# A Network Coding Based Hybrid ARQ Protocol for Underwater Acoustic Sensor Networks
## 5 Performance Evaluation
### Optimum Window Size  
In this section, we present the numerical simulation proof for the window size estimation algorithm proposed in Section 4.2. To verify the validity of the proposed algorithm more clearly, we firstly conduct simulations on a simple statical network model. All nodes in the network are fixed  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Maximum mobility range radius \(R\) & 300 m \\ Sound speed & 1500 m/s \\ Data packet size & 50 Bytes \\ Original data packet number in a block & 100 \\ Data rate (\(R_{s}\)) & 10 kbps \\ Carrier frequency & 10 kHz \\ Synchronous time & \(10/R_{s}\) \\ State transition delay & 1.5 s \\ Transmitting sound level & 123–125 dB \(re1\mu Pa\) @10 kHz \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation parameters.  
Figure 4: An example of six-node distribution.  
in place and each adjacent node pair has the same distance of 866 m. The transmitting sound level is fixed to 124 dB. The total number of nodes is six, Node 0 is the source node and Node 5 is the destination node. All the residual nodes act as relays. The packet error rate can be obtained from Equation (9) as discussed in Section 2.2. Particularly, the packet error rate vector is \(\mathbf{P}_{pac}=[0.2276\ 0.6021\ 0.8810\ 0.9835\ 0.9992]\) for sending packets from node 0 to nodes 1-5. Although some nodes are far apart from each other, they still have the opportunity to receive the encoded packets at a relatively high packet error rate.  
Figure 5a shows the convergence procedure of estimating the optimized sending block window size. To combat the high packet loss rate in an underwater environment, the source Node 0 has to send more encoded packets to ensure high probability of successful decoding at Node 1. Although the number of original data packets is 100, we need to send about 135 encoded packets to ensure that Node 1 can recover the original data packets with high probability. However, for Nodes 1-4, the number of sending encoded packets can be largely reduced with the help of node cooperation. For example, Node 2 can receive encoded packets from both Nodes 0 and 1. Thus, when it is the time slot for Node 2 to receive packets from Node 1, Node 1 can just transmit about 53 encoded packets, and it is sufficient for successful decoding at Node 2. However, it shows that Nodes 2 and 3 need to send about 71 packets, which is higher than Node 1. This is because Node 1 sends much less packets than Node 0, so Nodes 2 and 3 will receive less packets from their upstream nodes. All the estimation window size will become almost constant after sending 50 blocks, which means the proposed algorithm can optimize the window size within a short time. Figure 5b shows the calibration factor \(\Delta n_{i}^{\prime}\) factor at each node. The definition \(\Delta n_{i}^{\prime}\) can be referred to Section 4.2.1 and the change of \(\Delta n_{i}^{\prime}\) can represent the dynamic variation of lost packets. It is shown that, although the packet loss rate is fixed, the  
\(P_{pac}(2,3)\) begins to decrease. Thus, Node 1 needs to send more more packets and Node 2 can reduce the number of sending packets. The estimation becomes almost constant again after the movement ends. From the above simulations, we can conclude that the proposed window size estimation algorithm works well for both statical and mobile nodes. We can trace the changes of packet error rate in the underwater channel and update the optimized window size accordingly.