## 5 Network Calculus Basics  
The network calculus (NC) [48] approach is a powerful tool to represent the flows' timing properties in the domain of communication networks deterministically. These timing properties include strict upper- and lower-bound computations used for performance evaluations, such as end-to-end latencies, network use, and buffer requirements. In this paper, we use NC to determine the worst-case end-to-end latency for TT flows. Two appropriate curves have to be implemented, the arrival curve and the service curve, to describe the characteristics of flows and related nodes' availability. The analysis is mainly provided based on the min-plus, which uses convolution operation, as defined in the following formulas [48].  
\[(f\otimes g)(t)=\underset{0\leq\lambda\leq t}{\inf}\{f(t-\lambda)+g(\lambda)\}\]  
\[(f\oshellipsis g)(t)=\underset{\lambda\geq 0}{\sup}\{f(t+\lambda)-g(\lambda)\}\]  
where \(inf\) and \(sup\) mean infimum and supremum, i.e., maximal lower bound and minimal upper bound, respectively. The notations \(\otimes\) and \(\oshellipsis\) represent the convolution and deconvolution of min-plus operations, respectively.  
Figure 4: Overlapping configuration between two TT windows with different priority queues.  
The arrival curve \(\alpha(t)\) describes the arrival process \(R(t)\) of a stream. This process represents the cumulative function, at the ingress switching port, by counting the bits that reach the node until \(t\), as expressed in this inequality [48]:  
\[R(t)\leq\inf_{0\leq\lambda\leq t}\{R(\lambda)+\alpha(t-\lambda)\}=(R\otimes \alpha)(t)\;\forall\;\lambda<t\]  
The arrival curve is modeled using the input traces and the traffic configuration, as combined in the triple \(p\), \(j\), \(d\), where \(p\) represents the period of incoming streams, \(j\) denotes the jitter, and \(d\) is the lowest inter-arrival distance of traffic in the specified flow. Accordingly, the lower and upper arrival curves are formed as follows [48]:  
\[\alpha^{l}(\Delta)=\left\lfloor\frac{\Delta-j}{p}\right\rfloor \tag{2}\]  
\[\alpha^{u}(\Delta)=min\bigg{\{}\bigg{\lceil}\;\frac{\Delta+j}{p}\bigg{\rceil},\bigg{\lceil}\frac{\Delta}{d}\;\bigg{\rceil}\bigg{\}} \tag{3}\]  
The above bounds mean that in any time interval (\(\Delta\)), there arrive at least \(\alpha^{l}(\Delta)\) and at most \(\alpha^{u}(\Delta)\) stream events that are modeled by \(\alpha(t)\). The arrival curve \(\alpha(t)\) can be represented in a simple schematic diagram presented in Figure 5a. For the worst-case latency computations, the upper bound of the arrival curve is considered.  
The service curve \(\beta(t)\) describes the transmission availability of the network resources, as modeled in the departure process \(R^{*}(t)\) of a stream. This process represents the cumulative function, at the egress switch port, by counting the bits that can be served in the related node up to time \(t\), as expressed in the following inequality [48]:  
\[R^{*}(t)\leq\inf_{0\leq\lambda\leq t}\{R(\lambda)+\beta(t-\lambda)\}=(R\otimes \beta)(t)\;\forall\;\lambda<t\]  
In a given time interval \(\omega\), where the switch can serve the arrived streams by a rate \(C\), the service curve's lower and upper bounds are specified depending on network resource availability. Figure 5b shows an example of service curve bounds for a periodic time division multiple access (TDMA) resource.  
In each node, the traffic is expressed by the arrival curve at the ingress port and served by the egress port's service curve. The worst-case latency boundaries are obtained by  
Figure 5: Examples of upper and lower bounds: (**a**) the arrival curve and (**b**) the service curve for a periodic time division multiple access (TDMA) system.  
considering the maximum distance (\(D_{max}\)) between the upper-bound arrival curve (\(\alpha^{u}(t)\)) and lower-bound service curve (\(\beta^{l}(t)\)) [48], as represented in Figure 6.  
\[D_{max}Del\Big{(}\alpha^{u},\beta^{l}\Big{)} \tag{4}\]