This article has been accepted for inclusion in a future issue of this journal.Content is final as presented,with the exception of pagination IEEE/ACM TRANSACTIONS ON NETWORKING 1 minwes(|EFler-|EFlmin/V5-EFler),then C(Ta(w)(S))> C(Ta(S)).Additionally EFler-EFlmin> dTa-(ww】_2d(Tomin()】 1+△sp △p V5-EFcr √5 E YRxD) 21 der 2der (EFlcr-EFmin) 1 0 5(1+△sp) V5△p Fig.5.Transformation4 o shrinks=[to=['such that←'['≤f]←[≤ △+4p-2 (f r Oin this figure)and ensures that 1'2 g. -2v5△p(1+△sp) Lastly,observe in the worst case that Yi lies in Vi or V2,and Proof:Without loss of generality,let Vi=[-1,0]x there are some Shd C Sshd,C(Shd)>1/8C(Sshd),such that [0,],2=0,1]×[0,1],and(2,多，C)be the probability ShdUES H(w）andw∈Sha,du2>1/4.Therefore,. space4 of interest,where =Vi x V2 and C is the Lebesgue der>l/4.Define0≌（△p+△p-2/8v5Ap(1+△sp),and it follows that measure restricted on2.Given W=(x1,h,x2,2）∈2， define Ta:as To(w)=w'=(1,1,2,3) g=PrfYi is shaded}=C(Sshd)<8C(Shd) ≤8C(Ta(S%a)/ =-x1 头=1+(1-0)(2= (let S=Shd and f =1 in Lemma 12 门一22 2=r2 and because (Ta,=03) 欢=+(1-0)(h+2二m） ≤8C(T)(Shd)/ 1-D2 Intuitively,let E=(1,)and F=(2,4),then Ta linearly ≤茶cSa)=系= 8 卫 shrinks line segment EF to EF with 1,preserving its LetC0=i的/8，and we complete the proof. geometric topology.See Fig.5 for an example. Theorem 7:If the primary network employs an independent Let dw)be the distance from Yi to line segment EF relay protocol and△p>2,△sp≤（△p-2/2),then every and Sshd=w e:Yi is shaded),Strg =w e:secondary link with ranger(m)=o(Rmin(n))has at least on Yi is triggered).Assume Ssha is not empty and consider any average cu fraction of time to be unconstrained,where constant wb∈Sshd and the set H(b)={W∈2：W=Ta(wo),0> C11>0. 0}.Define 0cr and min such that Proof:Intuitively,due to the fact that the triggering(suc- cess)probability p is at least of the same order as the shading d(Tsw】=:0alEF=EFler (failure)probability g,as shown in Theorem 6,then if a sec- 1+△p ondary link is shaded for a significant fraction of time,this in- 2d(Toma (=0minEF=EFlmin. dicates that primary transmissions nearby(Bernoulli trials)are Ap intense and the link will also be triggered for a substantial frac- tion of time with high probability.The formal proof is presented According to Lemma 8,it is clear that cr and Omin uniquely below. exist and0cr>Amim.Moreover,.d∈H(o),d∈Ssha→ Without loss of generality,consider a time interval of unit IEFI>EFler;EFlmin≤IEFI≤EFler→w∈ length and a particular secondary link(Yj,YRx()).We only dis- Strg Therefore,we can introduce a mapping from the set of cuss the case that Yi is shaded by transmissions from some cell line segments with length(EFl,v5)to those with length Vi to V2 for a least some constant fraction of time,otherwise (EFmin,EF).where v5 is an upper bound of Fi.e.,the proof is trivial.This implies that the shading probability gis lower-bounded by g=e(1).and CaAp=e(1),where Ca is the number of flows that choose this route,and Ap =O(1/vn) is the per-node throughput of primary network.Then,from The- (uo)= EFln+(EFlr-VEFl) orem 6 we have the triggering probability p>c1og1 =(1). According to Lemmas 10 and 11,the fraction of candidate EF links that trigger Yi is at least pi pcs/2c9. Let I be the logical indicator function,and define Then,Vw∈Sshd,Th(w)(u）∈Strg,i.e.,we construct a mapping J=Iflow i chooses a link that triggersY.then between Sshd and Strg,and by invoking Lemma 12,the relation J is sum of ii.d.Bernoullian random variables with mean betweenpand g can be established.To that end,we first simply p2>pi.Denote E as expectation,and by applying Chernoff (w).It is obvious on a little thought that VS C Ssbd,if bounds,we get 4With abuse of notation,we use to denote a set and 1 an element instead of order when no confusion is caused. Pr<-Cuh<e-cm/s, (8)This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 10 IEEE/ACM TRANSACTIONS ON NETWORKING Fig. 5. Transformation ￾ shrinks to such that ￾ ￾ ￾ ￾￾ ( ￾  in this figure) and ensures that  . Proof: Without loss of generality, let , and be the probability space4 of interest, where and is the Lebesgue measure restricted on . Given , define as Intuitively, let and , then linearly shrinks line segment to with , preserving its geometric topology. See Fig. 5 for an example. Let be the distance from to line segment and is shaded , is triggered . Assume is not empty and consider any and the set . Define and such that According to Lemma 8, it is clear that and uniquely exist and . Moreover, , ; . Therefore, we can introduce a mapping from the set of line segments with length to those with length , where is an upper bound of , i.e., Then, , , i.e., we construct a mapping between and , and by invoking Lemma 12, the relation between and can be established. To that end, we first simply . It is obvious on a little thought that , if 4With abuse of notation, we use  to denote a set and  an element instead of order when no confusion is caused. , then . Additionally Lastly, observe in the worst case that lies in or , and there are some , such that and . Therefore, . Define , and it follows that is shaded let and in Lemma 12 and because is triggered Let , and we complete the proof. Theorem 7: If the primary network employs an independent relay protocol and , , then every secondary link with range has at least on average fraction of time to be unconstrained, where constant . Proof: Intuitively, due to the fact that the triggering (suc￾cess) probability is at least of the same order as the shading (failure) probability , as shown in Theorem 6, then if a sec￾ondary link is shaded for a significant fraction of time, this in￾dicates that primary transmissions nearby (Bernoulli trials) are intense and the link will also be triggered for a substantial frac￾tion of time with high probability. The formal proof is presented below. Without loss of generality, consider a time interval of unit length and a particular secondary link . We only dis￾cuss the case that is shaded by transmissions from some cell to for a least some constant fraction of time, otherwise the proof is trivial. This implies that the shading probability is lower-bounded by , and , where is the number of flows that choose this route, and is the per-node throughput of primary network. Then, from The￾orem 6 we have the triggering probability . According to Lemmas 10 and 11, the fraction of candidate links that trigger is at least . Let be the logical indicator function, and define flow chooses a link that triggers , then is sum of i.i.d. Bernoullian random variables with mean . Denote as expectation, and by applying Chernoff bounds, we get (8)