The Network Calculus theory has been designed to compute upper bounds on delay and backlog in data networks. A lot of results have been developed to address different aspects. However, they are not all compatible with each other since they make different assumptions on the continuity of a core aspect of the model (the cumulative curves). However, real systems may mix several mechanisms. When modeling such a system, one has to choose one continuity hypothesis and limit the analysis to a subset of existing results. This paper addresses the continuity problem and argues formally that continuity issues are mathematical details that can be solved as long as the min-plus properties are used (minimal and maximal service, shaping). Conversely, it gives a counter-example for properties based on strict service, requiring a generalisation of the backlogged interval notion.

Real-time systems often consist of numerous sub-systems engaged in extensive data exchange. Despite their complexity, a critical challenge lies in effectively incorporating real-time constraints within these systems. To address this challenge, designers typically conduct analyses to establish upper bounds on delays, ensuring they remain within the deadlines of incoming requests. However, adopting a pessimistic approach often results in over-dimensioning the systems. Then, to reduce the pessimism, we want to take into account the fact that a subsystem cannot propagate more requests/data than it can execute. This phenomenon is well-known in network analysis as it reduces the burst of data. As a consequence, this notion is easier to grasp in theories developed to compute delay bounds in networks. That is why we choose, in this paper, to perform the analysis using the Network Calculus theory, since it offers the possibility to easily aggregate flows (i.e. sum flows) and then take into account the phenomenon of smoothing the traffic. To handle tasks and networks, our model relies on packetization and workload scaling. In this paper, we improve some results regarding the already existing elements of Network Calculus and the aggregation. Also, we update and complete definitions and results related to workload scaling.

Cyclic Queuing and Forwarding (CQF) is a mechanism defined by IEEE TSN for providing low jitter in a deterministic network. CQF uses a common time cycle and two buffers per node output port: during one cycle incoming packets are stored in one buffer while packets in the other buffer are being transmitted; at the end of a cycle, the roles of the two buffers are exchanged. The cycle start times are determined by a time offset that may be different for every output buffer. A guard band at both cycle ends is devised in order to compensate for misalignment and timing inaccuracies. The proper operation of CQF requires that the guard band and the offsets are computed such that nodes are sufficiently time-aligned. First, we give necessary and sufficient conditions for this to be guaranteed. The sufficient conditions lend themselves to tractable computations and we show that they are close to optimal. Our conditions account for nonideal clocks and non-zero propagation times; we show that accounting for these two elements does matter. Second, we give a method for computing the minimal duration of the guard band, given prior choices of time offsets. Third, a judicious choice of time offsets can considerably decrease the required duration of the guard band: we give a practical algorithm, based on a Mixed Integer Linear Program, for computing offsets that minimize the guard band. We illustrate our results on several CQF network topologies with or without cyclic dependencies.