# Performance Analysis in Heterogeneous Networks with Spatiotemporal Traffic and Scheduling
## 4 Numerical and Simulation Results  
In this section, we present the MATLAB simulations that were used to verify the validity of the above analytical results. The extensive simulation results are given to illustrate the behavior of the system with respect to the change in its loading parameters. For simplicity, we analyze the throughput performance of a two-tier heterogeneous network. The first and second tiers are assumed to use RR and RS policy, respectively. We considered a simulation area of \(9\times 10^{6}\) km\({}^{2}\) with base stations and users scattered across the square plane, whose position remains static once they are settled. The total time slots for simulation was set to \(10^{5}\). The major parameters in our simulation were as follows, unless otherwise stated [7, 8]: the packet arrival rate was set to \(\xi=0.05\) packets/slot, the SIR is set as \(\theta=0\) dB, the path loss exponent \(\alpha\) is 3.8. This paper studies the performance of internal users of small base stations in heterogeneous networks. Because the data is not always available, the data arrival rate is set to 0.05 packets/slot which is the same setting in [7, 8]. The BS transmit powers in the first tiers is assumed to be 39 dBm, and the second tiers is respectively to be 24 dBm. The density of the BSs from the two tiers was respectively set to \(\lambda_{1}=10^{-5}\) km\({}^{-2}\), \(\lambda_{2}=\lambda_{1}\). We first simulated how bias factors influence the throughput performance of the two-tier network model. We set the bias factor of the first-tier \(b_{1}\) to 1 and analyzed the effect of the second-tier bias factor \(b_{2}\). To validate our derived results, the simulation results were carried out and repeated for over 10,000 iterations. The simulation parameters that were based on the Monte Carlo simulation method are shown in Figures 2 and 3.  
Figure 2 depicts the mean packet throughput of the two-tier network versus \(b_{2}\) which validates our analysis. The analytical results match the simulation results very well, which verifies the correctness of mathematical derivation. The figure shows that the average packet throughput increases first and then decreases with the increase in \(b_{2}\), i.e., an optimal bias factor \(b_{2}\) maximizes average packet throughput. This occurs because as \(b_{2}\) increases, more traffic will be offloaded to the second tier, and the first tier  
Figure 3: The mean packet throughput of the network versus \(\lambda_{u}/\lambda_{1}\). \(b_{1}=1\), \(b_{2}=4\).  
Figure 2: The mean packet throughputs of the network versus \(b_{2}\). \(b_{1}=1\).  
becomes less congested. Therefore, the first tier can provide higher packet service rate and higher packet throughput, while the throughput of the second tier is lowered due to increased burden. Given \(b_{2}\) is small, the throughput of first tier dominates the mean packet throughput. The significant increase in the first-tier throughput led to an increase in network throughput. When \(b_{2}\) increases, more traffic is transferred to the second tier, and the curve approaches the throughput of the second tier. Therefore, our analytical results can help with the design of an optimal bias factor for a network to maximize the mean packet throughput.  
Figure 3 depicts the variation of the mean packet throughput of the network with the density ratio \(\lambda_{u}/\lambda_{1}\). The mean packet throughput shows a decreasing tendency. This can be explained by a higher \(\lambda_{u}/\lambda_{1}\) leading to more congestion, which leads to a higher probability of BS activity and greater interference with typical users. When \(\lambda_{u}/\lambda_{1}\) increases, the mean packet delay is prolonged, which leads to the decline in the mean packet throughput. For achieving a certain packet throughput value, BS density can be designed for a given user density according to our analysis.