## 5 Simulation Results
### Chain Networks  
We next examine the throughput and delay performance of the proposed scheme using the chain topology, as shown in Figure 7(a). The distance between two adjacent nodes is set to 20 m. The forward flow (from Node 1 to Node \(N\)) is a CBR session with 1,000-byte packets, while the backward flow (from Node \(N\) to Node 1) is a CBR session with 700-byte packets. The data rates are set to 90 Kb/s, 80 Kb/s, 70 Kb/s, and 60 Kb/s for the 6-, 8-, 10-, and 12-node chain networks, respectively. These data rates are found via extensive simulations, which achieves the maximum throughput the the corresponding chain network. The mean transmission range is equal to 26.9 m, while the mean carrier sensing range is equal to 59.3 m. When the distance between the transmitter and the receiver is 20 m, the mean interference range is computed to be 35.6 m.  
Consider first the case that \(\beta=4\) and \(\sigma_{dB}=0.01\) dB. In this case, the received power is almost the same as the mean value, with low randomness exhibited. Since the mean transmission range if 26.9 m, a transmitter hardly reaches other nodes except for its two one-hop neighbors. For example, in the case that Node \(k+1\) and Node \(k+2\) concurrently transmit to Nodes \(k\) and \(k+3\), respectively, the success probabilities of the two transmissions are \(P_{DATA_{1}}\approx 1\), \(P_{DATA_{2}}\approx 1\), \(P_{ACK_{1}}\approx 1\), and \(P_{ACK_{2}}\approx 1\). And both transmissions are successful with high probability. Figure 7 shows the throughput and delay results for the chain network when \(\sigma_{dB}=0.01\) dB. We observe considerable throughput improvements achieved by the proposed protocol over the IEEE 802.11 MAC. We define the throughput improvement ratio as:  
\[R_{thput}=\frac{(\text{throughput\_proposed})-(\text{throughput\_802.11})}{( \text{throughput\_802.11})}\]  
The proposed scheme achieves 42.32%, 64.83%, 71.82%, and 47.32% throughput improvement ratios for the 6-, 8-, 10-, and 12-node chain networks, respectively. In addition, the average end-to-end delay  
Figure 5: Channel parameter estimation in an eight-node chain network. (a) Estimated channel variance \(\hat{\sigma}_{dB}\) over time for \(\beta\) = 4 and \(\sigma_{dB}\) = 2, 4, and 6. (b) Estimated path loss exponent \(\hat{\beta}\) over time for \(\sigma_{dB}\) = 4 and \(\beta\) = 2, 3, 4.  
is also drastically reduced for all the cases. We define the delay ratio as:  
\[R_{d}=\frac{(\text{delay\_proposed})}{(\text{delay\_802.11})}\]  
From Figure 8(b), we find that the proposed scheme achieves delay ratios of 19.49%, 28.63%, 22.98%, and 25.84% for the 6-, 8-, 10-, and 12-node chain networks, respectively. Most concurrent transmissions are successful in this case. We next consider the case of \(\beta=4\) and \(\sigma_{dB}=4\) dB. Due to the increased \(\sigma_{dB}\), the variation of the received power becomes larger. The one-hop distance may be longer in this case, a phenomenon also observed in the work on connectivity in the presence of log-normal shadowing [29, 30]. In the concurrent  
Figure 6: The three types of network topologies used in the ns-2 simulations. (a) Chain topology. (b) Grid topology. (c) Random topology. Figure 7: Simulation results for chain networks when \(\sigma_{dB}=0.01\) dB. (a) Throughput. (b) Delay. transmission scenario, the free and scheduled transmitters are regarded as the interfering node for each other's target receiver. Thus, the possibility of feasible concurrent transmission decreases. For example, when \(d\) = 20 and \(r\) = 40, the success probabilities are \(P_{DATA_{1}}=P_{DATA_{2}}=P_{ACK_{1}}=P_{ACK_{2}}=0.5376\). This is smaller than that when \(\sigma_{dB}\) = 0.01 dB. Consequently, there are fewer feasible concurrent transmissions when \(\sigma_{dB}\) gets large. Figure 8 shows the simulation results when \(\sigma_{dB}=4\) dB, where the achieved throughput improvement ratios are 12.21%, 17.27%, 25.10%, and 21.02%, respectively.