# 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  
#### 4.1.1 Frame Synchronous Coded-Slotted ALOHA  
We first analyze the impact of the marginal distribution of the arrivals in the key performance variables. To this end, we compare the results (Figure 4) between Poisson arrivals (memoryless) and discrete Lognormal arrivals with \(\mu=\sigma^{2}\). For frame-synchronous CSA, there is no noticeable difference between both cases when \(\mu=\sigma^{2}\). Clearly, performance of \(k\)-MUD CSA in this special case is unaffected or only mildly affected by the marginal distribution, and the key parameter in this respect is only the mean arrival rate (for fixed \(\ell\) and replica distribution \(\Lambda(x)\), of course). The reason for this behavior is that FS-CSA acts really as a gated system at the frame level: all the arrivals must await until the next frame begins, and then the replicas for each user's packets are uniformly distributed over the frame. Therefore, it seems logical that, if the variance in the number of packets is low, it is the average number of packets received during the previous frame, and not their marginal distribution, the fundamental measure of the system load and performance. Accordingly, we conjecture that the performance in this case is insensitive to the marginal distribution and is only a function of the system load, as expected. Contrarily, if we increase the variance, performance with the Lognormal marginal degrades substantially. Figure 5 depicts the PLR and average delay in two asymmetric cases, when \(\sigma^{2}=1.1\mu^{2}\) and with \(\sigma^{2}=4\mu^{2}\). We can see clearly that the PLR is now orders of magnitude higher for the same normalized traffic load, and that the average delay increases monotonically with \(\lambda_{a}/k\). Most likely, this happens because the SIC decoder cannot find any slot within the frame such that \(k\)-decoding can be used, and the recovery of the (replica) packets becomes trapped. In other words, for arrival processes with high variability--Poisson arrivals do not possess this property--the SIC decoder quickly hits the error floor region of the decoding process [28], which suggests that a larger window or a different, optimized degree distribution should be used in the system. Increasingwould also improve performance and shift the error floor region threshold, but at the cost of a high hardware complexity at the receiver. Figure 4: Simulated PLR (**top**) and Mean Delay (**bottom**) for FS-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 \(\mu=\sigma^{2}\). To test if the type of marginal distribution implies some change in the performance of the system, we also conducted simulation experiments with the discrete Pareto distribution shifted to zero, for \(\alpha=1.1\), \(\alpha=1.5\) and \(\alpha=1.9\) (\(\alpha\) is the rate of hyperbolic decay of the tail of the distribution). The results are shown in Figure 6. Now, for heavy-tailed marginal distributions, both the PLR and the average delay show a strong degradation, and are very sensitive to the value of \(\alpha\). We conclude from this set of experiments that the presence of heavy tails in the marginal distribution of the arrivals induces a notable change in performance for CSA, and that the performance degrades fast as the tail of the distribution becomes heavier, particularly for the average delay. A second test with the discrete Pareto distribution shifted to zero, but with \(\sigma^{2}=1.1\mu^{2}\) and \(\sigma^{2}=4\mu^{2}\), yields the results depicted in Figure 7. The numerical results are very similar, quantitatively and qualitatively, to those in Figure 5, confirming that the second-order moments of the arrival process cannot be ignored at all when calculating or predicting the system performance.