## 4 Description of the Proposed Topology Construction
### Design Issues of the Proposed Logical Topology
#### 4.5.1 Total Number of Chains in the System and Optimal Number of Nodes in a Chain  
The system can determine, _a priori_, the optimal number of chains (lower-level) for a particular system. This depends on several parameters, such as the positions of the sensor nodes, and the relative costs of computation _versus_ communication. The proposed topology was simulated for a data collection application using a network where a number of sensor nodes were randomly deployed. The value of the radio parameters of the transmitter and the receiver that were used in the simulation are \(E_{tx-elec}=E_{Rx-elec}=E_{elec}=50\) nJ/bit. The transmit amplifier was assumed to be 100 pJ/bit/\(m^{2}\). A computation cost of 5 nJ/bit/message to fuse 2000-bit messages was further assumed. In the experiment, the number of chains in the system was varied gradually in order to observe its impact on energy consumption, and delay. The experiment was repeated varying the total number of sensor nodes (100, 200, 300, 400, and 500). Figure 9 shows how the energy dissipation in the system varies with the number of chains in the system. Note that a zero chain means that no lower-level chain, and thus no higher-level chain, is constructed. In this situation, each sensor node directly transmits its sensed data to the BS. Also note that, 1 chain means there would be no higher-level chain, and 100 chains means there is actually no lower-level chain (because each chain has only one member) and only a single higher-level chain. Therefore, both 1 chain and 100 chains refer to the same system as PEGASIS. Figure 9 shows that for a 100-node network, energy dissipation is minimum when 5 to 10 chains are created. For a 200-node network, energy consumptions is minimum when 10-20 chains are created with 200 nodes in the network. For a network of 300 nodes in it, the energy consumption is minimum when 16 to 25 chains are created. For a 400-node network it is from 22 to 32, and finally for a 500-node network, 27-38 chains consume the least energy. However, a large number of chains would cause more overhead. Thus, for the proposed topology, the number of chain is maintained at 6%-8% of the sensor nodes. Therefore, for a target field of 200 sensor nodes deployed, 12 to 16 chains would be constructed. This calculation almost matches with the calculation provided in [41].  
The optimal number of nodes in a chain, denoted as \(C_{N}\), is the number of nodes that should be included in each chain during the chain construction phase. It can be argued that, if the number of nodes in a chain is fewer than \(C_{N}\), both the required time and energy dissipation increase in the network. On the other hand, if the number of nodes is more than \(C_{N}\), energy dissipation may decrease slightly, but the time requirement increases. Additionally, for the sake of even energy dissipation distribution, the lengths of the chains should be similar. Thus, in the proposed scheme, a similar number of sensor  
Figure 9: Normalised total system energy dissipated _versus_ number chains constructed.  
nodes are included for each chain. Since it is assumed that sensors are deployed randomly in the target field, creating chains of exactly the same number of sensor nodes may not be possible. However, the proposed scheme maintains a similar number of nodes in each chain. Thus for a target field of 100 nodes, the number of chains is between 6 to 8, and each of the chain contains around 17 to 12 sensor nodes respectively (\(C_{N}\) = 12 to 17).