Estimation of interference

Given the 152 links as defined above, there are a total of 11476 possible link pairs in our network. We ignore pairs in which the links share at least one endpoint, since such links will always interfere with one another. After removing such pairs, we are left with 9168 link pairs, which are still too many to be tested exhaustively. In this paper, we present results for 75 of these pairs, selected at random. We have also done some experiments with larger groups of link pairs, and have seen similar results.

For each selected link pair, we measured $LIR$ as follows. For each link in the pair, we measured the unicast throughput using 1000 byte UDP packets for 30 seconds. Immediately afterwards, we measured the aggregate throughput of the two links operating together, again using unicast UDP packets for 30 seconds. Using the definition in Equation (1) we calculated the LIR for this pair. Testing links in a pair in quick succession helps mitigate the impact of environmental variations. We repeated the experiment 5 times for each of our 75 link pairs. Thus the total duration of the experiment was just under 10 hours The median LIR value for these 75 link pairs are shown in Figure 2. Note that testing all 9168 pairs would have required more than 1100 hours.

Figure 2: Median LIR of 75 link pairs

First, note that we have several link pairs with intermediate LIR values between 0.5 and 1. In other words, interference is not a binary phenomenon. To compare this data with the binary predictions of $M1$, $M2$ and $M3$ models, we must pick a threshold, $\beta$. If $LIR < \beta$, we deem the links to have interfered in our experiment. If $LIR \geq \beta$, we deem that the links did not interfere.

Of the 75 node pairs, 24 pairs have LIR of 1. Thus, in each of these 24 pairs, the two links do not interfere with each other. Five other link pairs have $LIR$ values between 1 and 0.9. Given the minimal interference, we classify these link pairs as non-interfering as well. Thus, we set $\beta = 0.9$. With this threshold, we have 29 pairs in which links do not interfere, and 46 pairs in which the links do interfere.

We see that the $M1$ model is too pessimistic for our network, since we do have 29 non-interfering link pairs. On the other hand, the $M2$ model is too optimistic. The two links in each pair do not share an endpoint, so according to the $M2$ mode, none of the link pairs should show any interference. Yet, we have 46 link pairs in which the links do interfere.

Table 1: Performance of M3 model

	M3 Prediction
	Interference	No interference
Observed Interference	46	0
Observed No Interference	10	19

The $M3$ model is harder to verify. It is defined in terms of distance between nodes. We found that the predictions made using distance are quite inaccurate in our testbed. In an indoor testbed like ours, the radio signal propagation is also affected by office walls and other obstacles. There is no easy way to incorporate this information in the model. Therefore, we define a variant of the $M3$ model that does not rely on physical distance between nodes. We will say that a pair of links $L_{AB}$ and $L_{CD}$ interfere if there is a 2 hop (or shorter) path from $C$ to $B$, or from $A$ to $D$. In other words, the modified model says that a pair of links will interfere if the sender of one link is within two hops of the other link's receiver. Note that ``hop'' is just another term for a wireless link. This variant of the $M3$ model predicted that 56 of the 75 link pairs will show interference. In our experiments, we observed interference in only 46 of these 56 pairs. The other 10 pairs did not show interference in our experiments. On the other hand, the model predicted no interference for 19 pairs. We indeed did not observe interference in any of these 19 pairs. These numbers are summarized in Table 1. The conclusion is that the model is pessimistic: it errs on the side of predicting interference even when there is none.

It may appear that the model seems pessimistic because we used $\beta = 0.9$ to classify experimental observations, and it is too low a threshold. However, even if we use $\beta = 1$ to classify experimental observations (and hence classify more pairs as interfering), the model still incorrectly predicts interference in 7 pairs that do not see any interference.

The pessimistic nature of the model is probably due to the indoor setting of our testbed. In such an environment, the radio signal degrades much faster than it would in free space, thus limiting the overall interference. We also evaluated a 1-hop variant of the model, which turned out to be optimistic. We believe that it may be possible to modify the 1-hop variant further to provide better predictions. However, there is no guarantee that the predictions of the 1-hop model will be accurate in other environments. Furthermore, even the improved model will provide only binary predictions. In the following section, we present a measurement-based approach which automatically takes into account the impact of environmental factors, and is capable of predicting intermediate values of $LIR$.