# Improving Route Selections in ZigBee Wireless Sensor Networks
## 7 Performance Evaluation
### Symmetric Topologies
#### 7.1.3 Scenario S3: Two-Hop Scenario  
Consider now the topology of Figure 7 involving routes with more than 1 hop. In this scenario, sensors at node 3 at \((0,-140)\), node 4 at \((-130,0)\), node 5 at \((130,0)\), node 8 at \((-130,-60)\), and node 9 at \((130,60)\) send sensor measurements to the concentrator 0 at \((0,80)\); and node 1 at \((-35,0)\), node 2 at \((35,0)\), node 6 at \((-35,60)\), and node 7 at \((35,-60)\) are available for routing packets. Because of their distance, nodes 4 and 8 are hidden from nodes 5 and 9 and vice-versa; and nodes 4, 5, 8, and 9 are hidden from nodes 0 and 3 and vice-versa. Node 1 is not close enough to be within node 8's communication range; however, it can detect transmissions from node 8. Likewise, node 2 can detect transmissions from  
Figure 6: The median and the 15 th to 85 th percentiles of (**a**): the cumulative route cost of route 3,2,0 \((c_{3,2,0})\), (**b**): the standard deviation of the estimated costs measured at node 3 for the route 3,2,0 \((c_{3,2,0})\), (**c**): the ratio of times that route 3,2,0 was chosen \((r_{3,2,0})\), and (**d**): the number of times that node 3 had to retransmit for every 1000 messages for the symmetric scenario S2. node 9; node 6 can detect transmissions from node 4; and node 7 can detect transmissions from node 5. Consider also the same parameters as described in Section 5. For this scenario, assume that nodes 3, 4, 5, 8, and 9 generate the same average number of packets/s. Because of the symmetry of this scenario, the rates of interfering transmissions in nodes 1, 2, 6, and 7 are the same; and, as explained in Section 7.1.1, \(c_{3,6,1,0}\)=\(c_{3,7,2,0}\) and \(c_{3,6,2,0}\)=\(c_{3,7,1,0}\); however, \(c_{3,7,2,0}\)\(<\)\(c_{3,7,1,0}\). To see this, note first that node 7 is farther from node 1 than it is from node 2. Furthermore, and perhaps more importantly, node 7 is hidden from node 4's transmissions, while it is not hidden from node 5's transmissions, which means a lower probability of successfully transmitting to node 1; i.e., \(p_{7,1}<p_{7,2}\). Same argument follows to justify \(c_{3,6,1,0}<c_{3,6,2,0}\). Since \(c_{3,6,1,0}=c_{3,7,2,0}<c_{3,6,2,0}=c_{3,7,1,0}\), it is desirable to choose either routes 3,6,1,0 or 3,7,2,0. Note that the choice of a route is not done by node 3 alone. As explained in Section 2.2, a node does not select the whole route toward the concentrator; instead, it only selects the next hop node; and the next hop node then chooses its next hop node toward the concentrator. All nodes operate the same next hop selection procedure and we will refer to a procedure as _selecting_ a route as the route that resulted by the distributed operation of the procedure in the various nodes of the network. Figure 8a,b show the average of cumulative cost estimates \(\overline{c}_{3,7,x,0}\) and \(\sigma_{3,7,x,0}\). We use '\(x\)' in \(\overline{c}_{3,7,x,0}\) and \(\sigma_{3,7,x,0}\) because node 3 cannot differentiate a RREQ arriving through the route 3,6,1,0 from a RREQ arriving through the route 3,6,2,0; therefore, node 3 cannot estimate \(c_{3,7,1,0}\) or \(c_{3,7,2,0}\) separately. The figures showing \(\overline{c}_{3,6,x,0}\) and \(\sigma_{3,6,x,0}\) followed similar behavior as shown in Figure 8a,b, and were therefore, omitted. It is possible to see in Figure 8a,b that many of the conclusions reached for the single-hop scenario were also present in this two-hop scenario: the LQI-based procedure had the lowest variation in its cost estimates; however, it was not able to detect traffic increases; all procedures behaved similarly when the traffic load was low; and \(\sigma_{3,7,x,0}^{(URR)}\leq\sigma_{3,7,x,0}^{(LS)}\) as the traffic increased above 2 packets/s. Letting \(r_{3,6,1,0}\) and \(r_{3,7,2,0}\) respectively, denote the ratio of times that the resulting route was 3,6,1,0 and 3,7,2,0; the sum \(r_{3,6,1,0}+r_{3,7,2,0}\) represents the ratio of time that a procedure chose one of the best routes.