# A Power Control Algorithm Based on Chicken Game Theory in Multi-Hop Networks
## 2 System Model  
Figure 1 shows a typical ad hoc network, composed of virtual cells (VCs). In this model, many central ports act as gateways to networks. Every VC has its own gateway, and the location of the wireless ports are randomly distributed. Groups of VCs can work as a virtual base station. Because there is some shadowing decay, distance loss, and multi-path fading in networks, high transmission power is required by many VCs when they exchange messages directly with central ports. Multi-hop networks could reduce this problem. The signals in the network could be relayed via these VCs [16; 17; 18; 19; 20; 21]. A number of VCs can communicate smoothly with central ports with any downlink [22; 23; 24; 25; 26] or uplink [27; 28; 29; 30; 31; 32; 33] transmission of data. In order to judge the quality of the transmit power reduction, the numerical equations of transmit power are obtained below [34]. The ideal transmit power control (TPC) based on the signal-to-noise power ratio (SNR) measurements with ideal rake combining [35] is assumed here. A Rayleigh fading channel with an \(L\)-path is also assumed. In relation to a multi-hop relay with no diversity, the transmit power \(P_{t}(i)\) from port #\(i\) is shown in Equation (1). The system's supposition is that there is ideal rake combining, and the TPC works well enough to accurately value sending power. In addition, a fading channel with Rayleigh [35] is supposed as the \(L\)-path, so no diversity is applied in multi-hop relay networks [36; 37; 38; 39; 40; 41; 42]. Transmit power \(P_{t}(i)\) from port #\(i\) is expressed below:  
\[P_{t}(i)=\frac{P_{req}}{d_{i,j}^{-\alpha}10^{-\frac{\eta_{ij}}{10}}\sum\limits_ {l=0}^{L-1}\left|\xi_{i,j}(l)\right|^{2}} \tag{1}\]  
In Equation (1), \(P_{req}\) is the required received signal power. Between nodes #\(i\) and #\(j\), there is some definition, including shadowing decay \(\eta_{i,j}\) (dB), \(l\)-th path plurality path gain \(\xi_{i,j}\), path loss exponent \(\alpha\), and the distance \(d_{i,j}\). If the ensemble average operation is \(E[\bullet]\), then the result is \(E\left[\left|\xi_{i,j}\right|^{2}\right]=1/L\) independent plurality gaussian variables in which the zero-mean [43] is \(\left\{\xi_{i,j}\right\}\). In order to unify this article, _\"i\"_ and _\"j\"_ are used to symbolize nodes #\(i\) and #\(j\). Figure 1: Multi-hop (ad hoc) networks. To establish the complete, transmit power following the multi-hop maximum ratio combining (MHMRC) [29; 30; 31; 32; 33] route, an \(n\)-hop connection from the wireless terminal to the central port is investigated, where port \(\#i=0\) is the mobile terminal, port \(\#i=n\) is the central port, and port \(\#i=1\)-\(n\)\(-\)\(1\) is the intermediate port. The received power \(P_{r}(1)\) from the mobile terminal \(\#i=0\) is given by  
\[P_{r}(1)=P_{t}(0)d_{0,1}^{-\alpha}10^{-\frac{\tau_{0,1}}{10}}\sum\limits_{l=0} ^{L-1}\left|\xi_{0,1}(l)\right|^{2} \tag{2}\]  
Therefore, the mobile terminal transmit power \(P_{t}(0)\) can be written as  
\[P_{t}(0)=\frac{P_{req}}{d_{0,1}^{-\alpha}10^{-\frac{\tau_{0,1}}{10}}\sum\limits _{l=0}^{L-1}\left|\xi_{0,1}(l)\right|^{2}} \tag{3}\]  
For convenience during the numeric computation, Equation (3) has been included, although it should be noted that it is exactly the same as Equation (1) except that \(i=0\). This is because, if the numerical calculations start from 0, then the totaling is easier; it means that 1 is 1, and the current number does not need to be subtracted by 1. As for the ports where \(\#i=2\)-\((n-1)\), the receiving power \(P_{r}(i)\) of port \(\#i\) is the total of all the received power from all the previous ports and is derived by  
\[