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65536 observations Est., p = 2/3 log(65536) Partition, | | = 1111
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Figure 14.14 Effect of sensor network density (resolution) on boundary estimation. Column 1, noisy set of measurements; column 2, estimated boundary; and column 3, associated partition. (Reproduced by permission of IEEE [121].) In the naive approach, n j = n for all j, and therefore kn. In the hierarchical approach, rst consider the case when there is no boundary. We have already seen that in such cases the tree will be pruned at each stage high probability. Therefore, n = n/4 j and with j 2k n. Now if boundary of length C n is present, then n j n/4 j + C n. This produces a k(C + 2) n. Thus, we see that the hierarchical algorithm results in = O( n). Finally, a performance example is shown in Figure 14.14 [121]. 14.8 OPTIMAL TRANSMISSION RADIUS IN SENSOR NETWORKS In this section we discuss the problem of nding an optimal transmission radius for ooding in sensor networks. On one hand, a large transmission radius implies that fewer retransmissions will be needed to reach the outlying nodes in the network; therefore, the message will be heard by all nodes in less time. On the other hand, a larger transmission radius involves a higher number of neighbors competing to access the medium, and therefore each node has a longer contention delay for packet transmissions. In this section we discuss this tradeoff in CSMA/CA wireless MAC protocols. Even though ooding has some unique advantages it maximizes the probability that all reachable nodes inside a network will receive the packet it has several disadvantages as well. Several works have proposed mechanisms to improve ooding ef ciency. The broadcast storm paper by Ni et al. [131] suggests a way to improve ooding by trading robustness. The authors propose to limit the number of nodes that transmit the ooded packet. The main idea is to have some nodes refrain from forwarding their packet if its transmission will not contribute to a larger coverage. Nevertheless, the basic ooding technique is in wide use for a number of querying techniques for sensor networks (in large part because of its guarantee of maximal robustness), and in this section we focus on analyzing its MAC-layer effects and improving its performance by minimizing the settling time of ooding.
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Other studies have looked at the impact of the transmission radius in wireless networks. In Gupta and Kumar [132] the authors analyzed the critical transmission range to maintain connectivity in wireless networks and present a statistical analysis of the probability of connectivity. On the same line of work, Kleinrock and Silvester [133] analyze the minimum number of neighbors that a node should have to keep the network connected. In Takagi and Kleinrock [134], the authors describe a similar tradeoff for increasing the transmission radius: a shorter range implies fewer collisions and a longer range implies moving a packet further ahead in one hop. However, in that work the authors want to maximize a parameter called the expected one-hop progress in the desired direction, which essentially measures how fast a packet can reach its destination in point-to-point transmissions. All these studies were not analyzing a protocol like ooding, but instead trying to obtain an optimal transmission radius for other metrics such as connectivity, throughput or energy. In Ganesan et al. [135] an experimental testbed of 150 Berkeley motes [136] run ooding as the routing protocol. The study showed empirical relations between the reception and settling times parameters used in this section for different transmission ranges. In this section we discuss an optimal transmission radius. However, in this case the important metric is the amount of time that a ooded packet captures the transmission medium. To accomplish the goal of minimizing the settling time, the tradeoff between reception and contention times is studied including the interaction between the MAC-layer and network-level behavior of an information dissemination scheme in wireless networks. The network model is based on the following assumptions: (1) The MAC protocol is based on a CSMA/CA scheme. (2) All the nodes have the same transmission radius R. (3) The area of the network can be approximated as a square. (4) No mobility is considered. (5) The nodes are deployed in either a grid or uniform topology. In a uniform topology, the physical terrain is divided into a number of cells based on the number of nodes in the network, and each node is placed randomly within each cell. The analytical model is described by the following terms: (1) Reception time (TR ) average time when all the nodes in the network have received the ooded packet. (2) Contention time (TC ) average time between reception and transmission of a packet by all the nodes in the network. (3) Settling time (TS ) average time when all the nodes in the network have transmitted the ooded packet and signals the end of the ooding event. From these de nitions we observe that TS = TR + TC . If the transmission radius of the nodes is not carefully chosen, the ooded packet may take too long to be transmitted by all the nodes in the network, impacting overall network throughput. The more time the channel is captured by a ooding event, the fewer queries can be disseminated, and the less time the channel is available for other packet transmissions. We can state the relation between settling time and throughput Th in sensor networks as T h 1/TS . So, the goal
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