# 6G Goal-Oriented Communications: How to Coexist with Legacy Systems?
## 3 Coexistence of a Goal-Oriented and a Data-Oriented Communication System
### Evaluation of the System without Optimization
#### 3.4.6 The Goal Effectiveness and its Dependence on PER, Interference and Goal Cost  
The goal effectiveness in (15) depends on two performance indicators: _(i)_ the goal value (represented by the NREI), which, in our setting, directly and exclusively depends on the PER, and _(ii)_ the E2E delay, which depends on the PER, the interference caused by the DO user and the remote computation delay. As a first joint result, in Figure 7, we show heat maps representing the goal effectiveness (i.e., the goal effectiveness is represented by the different colors, from dark blue to yellow) as a function of PER (\(y\)-axis) and maximum delay threshold \(D_{\text{max}}\) (\(x\)-axis) for a fixed DO user data rate (i.e., fixed \(P_{d}=200\) mW) and for different goal value thresholds in three different plots, i.e., \(\Theta_{\text{th,1}}=-0.8\), \(\Theta_{\text{th,1}}=-0.5\) and \(\Theta_{\text{th,1}}=-0.4\), shown in Figure 7a-c, respectively (we remind that a higher threshold indicates a stricter constraint in the sense of goal achievability). Note that the heat maps are interpolated a posteriori for better visualization, while results have been obtained for the sub-selected PER and \(D_{\text{max}}\) values. In these figures, we also show different goal-effectiveness thresholds (i.e., \(\mathcal{E}_{\text{g,th}}=0.7\), \(\mathcal{E}_{\text{g,th}}=0.8\), \(\mathcal{E}_{\text{g,th}}=0.9\)), through contours plots. All the points interior to these contour plots represent a _goal feasibility region_ for each respective goal-effectiveness constraint, i.e., all the combinations of (fixed) PER and end-to-end delay, whose corresponding goal effectiveness exceeds a predefined threshold. However, these regions are subject to the fact that there is no adaptation of transmit power and target PER across time; whereas, as we will show in the sequel, much better results can be obtained in the optimized setting. From these plots, we can make the following considerations:  
* The goal effectiveness increases as the delay threshold (\(y\)-axis) increases (for each fixed target PER), while it does not necessarily decrease as a function of the PER, as expected and shown before in the disjoint plots, creating _goal effectiveness feasibility regions_, whose surface depends on \(\mathcal{E}_{\text{g,th}}\). * The goal effectiveness feasibility regions shrink as the goal value (NREI) threshold increases (see the difference between surface extensions in Figure 7a-c--looking at Figure 7c, we can even notice that no combination of fixed PER and \(D_{\text{max}}\) guarantees goal effectiveness above 0.9 for \(\Theta_{\text{th}}=-0.4\) in this setting). * Given a goal-effectiveness requirement, there always exists a minimum E2E delay threshold guaranteeing feasibility; whereas, below this threshold, it is infeasible (for any PER) to guarantee the requirement (examples of this point are shown by the red arrows in the figures). Moreover, each target PER experiences a different minimum \(D_{\text{max}}\) that can be guaranteed. The lower the PER is, the higher the minimum feasible delay is. Figure 6: Empirical probability density function of E2E delay for goal cost \(\mathcal{C}\approx 6\%\) (left-hand points of Figure 5). * As the delay threshold decreases, the feasible region in terms of PER also shrinks, i.e., with a lower delay constraint, higher PER values are needed to guarantee effectiveness; however, this is not always feasible due to the goal value constraint (see, e.g., Figure 7c). * As a consequence of the previous remark, as the PER decreases (\(y\)-axis), the minimum delay threshold to guarantee a target goal effectiveness increases, i.e., to guarantee lower PER (reliable communication), \(UE_{g}\) needs more time for offloading, resulting in more frequent delay outages.