# Performance Analysis of P2P Networks with Light Communication Links: The Static Managed Case
## 3 Centralized and Decentralized Networks
A seed can leave the system with rate \(\gamma\); in this case, the nodes have finished the download procedure and the user is no longer required to remain in the system, but it can choose to dwell for longer times to provide its resources to downloading peers. (In some cases, the system administrator offers rewards for seeds to dwell for longer times and share their resources, entailing a better system performance.) In this regard, note that usually \(\theta\
eq\gamma\). From this description, the rate at which _leechers_ are converted into _seeds_, \(\tau\), can be expressed as follows:  
\[\tau=min(cx,\mu(\eta x+y)) \tag{2}\]  
Figure 2: Peer-to-peer structure: download links and upload information between leechers and seeds. As Equation (2) describes, the total bandwidth in the system is the total number of peers, \(x+y\), uploading at rate \(\mu\), and accounting for the fact that not all leechers can share the file, i.e., there is an efficiency parameter \(\eta\) added to the leechers. Then, in case all leechers, \(x\), are downloading at the maximum rate, \(c\), the total required bandwidth from the network is \(cx\), and the rest of the bandwidth is not used. In case the leechers are not downloading at the maximum rate, the effective bandwidth used is \(\mu(\eta x+y)\). Then, the parameters involved in the P2P system are:  
\[\tau:\text{\it conversion rate}(\forall\tau\in\mathbb{R},\tau>0)\] \[c:\text{\it the downloading bandwidth}(\forall c\in\mathbb{R},c>0)\] \[\mu:\text{\it the uploading bandwidth}(\forall\mu\in\mathbb{R},\mu>0)\] \[\eta:\text{\it sharing \it effectiveness}(\forall\eta\in\mathbb{R},0< \eta<=1)\] \[x:\text{\it number of \it leechers}(\forall x\in\mathbb{N},x>=0)\] Figure 1:0  
And the P2P system can be classified into two conditions: _penury_ and _abundance_, whose relationship is described in Figure 1:1  
Also, the P2P network can be expressed in a system of differential equations, as shown below in (3) and (4) [34]:  
Figure 1:2  
Figure 1:3  
It is important to highlight that in Equation (3) we see that the number of leechers increases by arrivals and decreases by early departures and leech conversions to seeds while Equation (4) clearly depicts that the number of seeds only changes by leechers conversions or seed departures. Once the Equations (3) and (4) have been solved under stable conditions, the state in which the system is going to be studied can be defined according to the value of the rate \(\gamma\), according to the Figure 1:4  
These values of \(\gamma\) are very useful to provide rewards to peers to remain in the system for specific average times in order to achieve abundance conditions. However, these conventional models to describe the dynamics of the P2P system cannot apply in the case of static networks, such as the ones that can allow LiFi communications, due to the alignment requirements. Then, in our case, peers cannot enter or leave the system at any time, and the downloading procedure can only start when the administrator initiates the procedure. These differences require important modifications to the model, which will be explained in detail in further sections. \begin{table}
\begin{tabular}{c c c} \hline \hline
**State** & **Condition** & **Description** \\ \hline _Penury_ & \(cx>\mu(\eta x+y)\) & There is not enough bandwidth to download at the maximum rate \\ _Abundance_ & \(cx<\mu(\eta x+y)\) & There is enough bandwidth for all peers to download at the maximum rate \\ \hline \hline \end{tabular}
\end{table} 
Table 1: P2P system conditions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**State** & \(\tau\) & \(\gamma\) \\ \hline _Penury_ & \(\tau=cx\) & \(\gamma<\frac{nc}{c-\mu\eta}\) \\ _Abundance_ & \(\tau=\mu(\eta x+y)\) & \(\gamma>\frac{1}{c-\mu\eta}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Range of \(\gamma\) values for stable conditions.