## 5 TSN Scheduling Problem Formulation
### TSN Scheduling Optimization Objectives  
In this section, we discuss the relevance of optimization objectives. We analyze the importance of those objectives and we classify them into three groups according to their relevance. Figure 7 summarizes our classification effort. The first group concerns objectives that can be considered more as an acceptance criterion of the solution rather than an optimization objective. In the case that the solution  
Figure 7: Classification of optimization objectives by relevance. The optimization objectives in this figure use the same references as in Figure 3.  
satisfies the criteria, it can be used for the use case, and if not, then it is not convenient. For example, minimizing the end-to-end latency of the streams (e.g., [37]) is one of the earliest proposed objectives in the literature. Nevertheless, as long as the calculated schedule respects the delivery requirement of the stream (latency or deadline), the actual stream latency is not important, and consequently end-to-end latency can be seen as a constraint rather than an optimization objective. Other works such as [41] aim to minimize the schedule's length, which is typically between 8 and 1024 entries [41] to minimize the hardware resources used inside the switch. Unless the length of the schedule influences the performance of the forwarding behavior of the switch or minimizing the length of the schedule is targeting the applicability of the scheduling algorithm to more switches, this objective can, rather, be seen as a hardware constraint. Some works such as [58] aim to maximize the number of scheduled streams. However, for many industrial use cases, the number of HTS streams to be scheduled is given and has to be fully served. Consequently, this parameter can be seen as a hard constraint on the problem. On the other hand, maximizing the number of schedulable LTS streams might be important for some use cases, which require that the network is properly designed to provide enough resources. In this context, objectives such as the topology cost or the support of different techniques are not part of the optimization of the schedule, although they could be related to it. In Figure 7, we aim to cluster similar objectives. While some of them only require a simple yes/no question to answer (e.g., whether the solution algorithm supports multicast streams), others require the definition of a threshold (e.g., maximum allowed schedule synthesis time). In summary, minimizing the schedule length, the end-to-end latency, or the number of scheduled HTS streams, among other objectives in this first group, should be modeled as constraints rather than being part of the objective function of the optimization problem. Another group of objective functions can be classified as secondary as they aim to improve the performance of the overall system rather than guarantee the schedulability of the given HTS streams. For example, several works aim to distribute spare capacity and avoid future bottlenecks [59, 60]. Indeed, keeping spare capacity on all links maximizes the success probability in case HTS streams need to be scheduled during runtime once existing streams are already placed. For example, ref. [59] aims to minimize the absolute deviation of individual link utilization, while [58, 87] balances the load by minimizing the maximum load for all links. Another optimization objective is to maximize the opportunity for LTS traffic. Here, ref. [11] aims to minimize the average queuing delay for LTS streams by combining TAS and CQF. Ref. [24] aims to maximize the porosity of the schedule (the distance between HTS windows) to prevent back-to-back transmission of HTS streams and allow more BE traffic. Optimizing the assignment of queues to traffic classes is another important objective (e.g., ref. [38] aims to minimize the number of HTS queues). Minimizing bandwidth waste is also an important optimization objective since bandwidth is a costly resource [18].