# Performance Analysis in Heterogeneous Networks with Spatiotemporal Traffic and Scheduling
## 3 Mean Packet Throughput Analysis  
### Preliminaries  
Due to [PERSON]'s theorem [10], at time slot \(t\), the Signal-to-Interference Ratio (SIR) of the typical user, which is located at the origin, is:  
\[\gamma_{i,t}=\frac{h_{x_{i}}\|x_{i}\|^{-a}}{\sum\limits_{y_{k}\in\Phi,x_{i}}P_ {ik}\zeta_{y_{k},t}h_{y_{k}}\|y_{k}\|^{-a}}, \tag{1}\]  
where \(h_{x_{i}}\) and \(h_{y_{k}}\) represent small-scale fading of the transmission link and interfering link, respectively; \(\|x_{i}\|\) and \(\|y_{k}\|\) denote the serving distance and interfering distance, respectively; \(\Phi\) is the spatial realization of all the point processes, denoted as \(\Phi\): = \(\cup_{i\in\mathbb{N},\cdots,K}\Phi_{i}\); and \(\zeta_{y_{k},t}\in\{0,1\}\) denotes the transmission status ( transmission or not) of user at \(y_{k}\) at time slot \(t\), i.e., \(\zeta_{y_{k},t}=1\) shows the user is transmitting. \(P_{ik}^{c}=\frac{P_{ik}}{P_{ii}}\). Please note that the interference contains intra-tier and inter-tier interference.  
As shown in Figure 1, a heterogeneous network usually includes a macro station and several small base stations. The object of this study is a wireless terminal in a small base station, called typical  
Figure 1: A two-tier heterogeneous network with spatiotemporal model.  
user. Let \(\mu_{\Phi}\) be the transmission success probability of the typical user, which depends on the probability that the SIR of the typical user \(\gamma_{\Phi}\) is above a certain threshold \(\theta\) with the condition of \(\Phi\), i.e., \(\mu_{\Phi}=\mathbb{P}(\gamma_{\Phi}>\theta\mid\Phi)\). For deriving mean packet throughput, we provide a formal definition as follows:  
**Definition 1**.: \(A_{x}(t)\) _is the average number of data packets arriving at a transmitter \(x\) in time slot \([0,t]\), and \(D_{i,x}\) is sojourn time from the arrival of the \(i\)th packet to its successful sending. The definition of mean packet throughput is as follows:_  
\[\tau\triangleq\lim\limits_{R\rightarrow\infty}\frac{\sum\limits_{x\in\Phi \cap B(0,R)}\lim\limits_{t\rightarrow\infty}\frac{A_{x}(t)}{\sum\limits_{i =1}^{A_{x}(t)}D_{i,x}}}{\sum\limits_{x\in\Phi}\hat{X}_{\{x\in B(0,R)\}}}, \tag{2}\]  
Please note that the spatiotemporal characteristics of the network involve two aspects: (1) the service rate of the queueing depends on spatial distribution of users in the network; and (2) the active probability of each BS relies on the queueing status of the transmitters.  
From Lemma 1 and Lemma 2 in [8], we obtain the mean packet throughput and activity probability of a BS with \(N\) associated users under the RS strategy, subject to the realization of the point processing \(\Phi\). The expressions are as follows:  
\[\tau_{RS}=\left\{\begin{array}{ccc}\frac{\mu_{\Phi}/N-\xi}{1-\xi},&if&\frac{ \mu_{\Phi}}{N}>\xi\\ 0,&if&\frac{\mu_{\Phi}}{N}\leq\xi\end{array}\right., \tag{3}\]  
\[p_{a}^{RS}=\min\{N\xi/\mu_{\Phi},1\}. \tag{4}\]  
With the same method, the mean packet throughput and active probability of a BS with \(N\) associated users under the RR strategy condition on \(\Phi\) are given, respectively, as follows:  
\[\tau_{RR}=\left\{\begin{array}{ccc}\frac{\mu_{\Phi}/N-\xi}{1-\frac{N-1}{2}\mu _{\Phi}/N-\xi},&if&\frac{\mu_{\Phi}}{N}>\xi\\ 0,&if&\frac{\mu_{\Phi}}{N}\leq\xi\end{array}\right., \tag{5}\]  
\[p_{a}^{RR}=\min\{N\xi/\mu_{\Phi},1\}. \tag{6}\]