LTP: A HOP-BY-HOP DATA PROPAGATION PROTOCOL

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number of hops depends on the input (the positions of the grain particles). Reference 5, however, compares the ef ciency of protocols to the ideal case. A comparison with the best achievable number of hops in each input case will of course give better ef ciency ratios for protocols. To enable a rst step toward a rigorous analysis of smart dust protocols, reference 5 makes the following simplifying assumption: The search phase always nds a p (of suf ciently high battery) in the semicircle of center the particle p currently possessing the information about the event and radius R, in the direction toward W. Note that this assumption on always nding a particle can be relaxed in the following ways: (a) By repetitions of the search phase until a particle is found. This makes sense if at least one particle exists but was sleeping during the failed searches. (b) By considering, instead of just the semicircle, a cyclic sector de ned by circles of radiuses R R, R and also taking into account the density of the smart cloud. (c) If the protocol during a search phase ultimately fails to nd a particle toward the wall, it may backtrack. Reference 5 also assumes that the position of p is uniform in the arc of angle 2a around the direct line from p vertical to W. Each data transmission (one hop) takes constant time t (so the hops and time ef ciency of our protocols coincide in this case). It is also assumed that each target selection is stochastically independent of the others, in the sense that it is always drawn uniformly randomly in the arc ( , ). The above assumptions may not be very realistic in practice; however, they can be relaxed and in any case allow us to perform a rst effort toward providing some concrete analytical results. Lemma 1 [5]. The expected hops ef ciency of the local target protocol in the a-uniform case is E(Ch ) for large hopt . Also 1 E(Ch ) for 0

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Proof. Due to the protocol, a sequence of points is generated, p0 = p, p1 , p2 , . . . , ph 1 , ph , where ph 1 is a particle within W s range and ph is part of the wall. Let i be the (positive or negative) angle of pi with respect to pi 1 s vertical line to W. It is

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Since the (vertical) progress toward W is then

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From Wald s equation for the expectation of a sum of a random number of independent random variables (see reference 7), then E(h 1) E(cos i ) E(hopt ) = hopt E(h) E(cos i ) Now, i, E(cos i ) =

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E(h) 1 + = E(Ch ) sin hopt sin hopt Assuming large values for hopt (i.e., events happening far away from the wall, which is the most interesting case in practice since the detection and propagation dif culty increases with distance), we have (since for 0 it is 1 sin ) and 2 2 the result follows. 15.2.4 Local Optimization: The Min-two Uniform Targets Protocol (M2TP) Reference 5 further assumes that the search phase always returns two points p , and p , each uniform in ( , ), and that a modi ed protocol M2TP selects the best of the two points, with respect to the local (vertical) progress. This is in fact an optimized version of the Local Target Protocol. In a similar way as in the proof of the previous lemma, the authors prove the following result: Lemma 2 [5]. The expected hops ef ciency of the min-two uniform targets protocol in the a-uniform case is 2 2(1 cos )

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