# Adaptive Packet Coding for Reliable Underwater Acoustic Communications
## 7 Simulation Results and Discussions  
In this section, we analyze the performance of the proposed packet-coding scheme with the SR ARQ scheme. In the SR ARQ scheme, the transmitter sends \(W\) packets in each transmission and waits for the ACK messages. The transmitter retransmits the failed packets and the new packets to form a new group of packets in the next transmission. Unless otherwise specified, the simulation parameters of the system are presented in Table 1. We consider BPSK modulation and the UWA channels as a Markov Chain model. The communication link often exhibits ON/OFF behaviour in UWA communications. Therefore, we could model the channels as a two-state Markov Chain model [22; 24]: \"Bad (0)\" and \"Good (1)\". The BERs under the good and the bad channel conditions are \(e_{0}=1\times 10^{-3}\) and \(e_{1}=5\times 10^{-4}\), respectively. The transition matrix of the channel state is [24]  
\[P_{h}= \begin{bmatrix}P_{00}&P_{01}\\ P_{10}&P_{11}\end{bmatrix}  \tag{58}\]  
The steady-state probabilities are given by \(\pi_{0}=P_{10}/(P_{10}+P_{01})\), whereas \(\pi_{1}=1-\pi_{0}\). The steady-state probabilities could be used as the probabilities of CSI at the initial transmission, which does not have the feedback messages. In addition, we define a memory parameter \(\mu=1-(P_{10}+P_{01})\) to present the time correlation of channel states. Then, the transition matrix of the channel state can be computed by \(\pi_{0}\) and \(\mu\). In the simulation, we set \(\pi_{0}=0.43\) and \(\mu=0.79\) according to the experiment data given in [23]. The PEP \(P_{c}\) depends on the length of packet \(N_{b}\) and the BER \(e\); it can be calculated from the equilibrium distribution of the generalized Pareto renewal process [5]  
\[P_{c}=(\frac{1-\theta}{1-\theta+eN_{b}\theta})^{-1+\frac{1}{\theta}} \tag{59}\]  
where \(\theta\) is the shape parameter of the generalized Pareto distribution. In this paper, we use \(\theta=0.12\). For the J-ARQ scheme, the number of transmitted groups during a round-trip time is given by  
\[N_{g}=\frac{2T_{d}}{WT_{p}+2T_{g}+T_{\text{ACK}}} \tag{60}\]  
We evaluate the performance based on the throughput efficiency, which is given by  
\[\eta=(1-P_{E})\frac{N_{b}WT_{s}}{T_{\text{total}}} \tag{61}\]  
where \(P_{E}\) is the probability of packet loss; \(T_{\text{total}}\) is the total transmission time duration. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Symbol duration \(T_{s}\) & 0.2 ms \\ Transmission distance \(d\) & 5 km \\ The length of a packet \(N_{b}\) & 1000 \\ Guard interval \(T_{g}\) & \(T_{s}N_{b}\) \\ Time duration of ACK \(T_{ACK}\) & \(T_{s}N_{b}\) \\ Discount factor \(\gamma\) & 1.01 \\ The max number of transmissions \(M\) & 5 \\ Penalty term \(C\) & 1000 \\ Block size \(W\) & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. As shown in [5], the performance of the J-ARQ scheme depends on the number of source data. Thus, we first study the impact of the number of source data. Figure 3 illustrates the throughput efficiency as a function of the number of source data packets \(W_{s}\). From Figure 3, we find the SR-ARQ scheme has the lowest throughput efficiency in all schemes. The throughput efficiency performance of conventional J-ARQ increases with the length of source data. If the sender has enough source data, the J-ARQ scheme has the best throughput efficiency. The performance of the RLPC-ARQ scheme and RLPC-J-ARQ is independent of data length. The throughput efficiency of RLPC-J-ARQ is better than RLPC-ARQ since it takes full advantage of the transmission time. When the data length is short, the performance of J-ARQ is worse than that of the RLPC-J-ARQ scheme. In the RLPC-J-ARQ scheme, the transmission duration is adjustable. Thus, it has a better performance when \(W_{s}\) is not big enough. Figure 4 illustrates the throughput efficiency as a function of block size \(W\). In this simulation, \(W_{s}=400\). The throughput efficiency of all schemes increases with the increase of \(W\).