# Dynamic Cooperative MAC Protocol for Navigation Carrier Ad Hoc Networks: A DiffServ-Based Approach
## 5 Performance Analysis
### Saturated Throughput Analysis  
The OC mode in Scenarios 1 and 2 is within the direct transmission range, and its negotiation is partly based on the CSMA/CA in IEEE 802.11b DCF [6; 28]. Thus, the per-hop throughput can be deduced by [PERSON]'s Equation [29]  
\[\mathcal{S}_{1-hop}=\frac{P_{s}P_{tr}E[p]}{(1-P_{tr})\sigma+P_{tr}P_{s}T_{s}+ P_{tr}(1-P_{s})T_{c}} \tag{12}\]  
where \(P_{tr}\) denotes the probability that at least one node transmitting an RTS in a given time slot; \(P_{s}\) denotes the probability that one node transmitting an RTS in the given time slot; \(E[p]\) is the average packet payload size, \(T_{c}\) is the average time that the channel is sensed busy by each node during a collision. \(T_{s}\) represents the expected transmission time that a node transmits a packet without collision; due to the analysis of Scenarios 1 and 2 in Section 4.2, \(T_{s}\) can be calculated as  
\[T_{s}=\sum_{x\in\{1,2,5,5,11\}}P_{x}T_{dir}(x) \tag{13}\]  
where \(P_{x}\) is the probability for the data rate between \(C_{i}^{(j)}\) and \(C_{i}^{(j+2)}\) adopting \(x\) Mbps,  
\[P_{11}=\frac{d_{11}^{3}}{d_{1}^{3}},P_{5.5}=\frac{(d_{5.5}^{3}-d_{11}^{3})}{d_ {1}^{3}},P_{2}=\frac{(d_{2}^{3}-d_{5.5}^{3})}{d_{1}^{3}},P_{1}=\frac{(d_{1}^{3 }-d_{2}^{3})}{d_{1}^{3}} \tag{14}\]  
In Equation (13), \(T_{dir}(x)\) is the time duration of successful transmission for the packets, including RTS, CTS, DATA, and ACK, with the data rate \(x\) Mbps, and can be calculated as follows  
\[T_{dir}(x)=T_{rts}+T_{cts}+\frac{E[p]}{r_{x}}+T_{ack}+3T_{sifs}+4\sigma+T_{ difs} \tag{15}\]  
where \(\sigma\) is the propagation delay. During collision, each node senses that the channel is busy, and the collision time \(T_{c}\) can be deduced by  
\[T_{c}=T_{rts}+\sigma+T_{difs} \tag{16}\]The saturated throughput in Scenario 3 should take the overhead of cooperation and optimal relay selection into consideration. Then Equation (13) can be rewritten as follows  
\[T_{s}=\sum_{x\in\{1,2,5,5,11\}}P_{x}T_{relay}(x) \tag{17}\]  
where \(T_{relay}(x)\) is the transmission time between the pair (\(C_{i}^{(j)}\), \(C_{i}^{(j+2)}\)) with the transmission rate \(x\) Mbps, \(T_{relay}(11)=2\cdot T_{dir}(11)\), \(T_{relay}(5.5)=2\cdot T_{dir}(5.5)\). \(T_{relay}(2)\) can be calculated from the searching probability for \(C_{i}^{(j+1)}\) and the benefit by the relay node as follows:  
\[\begin{array}{l}T_{relay}(2)=T_{co\_oh}(P^{\prime}(d_{11},d_{11},2)+P^{ \prime}(d_{5,5},d_{11},2)+P^{\prime}(d_{5,5},d_{5,5},2))\\ +2P^{\prime}(d_{11},d_{11},2)E[p]/d_{11}+P^{\prime}{}_{a}(d_{5,5},d_{11},2) \cdot E[p]\Big{(}d_{11}^{-1}+d_{5,5}^{-1}\Big{)}+2P^{\prime}(d_{5,5},d_{5,5},2 )E[p]/d_{5.5}\\ +(1-P^{\prime}(d_{11},d_{11},2)-P^{\prime}(d_{5,5},d_{11},2)-P^{\prime}(d_{5,5},d_{5,5},2))E[p]/d_{2}\end{array} \tag{18}\]  
where \(T_{co\_oh}\) is the transmission time of a control message in optimal relay transmission.  
\[T_{co\_oh}=T_{diffs}+T_{rts}+T_{cts}+T_{ack}+4T_{sifs}+T_{back-off}+T_{sifs} \tag{19}\]  
Moreover, \(T_{relay}(1)\) can be derived similarly, and the saturated throughput in Scenario 3 can be calculated with Equation (12) by substituting Equation (17) into it.  
Through the deductions above, we complete the deduction of the saturated throughputs, which is applicable for the proposed DDC-MAC protocol under the three transmission scenarios concerned, and these results will be used as the basis for the calculation of saturated throughput in the following simulation-based analysis.