# 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
#### 3.4.3 Computation Delay Assumption  
In the absence of a model for the computing delay, we have empirically built a delay distribution from real-world experiments. The inference runs on a GPU NVIDIA(r) Tesla(r) V100. The CPU characteristics are the following: Intel(r) Xeon(r) Gold 6244 CPU@ 3.6 GHz, with 4 cores and 8 GB of memory. Then, we run batch inference for \(T=50\),000 independent realizations, we save the computing time and we retrieve a computing delay distribution, from which we can extract a random realization when evaluating performance. For the interest of the reader, the estimated probability density function of the computing delay is shown in Figure 3, for all values of PER, as (a priori) the distribution may change as a function of the PER. However, from the figure, it can be noted that the distribution is stable over PER values, showing only slight variations, probably caused by fluctuations in the CPU usage and background processes. Therefore, in the sequel, we will consider the computation delay distribution as independent from the target PER.  
We will now present the results in terms of goal effectiveness, as per its definition in (15). Let us recall that the goal effectiveness generally entails the goal value and other system constraints. For instance, in the case of edge inference, the goal value is represented by the NREI (cf. (14)), and the other system constraint is represented by the E2E delay (cf. (11) and (10), Figure 3). However, for the sake of smooth exposition, we will first present two separate results, i.e., the goal effectiveness by only taking into account the goal value--NREI (c.f. (14)), and the goal effectiveness by only taking into account the delay, respectively. We believe that this gentle introduction will help us with commenting on the results, and the reader with comprehending the conveyed message, via a clear explanation of the different sources of goal outages. The two notions of goal effectiveness will be then merged in Section 3.4.6 as a final result to coherently consider the complete definition in (15). Afterwards, an optimization problem will be proposed in Section 4.  
3.4.4. The Goal Effectiveness from the Goal-Value Perspective  
Let us evaluate the performance of goal effectiveness, only from the point of view of the goal value, i.e., the first term on the right-hand side of (15) (the inference reliability--negative relative entropy increase--NREI). In other words, we consider a received batch result with goal value higher than the predefined threshold, as a goal achievement, even if results are issued after the deadline \(D_{\text{max}}\). Let us first notice that the goal value, as per its definition in (14), is only affected by the PER, i.e., the errors generated by the wireless communication between the GO user and the GO AP. Then, in Figure 4, we show the goal effectiveness related to the goal value as a function of the PER for different thresholds \(\Theta_{\text{th}}\). First, we can notice how the probability of the goal value being above a threshold decreases as the PER increases and as more errors occur throughout the uplink communication phase. This result is also in line with Assumption 1 (Figure 2a). In addition, by increasing the threshold \(\Theta_{\text{th}}\) (recall that the average NREI assumes non-positive values), performance degrades, as expected.  
Figure 4: Probability of goal value (NREI cf. (14)) being above a predefined threshold, as a function of PER, for different threshold values.  
Figure 3: Empirical probability density function of computing delay, estimated through real experiments.  
As already mentioned, this result does not take into account the delay, which is a fundamental part of the goal (i.e., the edge inference service). In the following, we consider the goal effectiveness from the delay perspective then merge the two perspectives in Section 3.4.6.