# Self-Organized Fast Routing Protocol for Radial Underwater Networks
## 6 Performance Analysis  
We have developed a mathematical model to calculate the convergence delay, that is, how long it would take to create the routing paths, and compared it with the simulated convergence delay. The calculations are carried out with the help of MAPLE(r). The list of the symbols used in the following equations is given in Table 2.  
For the mathematical analysis, we will assumed that the number of nodes in all the strings is the same. First, we will calculate the duration of one timeslot. This is the amount of time reserved for a packet to reach from one node to another. This value depends on two factors: the packet size and the propagation delay. Equation (1) gives the size of the packets in bits. Equations (2) and (3) calculate the packet delay and the propagation delay respectively. Equation (4) calculates the guard time to take into account variations in propagation delay. The guard time is 5% of the total of the packet transmission delay and the propagation delay. Equation (5) adds packet transmission delay, propagation delay and guard time to obtain the duration for one timeslot. \[\text{PS}=37+8\text{ }^{*}\text{ND}, \tag{1}\]  
\[\text{PKD}=\text{PS}/\text{DR}, \tag{2}\]  
\[\text{PROD}=\text{DT}/\text{SPS}, \tag{3}\]  
\[\text{GT}=(\text{PKD}+\text{PROD})\text{ }^{*}0.05, \tag{4}\]  
\[\text{TS}=\text{PKD}+\text{PROD}+\text{GT}, \tag{5}\]  
Next, we developed the equations to calculate how long the gateway takes to find its neighbor nodes plus the time it takes the neighbor nodes to find their horizontal neighbors. Equation (6) calculates the length of the wait period. As mentioned in Section 4.1, the gateway waits one WP before transmitting SGN packets in order to avoid a collision with any of its neighbors, which might be broadcasting a beacon packet (see Equation (7)). Then the gateway sends an SGN packet three times. Equations (8) and (9) give the delay for the SHN and SHN_ACK packets and for the SGNRSP and SGNRSP_ACK packets, respectively  
\[\text{WP}=\text{TS}\text{ }^{*}(\text{TTS}-1), \tag{6}\]  
\[\text{D}_{\text{SGN}}=\text{WP}+3\text{ }^{*}\text{TS}, \tag{7}\]  
\[\text{D}_{\text{SHN}}=6\text{ }^{*}\text{WP}, \tag{8}\]  
Figure 12: Packet Header Format. \[\text{PS}=37+8\text{ }^{*}\text{ND}, \tag{1}\]0  
The delay for the pairs of SHN and SHN_ACK packets, RR and RR_ACK packets from the gateway to the neighbor nodes and RR_RSP/RRRSP_ACK packets is shown in Equation (10). Two WPs are required for packet sending and receiving and, since the packets will be sent three times, total WPs will be 2 * 3 = 6. \[\text{PS}=37+8\text{ }^{*}\text{ND}, \tag{1}\]1  
The delay for the pairs of RR and RR_ACK, CF_RR and CFRR_ACK and CFRR_BACK_ACK packets is shown in Equation (11):  
\[\text{PS}=37+8\text{ }^{*}\text{ND}, \tag{1}\]2  
The process to send the RR and CFRR packets proceeds from both sides of the network to the intermediate node. The number of intermediate nodes is even if the number of strings is even, otherwise there will be an odd number of intermediate nodes. Merging both scenarios, the process is repeated half times the number of strings and the result is rounded up to the next integer for an odd number of strings (Equation (12)). As mentioned in Section 4.2 the RR forwarding process is completed layer by layer and this process is repeated (ND-1) times (Equation (13)). \[\text{PS}=37+8\text{ }^{*}\text{ND}, \tag{1}\]3  
\[\text{PS}=37+8\text{ }^{*}\text{ND}, \tag{1}\]4  
If we add all the delays calculated up to this point, we obtain the total delay for RR packet forwarding from the gateway to all the nodes in the network (see Equation (14)). After the RR packet has been forwarded, the last nodes in the strings start sending the response. Since all the nodes of each layer send an RR_RSP packet at randomly selected timeslots, we simply multiply the delay for one pair of RR_RSP/RRRSP_ACK packets (Equation (15)) by the number of nodes in the strings. Finally, to get the total delay, we add Equations (14) and (15) to get Equation (16). Equation (17) is the expansion of Equation (16).