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Figure 4.6. The UDI model has two radii: a transmission radius (length 1) and an interference radius (length R 1). In this example, node v is not able to receive a transmission from node u if node x concurrently transmits data to node w even though v is not adjacent to x.
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protocols (e.g., coloring algorithms [15]) have been proposed to reduce interference by ensuring a certain hop distance between two senders. Concretely, it is assumed that only the k-neighborhood of a receiver u can interfere with u. Clearly, this is a stark simpli cation since in a UDG a (k + 1)-neighbor can be close to the receiver (see Figure 4.7). Model 4.3.5 (UDG with Hop Interference (UHI)). Nodes are located at arbitrary positions in R2 . Two nodes are adjacent if and only if their Euclidean distance is at most 1. Two nodes can communicate directly if and only if they are adjacent, and
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Figure 4.7. Example where UHI fails: Nodes v1 and vk+2 are separated by a path of k + 1 hops, but are close (distance 1 + ). Thus, concurrent transmissions of nodes v2 and vk+2 may interfere at v1 in spite of their large hop distance.
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if there is no concurrent sender in the k-hop neighborhood of the receiver (in the UDG). Observe that while the UHI model for every k sometimes overlooks interference terms which the UDI would take into account, the contrary does not hold. Theorem 4.3.1. By choosing R = k, and since a hop has at most length 1, the UDI model does not overlook any interference terms that UHI would have taken into account. The contrary does not hold (cf. Figure 4.7). Like UDI and UHI, also the protocol model (PM) is of type ONE (Model 4.3.3). However, the senders in the PM model adapt their transmission power according to DIST (Model 4.3.2) that is, depending on the distance between sender and receiver. Model 4.3.7 is a variation of the model introduced in reference 11. Model 4.3.6 (Protocol Model (PM)). Let u1 , u2 , ..., uk , be the set of nodes transmitting simultaneously to receivers v1 , v2 , ..., vk , respectively. The transmission of / ui is successfully received by vi if for all j = i, it holds that d(uj , vi ) > d(uj , vj ), where 1 is a given constant. That is, vi must not fall into a guard zone around any sender uj which is a factor (1 + ) larger than uj s transmission range. Many interference models distinguish between senders and receivers assuming that interference arises at senders and occurs at receivers. However, often receivers acknowledge messages and are therefore also senders. If the original messages are short (e.g., control messages), then the sender/receiver distinction may not make sense. By this observation, some models (e.g., reference 16) simply consider the interference of undirected links. Figure 4.8 depicts an example. Model 4.3.7 (Direction). DIR: This class of interference models distinguishes between senders and receivers (interference disks around senders). UNDIR: Interference originates from undirected links (interference pretzels around links).
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Figure 4.8. DIR vs. UNDIR: On the left, only the sender transmits data (interference disks around senders). On the right, there is no distinction between sender and receiver, and hence interference arises from the entire link ( pretzels around links).
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INTERFERENCE ISSUES IN WIRELESS SENSOR NETWORKS
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As in the case of connectivity models, the SINR, the UDI, and the UHI models can be extended with directional antennas and link failures, and hence Models 4.2.14 and 4.2.15 also apply here. Moreover, also the idea of quasi unit disk graphs (cf. Model 4.2.3) could be adopted. For example, the UDI can be quasi ed as follows: If two nodes are closer than a given threshold R1 , concurrent transmissions will always interfere; if the distance is larger than a second threshold R2 , there will be no interference. Finally, if the distance is between R1 and R2 , transmissions may or may not interfere. However, these models are often too complicated to be handled algorithmically. It is sometimes simpler to study general weighted interference graphs instead. That is, similar to connectivity graphs, the interference model is based on graphs; however, the edges are now weighted. Formally, in a weighted interference graph H = G(V, E, w), V represents the set of sensor nodes, E represents the set of edges, and w : E R+ is a function assigning a positive value to each edge. The weight denotes how large the interference between the corresponding nodes actually is. As in the SINR model, a transmission is received correctly if the ratio between received signal power and the amount (either the sum or the maximal interfering signal strength) of interfering traf c is smaller than a certain threshold. Model 4.3.8 (General Weighted Graph (GWG)). A weighted interference graph H is given. A receiver v successfully receives a message from a sender u, if and only if the received signal strength (the weight of the link between u and v in H) divided by the total interference (the sum or the maximum of the weights of the links of concurrently transmitting nodes with a receiver v in H) is above the threshold given by the signal-to-interference-plus-noise ratio. The general weighted graph model is quite pessimistic, because it allows for nonnatural network topologies. Again like in the BIG connectivity model we need a weighted graph model that captures the geometric constraints without making too many simplifying assumptions. Again, one approach is to assume that the nodes form a doubling metric (cf. UBG model of Section 4.2). Model 4.3.9 (Doubling Metric (DM)). The DM model assumes that the nodes form a doubling metric; that is, the set of nodes at a distance (which is now given by the weights of the edges) of at most r from a node u can be covered by a constant number of balls of radius r/2 around other nodes, for any r (cf. Model 4.2.9). Interference can be incorporated in various ways. For example, the amount of interference at a receiver u could depend on u s distance (in the doubling metric space) to the closest concurrently transmitting node (ONE model), or on the number of concurrent senders (SUM model). As a nal remark, note that so far we have only presented binary interference models: A message can be received either correctly or not at all. In practice, however, also the transfer rate at which messages can be transmitted can depend on interference: The larger the signal-to-noise ratio, the larger the available bandwidth. A WLAN 802.11, for example, exploits environments with less interference in order to transmit
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