throughput, which is depicted in Figure 2a. It shows that the equilibrium throughput is non-monotonic, even non-continuous with respect to the marking variations. Figure 2a is obtained by simulating the evolution of the piecewise linear system for several values of \(k_{1}\) from zero to seven. The value of the equilibrium throughput is plotted for every value of \(k_{1}\). Figure 1: A manufacturing cell and its Petri net model. (**a**) Manufacturing cell. (**b**) Manufacturing cell model. Figure 2: Equilibrium throughput of the Petri net model with respect to the initial marking and with respect to the firing rates. (**a**) Initial marking variation \(k_{1}\). (**b**) Firing rate variation \(\lambda_{2}\). With regard to the _firing rates variation_ and how the qualitative properties of the equilibrium throughput varies, the initial marking is fixed to \(\mathbf{m}_{0}=[10\;0\;0\;0\;0\;0\;0\;1\;6\;2\;1\;6]^{T}\) and the firing rate is \(\mathbf{\lambda}=[1\;\lambda_{2}\;1\;1\;1\;1\;1\;1]^{T}\). In this case, \(\mathbf{\lambda}(t_{2})=\lambda_{2}\) varies in the range \([0,\;4]\), leading to a variation in the equilibrium throughput, which is depicted in Figure 2b. This second figure shows that the equilibrium throughput is non-monotonic, even non-continuous with respect to the firing rate variations (there exists an abrupt change around \(\lambda_{2}=1\)). Figure 2b is obtained by simulating the evolution of the piecewise linear system for several values of \(\lambda_{2}\) from zero to four. The value of the equilibrium throughput is plotted for every value of \(\lambda_{2}\). It is worth to notice that these non-monotonic, non-continuous behaviors are undesirable. In both cases, increasing the number of resources or their speeds, the throughput of the unforced system (i.e., non-controlled) is reduced to zero, that is, the manufacturing system reaches a blocking situation. Moreover, these behavioral properties of TCPN systems are closely related to behavioral properties of discrete systems, even when the net is structurally live (i.e., there exists initial markings that make the system live). This example shows that the problem of non-monotonicity is originated when there are several tokens in \(\mathbf{m}_{0}(p_{1})\) (increasing the system raw material) or \(\mathbf{\lambda}(t_{2})\) is increased (there is a faster resource), allowing that \(t_{2}\) fires more often than \(t_{5}\). This reduces to zero the tokens in \(p_{8}\), \(p_{10}\), and \(p_{11}\); hence, \(t_{3}\) cannot be fired again and the TCPN is blocked.