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r-strip #0 Even r-strips are vertically alligned r-strip #1 r-strip #2 r-strip #3 Odd r-strips are shifted r-strip #4 r-strip #5 r-strip #6
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Figure 9.7. Illustration of the placement algorithm in a plane and a nite size region. Figure is redrawn from reference 32.
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The focus of reference 33 is on forming K-connected WSNs. K-connectivity implies that there are K independent paths among every pair of nodes. For K > 1, the network can tolerate some node and link failures and guarantee certain communication capacity among nodes. The authors study the problem of placing nodes to achieve K-connectivity at network setup time or to repair a disconnected network. They formulate the problem as an optimization model that strives to minimize the number of nodes required to maintain K-connectivity. They show that the problem is NP-hard and propose two approximation algorithms with varying degree of complexity and closeness to optimality. The algorithms are graph-theory-based. The idea is to compute a weighted complete graph on the same set of vertices (nodes) and then nd an approximate minimum-weight K-vertex-connected subgraph g. Finally, missing links (edges) in g are established by deploying the least number of nodes. Again in most WSNs, it is not necessary to achieve K-connectivity among sensors unless the base station changes its location frequently. Network lifetime has been the optimization objective for most of the published communication protocols for WSNs. The positions of nodes signi cantly impact the network lifetime. For example, variations in node density throughout the area can eventually lead to unbalanced traf c load that causes the rapid drain of the energy reserve of some sensors [26]. In addition, uniform node distribution may lead to the depletion of energy of nodes that are close to the base station at a higher rate than other nodes and thus shorten the network lifetime [34]. Some of the published work such as reference 27 which we discussed earlier has focused on prolonging the network lifetime rather than area coverage. The implicit assumption is that there is a suf cient number of nodes or that the sensing range is large enough such that no holes in coverage may result. Chen et al. [35] studied the effect of node density on network lifetime. Considering the one-dimensional placement scenario, the authors derived an analytical formulation
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NODE POSITIONING FOR INCREASED DEPENDABILITY OF WIRELESS SENSOR NETWORKS
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for the network lifetime per unit cost (deployed sensor). They also argued that the network lifetime is not growing proportionally to the increased node population and thus a careful selection of the number of sensors is necessary to balance the cost and lifetime goals. Considering the network to be functional until the rst node dies, an optimization problem was de ned with the objective of identifying the least number of sensors and their positions so that the network stays operational for the longest time. An approximate two-step solution is proposed. In the rst step, the number of sensors is xed and their placement is optimized for maximum network lifetime. They formulate such optimization as a multivariant nonlinear problem and solve it numerically. In the second step, the number of sensors is minimized in order to achieve the highest network lifetime per unit cost. A closed-form solution is analytically derived for the second step. Hou et al. [36] considered a two-tier sensor network architecture where sensors are split into groups; each is led by an aggregation-and-forwarding (AFN) node (Figure 9.8). A sensor sends its report directly to the assigned AFN, which aggregates the data from all sensors in its group. The AFNs and the base station form a second tier network in which an AFN sends the aggregated data report to the base station over a multihop path. The authors argue that AFNs can be very critical to the network operation and that their lifetime should be maximized. Two approaches were suggested to prolong the AFNs lifetime. The rst is to provision more energy to AFNs. The second is to deploy relay nodes (RNs) in order to reduce the communication energy consumed by an AFN in sending the data to the base station. The RN placement and energy provisioning problem was formulated as a mixed-integer nonlinear programming optimization. For a pool of an E energy budget and M relay nodes, the objective of the optimization is to nd the best allocation of the additional energy to
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