ENERGY-EFFICIENT ALGORITHMS IN WIRELESS SENSOR NETWORKS

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broadcast because of the argument above. Thus the sink will be reached if the whole network is operational. Lemma 4 [9]. PFR succeeds with probability 1 in sending the information from E to S given the event F . 15.3.5 The Energy Ef ciency of PFR Reference 9 considers the ctitious lattice G of the network area and let the event F hold. There is (at least) one particle inside each square. Now join all nearby particles of each particle to it, thus forming a new graph G which is lattice-shaped, but its elementary boxes may not be orthogonal and may have varied length. When G s squares become smaller and smaller, then G will look like G. Thus, for reasons of analytic tractability, Chatzigiannakis et al. [9] assume that particles form a lattice (see Figure 15.7). They also assume length l = 1 in each square, for normalization purposes. Notice, however, that when l 0, then G G and thus all results in this section hold for any random deployment in the limit. The analysis of the energy ef ciency considers particles that are active but are as far as possible from ES. Thus the approximation is suitable for remote particles. Reference 9 estimates an upper bound on the number of particles in an n n k (i.e., N = n n) lattice. If k is this number, then r = n2 (0 < r 1) is the energy ef ciency ratio of PFR. More speci cally, Chatzigiennakis et al. [9] prove the (very satisfactory) result below. They consider the area around the ES line, whose particles participate in the propagation process. The number of active particles is thus, roughly speaking, captured by the size of this area, which in turn is equal to |ES| times the maximum distance from |ES| (where maximum is over all active particles). This maximum distance is clearly a random variable. To calculate the expectation and variance of this variable, the authors in reference 9 basically upper bound the stochastic process of the distance from ES by a random walk on the line, and subsequently they upper bound this random walk by a well-known stochastic process (i.e., the discouraged arrivals birth and death Markovian process; see, e.g., reference 12). Thus they prove the following:

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E Sensor Particles

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Sink

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Figure 15.7. A lattice sensor network.

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EBP: THE ENERGY BALANCE PROTOCOL

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Theorem 15.3.1 [9]. The energy ef ciency of the PFR protocol is (( n0 )2 ), where n n0 = |ES| and n = N, where N is the number of particles in the network. For n0 = |ES| = o(n), this is o(1). 15.3.6 The Robustness of PFR To prove the following robustness result, the authors in reference 9 consider particles very near to the ES line. Clearly, such particles have large -angles (i.e., > 134 ). Thus, even in the case where some of these particles are not operating, the probability that none of those operating transmits (during the probabilistic phase 2) is very small. Thus, reference 9 proves the following. Lemma 5 [9]. PFR manages to propagate the crucial data across lines parallel to ES, and of constant distance, with xed nonzero probability (not depending on n, |ES|). 15.4 THE ENERGY BALANCE PROBLEM In order to save energy and keep the network functional for as long as possible, various approaches, including hop-by-hop transmission techniques [5, 10, 11], as well as clustering techniques [13] and alternating power-saving modes [14], have been proposed. All such techniques do not explicitly take care of the possible overuse of certain sensors in the network. As an example, note that in hop-by-hop transmissions toward the sink, the sensors lying closer to the sink tend to be utilized exhaustively (since all data pass through them). Thus, these sensors may die out very early, thus resulting in network collapse, although there may be still signi cant amounts of energy in the other sensors of the network. Similarly, in clustering techniques the cluster heads that are located far away with respect to the sink tend to spend a lot of energy. In this chapter, we present two protocols trying to balance energy dissipation among the sensors in the network: (a) the EPB (Energy Balance) protocol, introduced in reference 15, which probabilistically chooses between either propagating data one hop toward the sink or sending directly to the sink. The rst choice is more energy-ef cient, while the latter bypasses the critical (close to the sink) sectors. The appropriate probability for each choice in order to achieve energy balance is calculated in reference 15. (b) VTRP (Variable Transmission Range Protocol), proposed in reference 16, which implicitly contributes to energy balance by appropriately adapting (increasing) the transmission range, thus bypassing critical sensors and avoiding possible obstacles. 15.5 EBP: THE ENERGY BALANCE PROTOCOL 15.5.1 The Model and the Problem We assume that crucial events, which should be reported to a control center, occur in the network area. Furthermore, we assume that these events are happening at random

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