## 5 Modelling Rationale
### Modelling Assumptions
#### 5.3.2 Failure Model and Fault-Tolerance Coverages of the Modelled System
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Device** & **Unreliability** & **Fault** & **Prop.** & **Fault** & **Failure** & **Prop.** & **Stream** \\  & **Port** & **Prop.** & **Component** & **Mode** & **Prop.** & **Failure** \\ \hline TSN Talker & \(1-e^{-(\text{Tx}\times 1\times 10^{-4})}\) & Output & 50\% & Port/queue & Byzantine & 100\% & **Y** \\ \cline{2-7}  & Input & 50\% & - & - & - & **N** \\ \hline TSN Bridge & \(1-e^{-(\text{Rx}\times 1\times 10^{-4})}\) & - & 100\% & - & Byzantine & 100\% & **Y** \\ \cline{2-7} TSNListener & \(1-e^{-(\text{Rx}\times 1\times 10^{-4})}\) & Output & 50\% & - & - & - & **N** \\ \cline{2-7}  & Input & 50\% & Port/queue & - & 100\% & **Y** \\ \hline PTRF Talker & \(1-e^{-(\text{Tx}\times 15\times 10^{-4})}\) & Output & 50\% & PTRF transmission & TSN + buffer & 83.33\% & **Y** \\  & & & & counter \(>k\) & 8.33\% & \((\frac{k-1}{k})\text{N}/(\frac{1}{k})\text{Y}\) \\ \cline{2-7}  & Input & 50\% & - & - & - & **N** \\ \hline PTRF Bridge & \(1-e^{-((\text{Rx}\times\text{Fw}+\text{Tx})\times 1.5\times 10^{-4})}\) & Output & 50\% & PTRF transmission & TSN + buffer & 83.33\% & **Y** \\  & & & & counter \(>k\) & 8.33\% & \((\frac{k-1}{k})\text{N}/(\frac{1}{k})\text{Y}\) \\ \cline{2-7}  & Input & 50\% & PTRF reception & Fail to eliminate & 100\% & **Y** \\ \hline PTRF Listener & \(1-e^{-(\text{Rx}\times 1.5\times 10^{-4})}\) & Output & 50\% & - & - & - & **N** \\ \hline \hline \end{tabular}  
* Tx: time required for the port to transmit the frame. Rx: time required for the port to receive the frame. Fw: time required for the bridge to forward a frame from the input port to the output port. -: the value is irrelevant as it does not affect the result.  
\end{table} 
Table 1: Unreliability of each device, failure mode proportions and their impact on the stream. because, since the fault happens within the listener itself, it may affect the error-containment mechanisms that supervise the correct operation of the port. In the analyses we present here, which assume a single path, we consider that each end-system has one output port and one input port. Since the complexity of both ports is similar, we consider that when a fault happens in an end-system, it affects equiprobably its output and input ports. This is shown in the column Prop. of the left side of Table 1, where the proportion with which each port is affected is 50%. In any case, these proportions are parameters of our model that can be tuned to study the impact that modifying this proportion has on reliability. The analysis of the negative impact of temporary faults affecting PTRF-enabled bridges and end-systems is more complex. Unlike TSN, PTRF does provide time redundancy. Hence, as said before, faults can be tolerated in the corresponding path, as long as the listener receives at least one copy of each message edition through that path. To simplify the explanation from hereon, let us consider a single path. In order to model how faults can be tolerated when using PTRF, we need to differentiate which failure modes PTRF-enabled devices may exhibit on top of the ones exhibited by TSN devices. This is so because certain failure modes may exhaust the time redundancy provided by PTRF and, thus, exceed its fault-tolerance capacity. Table 2 describes the failure modes of PTRF. For this purpose, let us revisit Figure 4, which shows the extra components of a PTRF-enabled bridge coloured in blue. As already explained in Section 4, the blue components represent the extra hardware required for PTRF to create replicas during transmission and to eliminate surplus replicas upon reception. More specifically, the left side of the figure shows the components that eliminate surplus replicas upon reception. Let us refer to them as PTRF reception components. The right side of the figure presents the components that create and then queue in the output queues the frame replicas. Let us refer to them as PTRF transmission components.