# Schedulability Analysis in Fixed-Priority Real-Time Multicore Systems with Contention
## 4 Interference-Aware Schedulability Analysis for Fixed Priorities
### Interference-Aware Schedulability Analysis for Deadline Monotonic Scheduling
#### 4.2.1 Previous Definitions
In this case, K=0, which means that the receiving task is released when the deadline of the broadcasting task has expired and that it has not yet been released again. So, there is no interference. On the contrary, if the receiving task is released when the broadcasting task has been released and its deadline has not expired, there may be interference. Let us assume now that there exists \(t^{\prime}\) so that \(t^{\prime}=\alpha\cdot T_{j}\) and \(aT_{i}<t^{\prime}<aT_{i}+D_{i}\). In this case,  
\[t^{\prime}-T_{j}\left|\frac{t^{\prime}}{T_{j}}\right|=\alpha\  
To do so, we need the following definition and properties. Definition 3: Let \(A_{ij}\) be a binary matrix of \(N_{i}\times N_{j}\). The value of \(A_{ij_{mn}}\) indicates whether activation \(m_{th}\) of \(\tau_{i}\) falls within activation \(n_{th}\) of \(\tau_{j}\) or not in the following way:  
* \(A_{ij_{mn}}=1\): activation \(m_{th}\) of \(\tau_{i}\) overlaps with activation \(n_{th}\) of \(\tau_{j}\). * \(A_{ij_{mn}}=0\): activation \(m_{th}\) of \(\tau_{i}\) does not overlap with activation \(n_{th}\) of \(\tau_{j}\). This matrix serves as the basis to establish when an activation of task \(\tau_{i}\) is subject to the influence of activations of tasks with higher priority running in the same core. Property 1: If a pair of tasks \(\tau_{i}\) and \(\tau_{j}\) are allocated to different cores, \(M_{\tau_{i}}\
eq M_{\tau_{j}}\); then, \(A_{ij_{mn}}=0\ \forall m\in N_{i},\forall n\in N_{j}\). Property 2: \(A_{ij}=A_{ji}^{T}\ \ \ \forall\tau_{i},\tau_{j}\in\tau\). Property 3: \(A_{ij}=O\) if \(i=j\). We shall demonstrate the behavior of \(A\) with an example. We consider task set \(\tau^{\prime\prime}=[\tau_{0}^{\prime\prime},\tau_{1}^{\prime\prime}\tau_{2}^{ \prime\prime}]\), with \(\tau_{0}^{\prime\prime}=(1,2,3,1)\), \(\tau_{1}^{\prime\prime}=(2,5,5,0)\), and \(\tau_{2}^{\prime\prime}=(1,3,5,1)\), allocated to a dual-core system, where \(\tau_{0}^{\prime\prime}\) and \(\tau_{1}^{\prime\prime}\) are allocated to \(M_{0}^{\prime\prime}\), and \(\tau_{2}^{\prime\prime}\), to \(M_{1}^{\prime\prime}\) (Table 3). Figure 4 shows how tasks are allocated to cores and the relation of deadlines and periods. For simplicity, neither computation times nor interferences are depicted. Let us deduce the values of \(A_{01}\), \(A_{02}\), \(A_{10}\), \(A_{12}\), \(A_{20}\), and \(A_{21}\). If tasks are not allocated to the same core, \(A_{ij}\) are null matrices. Then, \(A_{02}=A_{12}=A_{20}=A_{21}=O\). Let us now calculate \(A_{10}\). \(A_{10}\) is the matrix that relates \(\tau_{1}\) and \(\tau_{0}\) and is composed of \(N_{1}\) rows and \(N_{0}\) columns, where \(N_{1}=H/T_{1}=3\) and \(N_{0}=H/T_{0}=5\). For each element \(a_{10_{ij}}\), it is evaluated if activation \(i^{th}\) of \(\tau_{1}^{\prime\prime}\) overlaps with activation \(j^{th}\) of \(\tau_{0}^{\prime\prime}\), and it may be deduced by observing Figure 4. For example, activation \(0\) of \(\tau_{1}^{\prime\prime}\) overlaps with activation \(1\) of \(\tau_{0}^{\prime\prime}\). Then, \(a_{10_{01}}=1\). But activation \(1\) of \(\tau_{1}^{\prime\prime}\) does not overlap with activation \(1\) of \(\tau_{0}^{\prime\prime}\). Then, \(a_{10_{11}}=0\). By evaluating the overlap among all activations of both tasks, \(A_{10}\) is obtained. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Task \(\tau^{\prime}\)** & **C** & **D** & **T** & **I** & **Core M** \\ \hline
0 & 1 & 2 & 3 & 1 & 0 \\
1 & 2 & 5 & 5 & 0 & 0 \\
2 & 1 & 3 & 5 & 1 & 1 \\ \hline \end{tabular}
\end{table} 
Table 3: System with task set \(\tau^{\prime\prime}\). \[A_{10}=\left( \begin{array}{cccccc}1&1&0&0&0\\ 0&0&1&1&0\\ 0&0&0&1&1\end{array} \right)\]  
The next section presents the definition of WCRT considering the interference produced in multicore systems, making use of the above definitions.