# 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
#### 4.2.3 Improved Saturation Access Delay Model  
We compute the saturation access delay \(D\) in the asymmetrical channel as follows:  
\[D=T_{s}+D_{s}+D_{c}+T_{slot} \tag{45}\]  
The access delay \(D\) consists of four parts: (1) \(T_{s}\) denotes the time for a successful transmission; (2) \(D_{s}\) represents the average time the channel is sensed to be busy because of a successful transmission by other stations; (3) \(D_{c}\) represents the average time a channel is sensed to be busy because of collisions; and (4) \(T_{slot}\) denotes the total time of idle slots, including the total back-off time of successful transmissions and collisions by each station.  
We obtain \(T_{s}\) and compute its three parts according to Equation (23). In the interval of two continuous successful transmissions by a station, the time for successful transmission by each other station is \(N_{s}T_{s}\), where \(N_{s}\) is the number of successful transmissions by other stations. We assume that all stations are within the communication range of one another, and can fairly share the channel. For a sufficiently long period of time, each station successfully sends data packets with the same probability. Hence, during the interval of two continuous successful transmissions in this station, each other station must have a successful transmission. If \(n\) is the total number of stations, then we have \(N_{s}=n-1\). We subsequently obtain:  
\[D_{s}=T_{s}(n-1) \tag{46}\]  
Let \(N_{c}\) be the count of continuous collisions giving us:  
\[P[N_{c}=i]=(1-P_{S}^{\prime})^{i}P_{S^{\prime}}^{\prime}(i=0,1,\ldots) \tag{47}\]The mean of \(N_{c}\) is  
\[\begin{array}{l}E[N_{c}]=\sum iP\{N_{c}=i\}\\ =\frac{1-P_{S}^{\prime}}{P_{S}}=\frac{1-(1-\tau_{2})^{n-1}-(1-\tau_{1})(n-1)\tau _{2}(1-\tau_{2})^{n-2}}{\tau_{1}(1-\tau_{2})^{n-1}+(1-\tau_{1})(n-1)\tau_{2}(1- \tau_{2})^{n-2}}\end{array} \tag{48}\]  
Considering the overall network, \(E[N_{c}]\) continuous collisions are observed during the period of time between two random continuous successful transmissions. According to the above-mentioned analysis, \(n\) successful transmissions are observed during time \(D\); thus, we obtain:  
\[D_{c}=nE[N_{c}]T_{c}=nT_{c}\frac{1-(1-\tau_{2})^{n-1}-(1-\tau_{1})(n-1)\tau_{2 }(1-\tau_{2})^{n-2}}{\tau_{1}(1-\tau_{2})^{n-1}+(1-\tau_{1})(n-1)\tau_{2}(1- \tau_{2})^{n-2}} \tag{49}\]  
Let \(N_{slot}\) be the count of continuous idle slots in a back-off interval. The probability that \(N_{slot}\) is a random integer can be written as follows:  
\[P[N_{slot}=i]=(P_{I})^{i}(1-P_{I}),i=0,1,\ldots \tag{50}\]  
The mean of \(N_{slot}\) is  
\[E[N_{slot}]=\frac{P_{I}}{1-P_{I}} \tag{51}\]  
A back-off interval can be found before each successful transmission or collision; hence, \(n\) successful transmissions and \(N_{c}\) collisions exist during time \(D\).  
The total time of the idle slot is  
\[\begin{array}{l}T_{slot}=(E[N_{c}]+n)E[N_{slot}]T_{I}\\ =T_{I}\cdot[\frac{(1-\tau_{1})(1-\tau_{2})-(1-\tau_{1})(1-\tau_{2})^{n}+n\tau_ {1}(1-\tau_{1})(1-\tau_{2})^{n}+(n-1)^{2}(1-\tau_{1})^{2}\tau_{2}(1-\tau_{2}) ^{n-1}}{\tau_{1}(1-\tau_{2})-\tau_{1}(1-\tau_{1})(1-\tau_{2})^{n}+(1-\tau_{1}) (n-1)\tau_{2}(1-\tau_{1})^{2}(n-1)\tau_{2}(1-\tau_{2})^{n-1}}]\end{array} \tag{52}\]