# Performance Analysis in Heterogeneous Networks with Spatiotemporal Traffic and Scheduling
## 2 System Model  
In this paper, we consider interference performance under direct data transmission in a downlink \(K\)-tier heterogeneous network. The research on heterogeneous networks dates back to the BARWAN (Bay Area Research Wireless Access Network) project initiated by the University of California, Berkeley in 1995. The project leader, [PERSON], first mentioned the integration of different types of networks that overlap each other to form heterogeneous networks. Network to meet the business diversity needs of future terminals. Each converged network is called a layer of heterogeneous network, and a layer of heterogeneous network usually contains a base station and several mobile terminals. In the following decades, heterogeneous networks have attracted widespread attention in the field of wireless communications, and have also become the development direction of next-generation wireless networks. Locations of BSs from the \(i\)th (\(i=1,2,...,K\)) tier formed an independent homogeneous PPP \(\Phi_{i}\) with density \(\lambda_{i}\) and the users formed another PPP \(\Phi_{u}\) with density \(\lambda_{u}\). The BS transmission power of the \(i\)th tier is \(P_{ti}\). The propagation channel suffers from small-scale Rayleigh fading with unit mean power and large-scale path loss, which can be written as \(l\left(r\right)=r^{-\alpha}\), with \(\alpha>2\) being the path loss exponent according to the power law model. The Rayleigh fading channel is a statistical model of the radio signal propagation environment. In the wireless communication channel environment, after the electromagnetic waves reach the receiver through multiple paths such as reflection, refraction, and scattering, the intensity of the total signal obeys the Rayleigh distribution. At the same time, due to the movement of the receiver and other reasons, characteristics such as signal strength and phase are fluctuating, so it is called Rayleigh fading. Rayleigh fading is applicable to the case where there is no direct signal from the transmitter to the receiver, and it is especially suitable for complex heterogeneous network scenarios. To balance the load among different tiers, we assume the \(i\)th tier has the bias factor \(b_{i}\). Each user is associated with the BS, providing the maximum average biased received power. The universal frequency reuse mode is adopted, and users suffer from both intra-and inter-tier interference. The general frequency reuse mode is to reuse frequencies and was first used in GSM networks. Frequency multiplexing means that the same frequency covers different areas. These areas using the same frequency need to be separated from each other by a certain distance to suppress the same frequency interference to within the allowable index. In order to make full use of frequency resources, frequency reuse technology has also become an important research content of next-generation wireless networks. We introduce discrete time queueing model to capture the sporadic wireless data generation conditions. In particular, time is divided into a series of equal time intervals, i.e., time slots. Packet arrivals at each user follow an independent Bernoulli process at a rate of \(\xi\in[0,1]\) (packets/slot). Under normal circumstances, the data in the network is bursty. To facilitate the analysis and analysis of system performance, it is assumed that the arrival of the data packet follows the Bernoulli process. The size of each packet is unified. We assume the BS is equipped with an infinite buffer and all the incoming packets of a cell follow first come first serve (FCFS). In a heterogeneous network, all nodes will try to send data in the same time slot. Whether the data is successfully transmitted depends on the data buffer and channel conditions. At the beginning of the time slot, the base station with data packets waiting to be sent starts a transmission attempt.