# Network Performance Optimization for Low-Voltage Power Line Communications
## 4 Network Performance Optimization Based on Dynamic Game Theory
### Performance Optimization Model Based on Game Theory
#### 4.2.2 Improved Bandwidth Utilization Model  
Suppose that the sender and the receiver send data packets with probabilities \(\tau_{1}\) and \(\tau_{2}\), respectively. The station sends the successful probability of the data packet in any slot as follows:  
\[P_{S1}=\tau_{1}(1-\tau_{2})^{n-1} \tag{28}\]  
The remaining (\(n-1\)) stations send the successful probability of the data packet:  
\[P_{S2}=(1-\tau_{1})(n-1)\tau_{2}(1-\tau_{2})^{n-2} \tag{29}\]  
The bandwidth control ratio factor \(r\) is defined as:  
\[r=P_{S1}/P_{S2} \tag{30}\]  
We obtain the relationship between \(\tau_{1}\) and \(\tau_{2}\) as follows:  
\[\tau_{1}=\frac{rn\tau_{2}}{1+(rn-1)\tau_{2}} \tag{31}\]  
The station sends the successful probability sum of the data packet:  
\[P^{\prime}_{S}=(P_{S1}+P_{S2})(1-P^{\prime}_{e}) \tag{32}\]  
where \(P^{\prime}_{e}\) denotes the probability of the frame errors caused by noise interference and channel fading and depends on the data packet length, transmission rate, and bit error rate in the PHY layer; \(P_{e\_data}\) is the frame error rate (FER) for the data frame; \(P_{e\_ack}\) is the FER for the ACK frame; and \(L\_ack\) denotes the length of the ACK.  
\[P^{\prime}_{e}=1-(1-P_{e\_data})(1-P_{e\_ack}) \tag{33}\]  
\[P_{e\_data}=1-(1-BER)^{E[DATA]} \tag{34}\]  
\[P_{e\_ack}=1-(1-BER)^{L\_ack} \tag{35}\]  
The idle probability is given by:  
\[{P_{l}}^{\prime}=(1-\tau_{1})(1-\tau_{2})^{n-1} \tag{36}\]  
The collision probability is given by:  
\[P^{\prime}_{c}=1-P^{\prime}_{S}-{P_{l}}^{\prime} \tag{37}\]  
The improved bandwidth utilization utility model is given by:  
\[\eta^{\prime}(r_{s})=\frac{P^{\prime}_{S}E[payload]}{(P^{\prime}_{S}T_{s}+P_{ l}^{\prime}T_{l}+P^{\prime}_{c}T_{c}+P^{\prime}_{e}T_{c})B}=\frac{E[payload]}{ \left\{T_{s}+T_{l}[(1-\frac{T_{l}}{T_{l}})\frac{P^{\prime}_{s}}{P^{\prime}_{S} }+\frac{T_{c}}{P_{S}T_{l}}-\frac{T_{c}}{T_{l}}]\right\}B} \tag{38}\]The improved bandwidth utilization is maximized when the following quantity is minimized:  
\[\min\{(1-\frac{T_{c}}{T_{I}})\frac{{P_{I}}^{\prime}}{P_{S}^{\prime}}+\frac{T_{c}} {P_{S}T_{I}}-\frac{T_{c}}{T_{I}}\} \tag{39}\]  
Taking the derivative of Equation (39) with respect to \(\tau_{2}\) and setting it equal to zero, we obtain:  
\[\partial[(1-\frac{T_{c}}{T_{I}})\frac{{P_{I}}^{\prime}}{P_{S}^{\prime}}+\frac{ T_{c}}{P_{S}T_{I}}-\frac{T_{c}}{T_{I}}]/\partial\tau_{2}=0 \tag{40}\]  
We obtain the following equation after some simplifications:  
\[(1-\tau_{2})^{n+1}(1-\tau_{1})^{2}(T_{I}-T_{c})+\tau_{2}(1-\tau_{1})^{2}(1- \tau_{2})^{n}(T_{I}-T_{c})+T_{c}(1-\tau_{2})(1-2\tau_{1})-T_{c}\tau_{2}(1-\tau_ {1})(n-1)=0 \tag{41}\]  
We obtain the following by substituting Equation (31) in Equation (41):  
\[(1-\tau_{2})^{n+2}(T_{I}-T_{c})+\tau_{2}(1-\tau_{2})^{n+1}(T_{I}-T_{c})+T_{c}( 1-\tau_{2})^{2}-T_{c}\tau_{2}(1-\tau_{2})(n-1)-T_{c}nr\tau_{2}^{2}(nr-n+1)=0 \tag{42}\]  
If \(\tau_{2}\ll 1\), then  
\[(1-\tau_{2})^{n}\approx 1-n\tau_{2}+\frac{n(n-1)}{2}\tau_{2}^{2} \tag{43}\]  
This equation is expressed as:  
\[\begin{array}{c}\frac{n(n-1)}{2}\tau_{2}^{3}(T_{I}-T_{c})+\tau_{2}^{2}(T_{I }-T_{c})[\frac{n(n-1)}{2}+T_{c}(n-1)]+T_{I}-\\ \tau_{2}[(T_{I}-T_{c})n+T_{c}+T_{c}(n-1)^{2}r^{2}+T_{c}(n-1)^{2}r+T_{c}(n-1)r- (n-1)T_{c}]=0\end{array} \tag{44}\]  
We obtain the only solution \(\tau_{2-option}\) to Equation (44) from the Morigane formula, then obtain \(\tau_{1-option}\) when \(r\) is given.