IEEE TRANSACTIONS ON AUTOMATIC CONTROL using Lemma 4.1,we have log(R.H.S.of (15)) =(n(s(1-a+aam,m)》 for all sufficiently large n,considering =o(r).Therefore, >Pr({P;is not covered})=e(me-logn-loglogn-E)20e- PEM (16 Fig.5.Coverage Under 1-Dimensional Random Walk Mobility Model. for any fixed 1 and all nN(E,0). Let t(n)=kry(n)and to(n)=kro(n).And we can similarly obtain the lower bound 2)Necessary ESR for Full Coverage of Dense Grid:Let denote the event that the dense grid M is not fully covered in the given time slot r.We present the following proposition P({Pi,P;are not covered}) regarding the necessary condition on ESR. (1-4rr(n)(1-2(1+Sy)kr,(m) Proposition 4.2:In the mobile heterogeneous WSNs with y=1 1-dimensional random walk mobility model,if ro= e-2n∑-i(cwtw(n) (17) logntloglognts(n)and the density of the dense grid M is m =n logn,then and the upper bound lim inf P,(C7)>e-6-e-26. P({Pi,Pj are not covered}) whereξ=limn→+o(n). sira(-a+ro)°"+i-a+ow,m】 Proof:The technique used in this proof is similar to that used in the proof of Proposition 4.1,and we present the main steps here.We begin with the case thatrogno ≤Πrmj(1-m,(m)+Π(1-r,(m)n kn y=1 for a fixed E. e-2n∑-1(cvtv(m) P,(g)≥ >P({P:is not covered}) PEM Combining (17)and (18),we have P≠P P({Pi,P;are not covered}).(14) P≠P P,P,EM P(P P;are not covered}) P,P;EM Based on (13),we can evaluate the first term of on the right m2e-2n∑y-1(m》=e-25 (18) hand of side of (14)and have Thus,using (16)and (18)into (14),we obtain P(P:is not covered}) =Ⅱ1-(rm+m,m》 P,(gr)≥0e-f-e-2 (19) y=1 (15) Taking into account the case that is a function of n,the =Π(1-(1+Sg)kr,)” conclusion still holds. ◆ From Proposition 4.2.we know thatro is where Let we can easily obtain necessary to achieve the full coverage of M. =o(r).Taking the logarithm of the right side of (15)andIEEE TRANSACTIONS ON AUTOMATIC CONTROL 8 Fig. 5. Coverage Under 1-Dimensional Random Walk Mobility Model. 2) Necessary ESR for Full Coverage of Dense Grid: Let Gτ denote the event that the dense grid M is not fully covered in the given time slot τ . We present the following proposition regarding the necessary condition on ESR. Proposition 4.2: In the mobile heterogeneous WSNs with 1-dimensional random walk mobility model, if r = log n+log log n+ξ(n) κn and the density of the dense grid M is m = n log n, then lim inf n→+∞ Pτ (Gτ ) ≥ e−ξ − e−2ξ. where ξ = limn→+∞ ξ(n). Proof: The technique used in this proof is similar to that used in the proof of Proposition 4.1, and we present the main steps here. We begin with the case that r = log n+log log n+ξ κn for a fixed ξ. Pτ (Gτ ) ≥ Pi∈M Pτ ({Pi is not covered}) − P i=Pj Pi,Pj∈M Pτ ({Pi, Pj are not covered}).(14) Based on (13), we can evaluate the first term of on the right hand of side of (14) and have Pτ ({Pi is not covered}) = u y=1  1 − (πr2 y(n) + κry(n)) cyn = u y=1  1 − (1 + ζy)κry(n) cyn . (15) where ζy = πry(n) κ . Let ζ = u y=1 cyζy, we can easily obtain ζ = o(r). Taking the logarithm of the right side of (15) and using Lemma 4.1, we have log(R.H.S. of (15)) = u y=1  cyn  log  1 − (1 + ζy)κry(n) ∼ − u y=1  cynκry(n) = − log n − log log n − ξ for all sufficiently large n, considering ζ = o(r). Therefore, Pi∈M Pτ ({Pi is not covered}) = Θ(me− log n−log log n−ξ) ≥ θe−ξ (16) for any fixed θ < 1 and all n>N(ξ, θ). Let ty(n) = κry(n) and t(n) = κr(n). And we can similarly obtain the lower bound Pτ ({Pi, Pj are not covered}) ≥ u y=1  1 − 4πr2 y(n) (1 − 2(1 + ζy)κry(n))cyn ∼ e−2nu y=1(cyty(n)), (17) and the upper bound Pτ ({Pi, Pj are not covered}) ≤ u y=1 πr2 y(n)  1 − (1 + ζy)κry(n) cyn + u y=1  1 − (1 + ζy)κry(n) cyn ≤ u y=1 πr2 y(n)  1 − κry(n) cyn + u y=1  1 − κry(n) cyn ∼e−2nu y=1(cyty(n)). Combining (17) and (18), we have P i=Pj Pi,Pj∈M Pτ ({Pi, Pj are not covered}) ∼ m2e−2nu y=1(cyty(n)) = e−2ξ. (18) Thus, using (16) and (18) into (14), we obtain Pτ (Gτ ) ≥ θe−ξ − e−2ξ. (19) Taking into account the case that ξ is a function of n, the conclusion still holds. From Proposition 4.2, we know that r ≥ log n+log log n κn is necessary to achieve the full coverage of M.