## 5 Modelling Rationale
### Reliability Metric  
In order to build up a model that quantifies the reliability, first we need to particularise this general definition into a reliability metric which the model must calculate. In this sense, the reliability metric must be aligned with the objective of the model. That is, we must decide and state clearly what we are measuring.  
It is important to note that we do not model the TSN network of any specific system. Instead, we want to quantify the reliability achievable by _any_ TSN network. This implies that the reliability metric has to be defined in such a way that it is general enough to be applicable to the network of any system, regardless of its specific characteristics. Thus, we propose a reliability metric to quantify the reliability of streams.  
We must note that in this work we only consider time-triggered traffic. Therefore, in our work a stream is a virtual communication channel that consists of a talker sending a message edition every \(T\) units of time to one or several listeners through a given number of paths; where \(T\) is the stream period. In the case of using standard TSN, the talker only sends one copy of each message edition per period (one copy through each path in case of having multiple paths). In the case of PTRF, the talker sends several replicas of each message edition through each path as explained in Section 4.  
On top of that, we can specify in the model the number of listeners that must receive each message edition in order for the system to work correctly. In this way, we can take into account systems that only require a subset of the intended end-systems to communicate correctly to provide a correct service, e.g., systems that use node redundancy to tolerate faults and thus can operate with a subset of end-systems.  
At this point we can define our reliability metric as the probability that for each stream a subset of the subscribed listeners receives at least one copy of each message edition during a given interval of time called mission time.  
In the particular case of the analyses that we show in this article, we do not consider space redundancy. This is so because in this article we want to prevent the reliability benefits of using space redundancy from masking the reliability benefits of the time redundancy provided by PTRF. Thus, our reliability metric is calculated taking into account one path. Furthermore, we only analyse the impact of the reliability for one talker that communicates with a single listener, as this allows us to draw significant conclusions while minimising the factors that can interfere with our analysis. In the future we plan to evaluate networks with several streams, each with several listeners and traffic types.  
Finally, it is important to note that this metric alone does not quantify the reliability of a whole distributed system based on TSN. To achieve such a quantification, it would be necessary to model the particularities of a given system, as well as to include all the reliability mechanisms the system is provided with, e.g., the mechanisms to replicate the talker and listener themselves. Thus, our metrics and models have to be understood as a relevant piece that can help in quantifying the reliability of a whole system, but not as the only piece that is necessary for doing so.