# Performance Analysis in Heterogeneous Networks with Spatiotemporal Traffic and Scheduling
## 3 Mean Packet Throughput Analysis
### Mean Packet Throughput of the Heterogeneous Network with UFR  
Use the same method as in [11, 12], we know that the probability mass function (PMF) of the user number per cell for a single-tier PPP network with density \(\lambda\) is  
\[\mathbb{P}\left(N=n\right)=\frac{\Gamma\left(n+q\right)}{\Gamma\left(n+1 \right)\Gamma\left(q\right)}\frac{{\lambda_{u}}^{n}\left(q\lambda\right)^{q}} {\left(\lambda_{u}+q\lambda\right)^{n+q}}, \tag{7}\]  
where \(\Gamma(\cdot)\) is the gamma function, \(q=3.575\). For a \(K\)-layer heterogeneous network, the area distribution of Voronoi cells in a single-layer PPP network with a density of \(\lambda_{i}/a_{i}\)[13] can be used to accurately estimate the area distribution of weighted Voronoi cells in the \(i\)th tier, where \(a_{i}\) is the probability that a typical user is associated with the \(i\)th tier. The expression is as follows:  
\[a_{i}=\frac{\lambda_{i}(P_{ti}b_{i})^{2/\alpha}}{\sum\limits_{k=1}^{K}\lambda _{k}(P_{tk}b_{k})^{2/\alpha}}. \tag{8}\]Then, the probability mass function (PMF) of \(n\) user per cell in the \(i\)th tier is as follows:  
\[ \begin{array}{l}\mathbb{P}\left(N_{i}=n\right)=\frac{\Gamma(n+4)}{\Gamma(n+1) \Gamma(q)}\frac{\left(\lambda_{s}\right)^{n}\left(q\lambda_{i}/a_{i}\right)^{q }}{\left(\lambda_{s}+q\lambda_{i}/a_{i}\right)^{n+q}}\\ =\frac{\Gamma(n+q)}{\Gamma(n+1)\Gamma(q)}\frac{\left(\frac{q\lambda_{i}}{a_{i }}\right)^{q}\left(q\lambda_{i}\right)^{q}}{\left(\lambda_{s}\lambda_{s}+q \lambda_{i}\right)^{n+q}}.\end{array}  \tag{9}\]  
**Lemma 1**. : _The expression of the average active probability of the \(i\)th tier is as follows:_  
\[p_{ai}=\sum_{k=1}^{\lfloor\frac{n_{\Phi_{i}}}{\xi}\rfloor}\frac{k\xi}{\mu_{ \Phi_{i}}}\cdot\mathbb{P}\left(N_{i}=k\right)+1-\sum_{k=0}^{\lfloor\frac{n_{ \Phi_{i}}}{\xi}\rfloor}\mathbb{P}\left(N_{i}=k\right). \tag{10}\]  
_where \(\mu_{\Phi_{i}}\) represents the probability of success transmission of the \(i\)th tier._  
Proof. : We assume \(M_{i}=\lfloor\frac{n_{\Phi_{i}}}{\xi}\rfloor\). According to [PERSON] theorem, \(M_{i}\) is then the maximum average number of users per cell in the \(i\)th tier to maintain the stability of the cell queue states. For cells of the \(i\)th tier that have \(k\left(k=1,2,\cdot\cdot\cdot,M_{i}\right)\) associated users, they are active with probability \(k\xi/\mu_{\Phi_{i}}\). For cells that have more than \(M_{i}\) users, the queueing status is unstable, which means that the BSs are always active. By deconditioning Equations (4) or (6) on \(N\), we obtain get the BS active probability in Equation (10). According to the above derivations, we are now ready to derive the mean packet throughput via queueing theory and stochastic geometry. **Theorem 1**. : _Under UFR mode, Equation (11) gives the mean packet throughout of the network in which \(K_{1}\)tiers adopt RS policy and \(K_{2}\) tiers adopt RR policy. In Equation (11), \(I_{1}\) and \(I_{2}\) are the collections of the tiers that adopt RS and RR, respectively; \(K_{1}=\left|I_{1}\right|\), \(K_{2}=\left|I_{2}\right|\), \(\mid\cdot\mid\) denotes the cardinality of the set; the success transmission probabilities of all tiers satisfy the following equations:_  
\[\tau=\sum_{i\in I_{1}}a_{i}\sum_{k=1}^{\lfloor\frac{n_{\Phi_{i}}}{\xi}\rfloor }\frac{\mathbb{P}(N_{i}=k)(\mu_{\Phi_{i}}/k-\xi)}{1-\xi}+\sum_{i\in I_{2}}a_{i }\sum_{k=1}^{\lfloor\frac{n_{\Phi_{i}}}{\xi}\rfloor}\frac{\mathbb{P}(N_{i}=k )(\mu_{\Phi_{i}}/k-\xi)}{1-\frac{k-1}{2}\cdot\mu_{\Phi_{i}}/k-\xi}, \tag{11}\]  
\[ \begin{array}{l}\left\{\begin{array}{l}\mu_{\Phi_{1}}=\frac{1}{1+a_{1} \prod\limits_{k=1}^{K}\left[p_{ak}\cdot\hat{\lambda_{k}}Z(\theta,\alpha,\hat{ \rho_{k}})\hat{p_{k}}^{\frac{2}{2}}\right]},\\ \mu_{\Phi_{2}}=\frac{1}{1+a_{2}\prod\limits_{k=1}^{K}\left[p_{ak}\cdot\hat{ \lambda_{k}}Z(\theta,\alpha,\hat{\rho_{k}})\hat{p_{k}}^{\frac{2}{2}}\right]}, \\ \vdots\\ \mu_{\Phi_{K}}=\frac{1}{1+a_{K}\prod\limits_{k=1}^{K}\left[p_{ak}\cdot\hat{ \lambda_{k}}Z(\theta,\alpha,\hat{\rho_{k}})\hat{p_{k}}^{\frac{2}{2}}\right]}. \end{array}\right.\end{array}