probability P(D)for a given cluster C;and a given access minimum distance between any point outside T and inside point Oh in timeslot A.Then extend the analysis to na access C;is larger than r.Therefore,we obtain points during k time slots which is exactly P(F). The related positions of Hi and Oh at one timeslot A are P(T)≥1-S()=1-x(r+R)2, (4-4) illustrated in Figure 4.1.As in Figure 4.1(a),we denote the region centered around Hj with radius (r-R)as A.If where S(T)is the area of Qh∈ThA during入，Cj can connect to Oh no matter where the cluster members are.This is because the distance between Let T denote the event that no cluster member in Cj can any point in and Cj is no larger than r.But if connect to any access point during timeslot A.Then,we have the cluster connectivity to Oh cannot be guaranteed because P)=((T). (4-5) the cluster members may appear in the dead zone shown in Figure 4.1(a).Thus,we have P(D)≤1-P() (4-2) Obviously,we have TF.This is because T is ≤1-π(r-R)2 the sufficient condition of And note thatis not the sufficient condition of T.Because even if C is disconnected, each access point might still cover some cluster members in Ocoverage of h⊙Ci Cj.Hence, P()=((r)≥(PT)” (4-6) ≥(-r+) Therefore,with Egn.(4-3)and Egn.(4-6),we can obtain Lemma 1. ■ dead zone 2)Investigation of the Necessary Condition:Lemma 2 will Fig.4.1:Related positions of Hi and Oh at one timeslot.(a) show that m.P(Fj)is bounded from 0. Left figure.(b)Right figure. Lemma 2:If r=V osm+,a+26<0,<a≤1，fr For the upper bound,we consider the eventsF and D. kana any fixed 0<0<1 and sufficiently large n,we have F is the event that cluster Cj is disconnected during one time slot A.Note that for F,Cj can be covered by several m(-+刚) ≥0e-s (47) access points.D is the event that for any access point,a single access point can not cover Cj.Therefore,it is obvious where r is the transmission range and R-(nB). that FD.And we have Proof:Because ofr==(re).we have P(F)=P(∩F) r=w(R).Taking the logarithm of the left hand side and the λ= power series expansion for log(1-z),we have =(e) log(L.H.S.of Eqn.(47)）】 ≤(eD) (4-3) logm kna log (1-(r +R)2) (48) =(eD)) 2 log m-kn (π+R2） +m) ≤-r-), =1 where o(n)is equal to where the fourth equality is due to the independent distribution of access points and the last inequality follows from(4-2). 6(n)= 、(+R2) For the lower bound,an illustration is shown in the right =3 figure of Fig.4.1.Denote the region centered around Hj with (49) radius(r+R)as ThA.Let ThA denote the event that no cluster a+R) ≤ member in C;can connect to the access point h during 3 timeslot入.During入，Qh∈maybe able to connect to Note that the second inequality is due toπ(r+R)2≤，for some.But if生T入，no can reach it because the all sufficiently large n.5 probability P(Dhλ j ) for a given cluster Cj and a given access point Qh in timeslot λ. Then extend the analysis to n α access points during k time slots which is exactly P(Fj ). The related positions of Hj and Qh at one timeslot λ are illustrated in Figure 4.1. As in Figure 4.1(a), we denote the region centered around Hj with radius (r − R) as J hλ j . If Qh ∈ J hλ j during λ, Cj can connect to Qh no matter where the cluster members are. This is because the distance between any point in J hλ j and Cj is no larger than r. But if Qh ∈ J/ hλ j , the cluster connectivity to Qh cannot be guaranteed because the cluster members may appear in the dead zone shown in Figure 4.1(a). Thus, we have P(D hλ j ) ≤ 1 − P(J hλ j ) ≤ 1 − π(r − R) 2 . (4-2) Fig. 4.1: Related positions of Hj and Qh at one timeslot. (a) Left figure. (b) Right figure. For the upper bound, we consider the events F λ j and Dλ j . F λ j is the event that cluster Cj is disconnected during one time slot λ. Note that for F λ j , Cj can be covered by several access points. Dλ j is the event that for any access point, a single access point can not cover Cj . Therefore, it is obvious that F λ j ⊆ Dλ j . And we have P(Fj ) = P( \ k λ=1 F λ j ) = € P(F λ j ) Šk ≤  P(D λ j ) k = € P(D hλ j ) Šn α k ≤  1 − π(r − R) 2 knα , (4-3) where the fourth equality is due to the independent distribution of access points and the last inequality follows from (4-2). For the lower bound, an illustration is shown in the right figure of Fig. 4.1. Denote the region centered around Hj with radius (r+R) as TÜhλ j . Let T hλ j denote the event that no cluster member in Cj can connect to the access point Qh during timeslot λ. During λ, Qh ∈ TÜhλ j maybe able to connect to some ψ λ jκ. But if Qh ∈/ TÜhλ j , no ψ λ jκ can reach it because the minimum distance between any point outside TÜhλ j and inside Cj is larger than r. Therefore, we obtain P(T hλ j ) ≥ 1 − S(TÜhλ j ) = 1 − π(r + R) 2 , (4-4) where S(TÜhλ j ) is the area of TÜhλ j . Let T λ j denote the event that no cluster member in Cj can connect to any access point during timeslot λ. Then, we have P(T λ j ) = € P(T hλ j ) Šn α . (4-5) Obviously, we have T λ j ⊆ Fλ j . This is because T λ j is the sufficient condition of F λ j . And note that F λ j is not the sufficient condition of T λ j . Because even if Cj is disconnected, each access point might still cover some cluster members in Cj . Hence, P(Fj ) =  P(F λ j ) k ≥  P(T λ j ) k ≥  1 − π(r + R) 2 knα . (4-6) Therefore, with Eqn. (4-3) and Eqn. (4-6), we can obtain Lemma 1. 2) Investigation of the Necessary Condition: Lemma 2 will show that m · P(Fj ) is bounded from 0. Lemma 2: If r = Èγ log n+ξ kπnα , α + 2β < 0, γ k < α ≤ 1, for any fixed 0 < θ < 1 and sufficiently large n, we have m  1 − π(r + R) 2 knα ≥ θe−ξ (4-7) where r is the transmission range and R = Θ(n β ). Proof: Because of r = Èγ log n+ξ kπnα = Θ(rc), we have r = ω(R). Taking the logarithm of the left hand side and the power series expansion for log(1 − x), we have log€ L.H.S. of Eqn.(4-7) Š = log m + knα log € 1 − π(r + R) 2 Š = log m − knα X 2 i=1 € π(r + R) 2 Ši i + δ(n)  , (4-8) where δ(n) is equal to δ(n) =X∞ i=3 € π(r + R) 2 Ši i ≤ € π(r + R) 2 Š2 3 . (4-9) Note that the second inequality is due to π(r + R) 2 ≤ 1 2 , for all sufficiently large n