# Study of Coded ALOHA with Multi-User Detection under Heavy-Tailed and Correlated Arrivals
## 4 Numerical Experiments
### Impact of the Marginal Distribution of the Arrival Process
#### 4.1.2 Frame Asynchronous Coded-Slotted ALOHA  
We now turn our attention to the performance of frame asynchronous CSA. In this case, the frameless structure suggests that the effects of LRD and/or heavy-tailed distributions could be sharper, since correlations in the input traffic are not affected by the finite frame structure, only by the memory used by the decoder. Thus, we analyze several tests to confirm this hypothesis. First, we change the marginal distribution of the point arrivals only, considering both Poisson and Lognormal distributions with \(\mu=\sigma^{2}\). Figure 8 confirms that, in this particular case with average and variance having the same value, the marginal distribution (both are light-tailed) has the effect of shifting the \(\mathsf{PLR}\) and average delay to the right, i.e., system overload appears later with the discrete Lognormal marginal. Actually, it is quite remarkable that, for \(k=1,2\), we do not see the exponential increase of \(\mathsf{PLR}\) for \(\lambda_{a}/k\leq 1\) in the case of Lognormal marginals, suggesting that the threshold for overloading the SIC receiver is farther and appears at higher arrival rates. It is reached for \(\lambda_{a}/k\approx 0.9\) when \(k=3\), instead. A second test if to check whether the parameters of the Lognormal distributions have any impact on the dynamics of the system. To verify this, we simulated two types of arrivals with marginal Lognormal with \(\sigma^{2}=1.1\mu^{2}\) and \(\sigma^{2}=4\mu^{2}\). The results are displayed in Figure 9, and we see that the \(\mathsf{PLR}\) is roughly one or two orders of magnitude higher when the variance is larger. The average delay is also larger when the variance increases, and there is not a significant difference in average decoding delay if \(k=1,2,3\), at least until the arrival rate reaches the threshold of system overload. Next, we want to investigate how the existence of heavy tails in the input point process influences the packet loss and the delay. In order to gain some insights, we simply simulated the system changing the input process and using three discrete Pareto marginal  
Figure 8: Simulated PLR (**top**) and Mean Delay (**bottom**) for FA-CSA versus \(\lambda_{a}/k\), with \(\Lambda(x)=0.86x^{3}+0.14x^{8}\), \(n=200\) and \(k=\{1,2,3\}\), and marginal distribution Poissonian (**left**) and Lognormal (**right**), in both cases \(\mu=\sigma^{2}\). Figure 9: Simulated PLR (**top**) and Mean Delay (**bottom**) for FA-CSA versus \(\lambda_{a}/k\), with \(\Lambda(x)=0.86x^{3}+0.14x^{8}\), \(n=200\) and \(k=\{1,2,3\}\), and marginal distribution Lognormal, with \(\sigma^{2}=1.1\mu^{2}\) (**left**) and \(\sigma^{2}=4\mu^{2}\) (**right**). distribution for the arrivals, with parameters \(\alpha=1.1\), \(\alpha=1.5\), and \(\alpha=1.9\), respectively. Remember that the three cases have infinite variance, since \(\alpha<2\). According to the results shown in Figure 10, for heavy-tailed arrivals and unbounded second moment, the \(\mathsf{PLR}\) is high but not very much sensitive to the normalized load \(\lambda_{a}/k\). As for the average delay, notable differences among the cases \(k=1,2,3\) arise only for the slowest decay rate of the pdf tail (top-right panel in Figure 10) whereas for faster decay rates the behavior is very similar with respect to the normalized loads. If one ignores for the moment the variations due to the intensity of the arrival rate \(\lambda_{a}\), then Figure 10 allows us to conclude that even for heavy-tailed input traffic \(k\)-MUD for \(k>1\) has a good performance, since both the packet loss ratio and the average delay are almost independent of the value of \(k\). Alternatively, we obtain approximately the same performance for arrival rates \(\lambda_{a}=\lambda k\), and \(k=1,2,3,\dots\). This quasi-linear response is perhaps the main advantage in CSA, and our results show that it is preserved even for different marginals, unless the input traffic has a very long tail in its marginal distribution.