The results from the previous section show that for energy-efficient position-based routing, the optimal links are almost exclusively those with above-threshold PRR, at least in cases where the control packets are of similar size to the data ones.
This fact means that ARB is more energy efficient than PRR×distance but also leads to a strong justification of the UDG model. Given that the optimal links are almost always ones with high enough PRR it is reasonable to argue that from the perspective of the routing protocol there are only two types of links: links with above threshold PRR that are deemed acceptable and may be considered for routing and all other links which are unacceptable and should be ignored.
The acceptable links have effectively perfect reception rates since no ARQ or retransmissions are ever required on them. On the other hand the unacceptable
3.3. THE UDG MODEL AS AN APPROXIMATION OF ARB 91
links may as well have 0% reception rates since they are excluded from the routing algorithm’s consideration. This is exactly the binary link status assumption of the UDG model and so the results from the previous section justify one key element of the model.
However, there is another element of the UDG model which still requires justi-fication, namely the assumption that the cross-over point between perfect links and others is at some fixed distance. This assumption is plainly a simplification of the reality and cannot be true for individual links. Nevertheless, in this section I will argue and then show that the standard UDG model with a fixed transmis-sion radius will serve as a good approximation to the performance of the ARB strategy which would then justify the use of UDG. Caution would still be needed because although it would be appropriate to use UDG in simulations because of its simplicity, the actual routing strategy used in the real network would have to be ARB. That is, the links that a node considers for routing must be determined by ARB rather than UDG in any real network.
The reasoning behind the expectation that UDG would approximate ARB starts from a return to the initial justification for UDG given by Stojmenovic et al.
mentioned above in Section 3.1.13, namely that the expected packet reception rate is dependent on distance only and is closely matched by UDG. Under the ARB strategy, the chosen links must all have a similar PRR value (assuming that the value of q is relatively high) because they must have a PRR at least equal to q. The Gaussian relationship between link PRR and link length means that the links with a given PRR will be normally distributed around a certain length.
With enough links the average link length would converge to a fixed value which could then be taken as the transmission radius of the UDG model.
As an initial test of this reasoning, Monte Carlo simulations of a simple chain topology were conducted in line with the approach of Seada et al. [SZHK04].
The source and destination nodes were placed 1,000m apart with nodes evenly spaced between them. The parameter values were as summarised in Table 3.1 with the exception that q = 1.0 to guarantee packet delivery. The distance between the nodes was varied to examine the effect of density and the average energy consumption of 50 runs was recorded. In the case of the UDG model, the transmission radius was tuned in order to match, as closely as possible, the average hop length of ARB.
0 1 2 3 4
Distance Between Nodes (m)
0 10 20 30 40 50 60 70
Average Energy Consumption
ARB UDG
Figure 3.8: The UDG model applied to a simple chain topology is a close approx-imation to the optimal ARB strategy, although at low densities the two become less similar.
Fig. 3.8 shows the results which confirm that UDG is a close approximation of the ARB strategy. The difference between the two is very low at high densities (<5%) but is higher at low densities (up to 9% difference when the distance be-tween nodes is 4m). The divergence bebe-tween the two is very strongly related to the density, as indicated by the distance between nodes (r = 0.90, p = 0.002). At low densities the two methods produce virtually identical results but at higher densities the two diverge although this is perhaps related to the increase in uncer-tainty in the ARB results. As can be seen from the graph, the confidence interval values increase with an increase in inter-node distance (r = 0.905, p = 0.002) and the UDG average falls well within the interval.
A further set of simulations, similar to those carried out for ARB described above, consider a network of nodes randomly and uniformly deployed. The method is as described previously and the results are shown in Fig. 3.9. The results are similar to those found in the chain topology. The UDG model is a close approximation of the ARB strategy with the difference between them being no more than 9% which falls within the 95% confidence interval of the performance of ARB. Indeed, over the range of densities examined, there is no statistically significant difference in