QoS-Oriented Distributed Opportunistic Scheduling for Wireless Networks with Hybrid Links

Transcription

1 Globecom Wirele Neworking Sympoium QoS-Oriened Diribued Opporuniic Scheduling for Wirele Nework wih Hybrid Link Wenguang Mao, Shanhan Wu, and Xudong Wang UM-SJU Join Iniue, Shanghai Jiao ong Univeriy, Shanghai, China {maowenguang1987, wuhanhan, wxudong}@ju.edu.cn Abrac In hi paper opporuniic cheduling i udied for a wirele nework wih hybrid link ha have differen characeriic and qualiy of ervice (QoS) requiremen. More pecifically, we conider wo ype of link coexiing in a nework: 1) ecure link for criical meage demanding phyical layer ecuriy; 2) regular link for oher meage. Since he ranmiion rae of a ecure link i uually much lower han ha of a regular link, fair haring of he ranmiion opporuniy beween wo ype of link ha ome drawback: 1) if ecure link generae more packe, he nework hroughpu i low; 2) if regular link generae more packe, QoS of ecure link may no be aified. o effecively chedule ranmiion in a nework wih hybrid link, a QoS-oriened diribued opporuniic cheduling cheme i developed o maximize he overall yem hroughpu under he conrain of QoS requiremen of ecure link. I i derived baed on he opimal opping heory. Performance reul how ha QoS of ecure link can be gracefully guaraneed wihou necearily compromiing he overall hroughpu of hybrid link. I. INRODUCION I i common ha wirele link in he ame nework have heerogeneou characeriic uch a ranmiion rae and QoS requiremen. Such link are called hybrid link in hi paper. One facor leading o link heerogeneiy i differen applicaion-pecific requiremen for differen link. For example, a wirele nework may adop phyical layer ecuriy o enure perfec ecrecy [1] in ome link (called ecure link). Since perfec ecrecy come a he co of degrading channel capaciy [2]-[3], ecure link have much lower ranmiion rae a compared wih oher link (called regular link). Due o ecuriy concern, hee link may alo demand ringen QoS guaranee. Packe ranmiion in a nework wih hybrid link can be conduced in wo differen way: 1) following a random acce medium acce conrol (MAC) proocol; 2) baed on a cheduling cheme. he former approach i imple and eay o implemen, bu lead o low hroughpu. hu, he laer approach i neceary for improving hroughpu. Among exiing cheduling cheme, opporuniic cheduling i conidered a he mo effecive oluion o exploi differen ranmiion rae o improve overall yem hroughpu [4][5]. he key idea of hee cheme are he ame: given a ranmiion he reearch work i uppored by Naional Naural Science Foundaion of China (NSFC) No , he Miniry of Educaion (MOE) Program for New Cenury Excellen alen, and Orienal Scholar Program of Shanghai Municipal Educaion Commiion. he auhor would like o hank hee ponor for heir generou uppor. opporuniy, if a link wih he highe ranmiion rae i eleced, maximum hroughpu can be achieved. However, hee cheme require he knowledge of inananeou channel qualiie of ome/all link in he nework, which i demanding in many communicaion yem. Hence hee cheme are difficul o implemen. A diribued opporuniic cheduling i propoed in [6] which ha meri of boh approache: 1) i i baed on random acce mechanim and hence eay o implemen; 2) i exploi he idea of opporuniic cheduling and hence ignificanly improve he yem hroughpu. However, hi cheme canno guaranee QoS requiremen for a pecific ype of link and a he ame ime maximize he overall hroughpu: 1) nework cenric delay conrain in [6] i applied o all link inead of one pecific ype of link; 2) if individual delay conrain for each link are applied, uercenric oluion i adoped in [6] and he overall hroughpu i no maximized. In order o rea hybrid link eparaely and alo uppor QoS requiremen of a pecific ype of link, a new diribued opporuniic cheduling cheme i propoed in hi paper. I i developed baed on he opimal opping heory and conidering wo ype of link: ecure link and regular link. Compared wih exiing opporuniic cheduling cheme, he new cheduling cheme i diinc wih following feaure: 1) he overall yem hroughpu i maximized under variou QoS conrain for a pecific link ype (e.g., ecure link); 2) i can be implemened a a double-hrehold cheduling policy, i.e., one hrehold for each link ype, and hen a link ake i ranmiion opporuniy baed on hi hrehold. Simulaion are carried ou o evaluae he new opporuniic cheduling cheme. Performance reul how ha QoS of ecure link can be gracefully guaraneed wihou ignificanly compromiing he overall yem hroughpu. he re of he paper i organized a follow. he yem model for our opporuniic cheduling i explained in Secion II. he QoS-oriened opporuniic cheduling cheme i derived in Secion III. Performance reul are preened in Secion IV. he paper i concluded in Secion V. II. SYSEM MODEL In hi paper, we focu on an ad hoc nework where node can hear each oher. Moreover, we conider he following wo ype of link beween communicaion node: 1) ecure link for ranmiing criical meage wih phyical layer ecuriy; 2) regular link for oher meage. We aume ha each /13/$ IEEE 4524

2 Globecom Wirele Neworking Sympoium Fig. 1. An example of he channel conenion proce node ha he capabiliy of loed random acce wih carrier ening and can collec he informaion of rae diribuion and channel conenion probabiliie of oher node. When a node uccefully conend he channel, inead of proceeding o packe ranmiion direcly, i need o deec curren channel qualiy and follow a deciion rule o deermine wheher o ranmi a packe: if curren channel qualiy i low, he node kip he ranmiion opporuniy o avoid ha he wirele medium i occupied by a low-rae ranmiion. he deciion rule i derived baed on he opimal opping heory uch ha he profi from ending a packe i greaer han ha from kipping he opporuniy. A hown in Fig. 1, a node conend a channel lo by lo. Afer ucceeding in conenion, i ue he deciion rule o decide if a packe ranmiion can ar. If he profi doe no juify a new packe ranmiion, he node give up hi ranmiion opporuniy, and hen repea he conenion proce. he conenion proce end once he node decide o end a packe afer a ucceful conenion. Noe ha a node may conend he channel for eiher ype of link (ecure or regular), depending on which packe are o be en in he head of he queue. For he clariy of preening he opporuniic cheduling cheme, everal parameer are defined here. A hown in Fig 1, a ime lo ha a lengh of, and daa ranmiion ime i. here are M node in he nework. For a node m, i probabiliie of conending he channel for ending packe in i ecure link or regular link are denoed a p,m and p r,m, repecively. he probabiliy of ucceful conenion for ecure link of node m, denoed a P,m,igivenby P,m = p,m (1 p,i p r,i ). (1) i m hu, he probabiliy of ucceful conenion for all ecure link, denoed a P, i given by P,m. P r,m and P r are defined in a imilar way. he ranmiion rae on ecure link and regular link of node m follow he diribuion F,m (r) and F r,m (r), repecively. A hown in Fig. 1, if a node uccefully conend he channel a ime N and decide o ranmi a packe, hen we call N a opping ime. Given a opping ime N, N denoe he ime for hi ranmiion round, including conenion period and packe ranmiion ime, and R N i he ranmiion rae of he node. If he ranmiion in hi round i on a ecure link, R i ued o denoe he ranmiion rae. he diribuion of R i given by F (r) = P,m F,m (r). (2) P m Alo, denoe he waiing ime for ecure link unil he ranmiion on one of hee link i finihed a hown in Fig. 1. Evidenly, he elecion of opping ime ha influence on. Moreover, R r, r, and F r (r) are defined for a regular link in a imilar way. III. OPPORUNISIC SCHEDULING UNDER VARIOUS QOS CONSRAINS In hi ecion, we develop a new diribued opporuniic cheduling cheme o maximize overall yem hroughpu under variou QoS conrain for ecure link. he cheme i derived baed on he opimal opping heory, conidering hree cenario: 1) he hroughpu of ecure link ha a minimum requiremen (i.e., hroughpu conrain); 2) he delay of ecure raffic ha a maximum hrehold (i.e., delay conrain); 3) boh delay and hroughpu conrain are pecified. A. Opporuniic Scheduling wih hroughpu Conrain We define ζ 1 a he e of opping ime a follow: ζ 1 {N : N 1,E[ N ],θ α}, where θ denoe he hroughpu of ecure link, and α denoe minimum hroughpu requiremen. he maximizaion of overall hroughpu under he ecure hroughpu conrain can be formulaed a E[R N ] max N ζ1, ubjec o θ = E[R ] α. E[ N ] E[ ] For convenience, we define he opimal opping ime N and opimal hroughpu x a follow N E[R N ] argmax N ζ1, E[ N ] x E[R N ] up N ζ1. E[ N ] Opporuniic cheduling baed on he opimal opping heory i alo udied in [6], bu he oluion herein i no applicable in our cenario for wo reaon. Fir, here exi wo differen ype of link in our problem, which require differen opping policie for each of hem. Alo, he conrain preened in our formula i applied o one ype of link inead of o all link, which i differen from ha in [6]. Hence, a new derivaion i needed. According o heorem 6.1 in [7], he opimizaion problem formulaed previouly i equivalen o max N ζ1 E[R N ] x E[ N ], ubjec o αe[ ] E[R ] 0. Baed on Lagrange dualiy heory [8], he opimizaion problem i convered o max N ζ1 E[R N ] x E[ N ] λ(αe[ ] E[R ]), 4525

3 Globecom Wirele Neworking Sympoium where λ i he Lagrange muliplier. By olving hi problem, he following propoiion can be derived. Propoiion 3.1: he opimal opping rule for ecure link and regular link i a double-hrehold policy. he hrehold for ecure link i φ, and ha for regular link i φ r. If a ecure link uccefully conend he channel, i doe no kip he ranmiion opporuniy only when R φ ;ifa regular link uccefully conend he channel, i ake he ranmiion opporuniy only when R r φ r. he doublehrehold policy can be expreed a N =min{n 1:R φ or R r φ r }, where φ and φ n are defined a { φ = x +λα, φ r = x (3) + λα. he proof can be found in Appendix A. In Propoiion 3.1, he opimal hrehold φ and φ r are expreed in erm of (x,λ). Hence, furher calculaion of φ and φ r require he knowledge of (x,λ). he procedure for deermining (x,λ) i preened a follow. According o he definiion of profi given in Appendix A. he maximum expeced profi L can be expreed a L = P E[max(R + λr x λα, L ) k(x + λα)] + P r E[max(R r + λe[r ] x λα λαe[ ],L ) k(x + λα)], (4) where k i he number of ime lo before he fir ucceful channel conenion. Noe ha he fir expecaion in he righ hand ide of Eq. (4) denoe he maximum expeced profi if he fir ucceful channel conenion i won by a ecure link, while he econd expecaion and for he maximum expeced profi if a regular link ake he fir ucceful channel conenion. Since L i zero a explained in Appendix A, Eq. (4) can be implified a (x + λα) = P (1 + λ)e[(r φ ) + ]+P r E[(R r φ r ) + ], where ( ) + i defined a max{, 0}. In addiion, according o KK condiion, we have λ(e[r ] αe[ ]) = 0. hu, (x,λ) can be deermined by he following equaion { (x +λα) = P (1 + λ)e[(r φ ) + ]+P r E[(R r φ r ) + ], λ(e[r ] αe[ ]) = 0, where E[R ]= φ rdf(r), df(r) φ E[ ]= +Pr(1 Fr(φr)) P (1 F (φ )) +. Noe ha (φ,φ r ) can be eliminaed baed on Eq. (3). Alo, numerical mehod are required o olve above equaion. I i neceary o emphaize ha he hroughpu conrain α i effecive only when i fall ino a pecific range. If α i oo mall, he opimal hrehold derived from previou equaion will be equal o hoe in unconrained cae. In hi cenario, he hroughpu conrain i inacive. If α i oo large, he conrain canno be aified even if kipping all regular ranmiion. o characerize he lower bound and he upper bound for α, we have he following propoiion. Propoiion 3.2: he effecive range for he hroughpu requiremen α i given by θ L α θ U, where θ U i he opimal hroughpu of ecure link when φ r =, and i can be deermined by P rdf θ U θ = U (r) + P (1 F (θ U )). Alo, θ L i he hroughpu of ecure link when hrehold for ecure link and regular link are e o he opimal value φ for he unconrained cae, and i i given by P θ L rdf φ = (r) + P (1 F (φ )) + P r (1 F r (φ )). he proof can be found in Appendix B. B. Opporuniic Scheduling wih Delay Conrain In hi ubecion, an opporuniic cheduling cheme wih delay conrain i udied. Noe ha he delay of ecure link i defined a he ime beween wo ucceive ranmiion on ecure link. We define ζ 2 a he e of opping ime a follow: ζ 2 {N : N 1,E[ N ],σ β}, where σ and for he average delay of ecure link, and β denoe he delay requiremen. hu, we formulae he problem a E[R N ] max N ζ2, ubjec o σ = E[ ] β. E[ N ] A dicued in Secion III-A, he previou opimizaion problem i equivalen o max N ζ2 E[R N ] x E[ N ] μ(e[ ] β), where μ i he Lagrange muliplier. By olving hi problem, a double-hrehold policy can be derived a { φ = x + μ μβ, φ r = x + μ. he derivaion and he calculaion of he opimal hrehold in hi cenario follow imilar framework preened in Secion III-A and Appendix A. In addiion, imilar wih he hroughpu conrain, he delay requiremen β alo ha an effecive range a decribed in he following propoiion. Propoiion 3.3: he delay conrain β ha a lower effecive bound β L and an upper effecive bound β U. β L i he minimal poible average delay for ecure link, and i given by β L = P

4 Globecom Wirele Neworking Sympoium β U i he average delay for ecure raffic when hrehold for ecure link and regular link are e o he opimal value for he unconrained cae, and i can be deermined by β U = + P r(1 F r (φ )) P (1 F (φ +. )) Proof: For lower bound, include he ime period P for a lea one round ucceful channel conenion and packe ranmiion ime. Hence, he minimum achievable delay requiremen i P +. For upper bound, if he delay requiremen β i greaer han β U, hen he opimal oluion for he unconrained cae i locaed in he feaible domain, which mean ha hi opimal oluion i alo he opimal one for he conrained problem. In hi cae, he delay conrain i inacive. C. Opporuniic Scheduling wih hroughpu and Delay Conrain o aify he conrain of delay and hroughpu, we define ζ 3 a ζ 3 {N : N 1,E[ N ],θ α, σ β}. hen, he maximizaion of overall hroughpu under double conrain can be formulaed a E[R N ] max N ζ3, E[ N ] ubjec o θ = E[R ] α and σ = E[ ] β. E[ ] A dicued in Secion III-A, he above opimizaion problem i equivalen o max N ζ3 E[R N ] x E[ N ] λ(αe[ ] E[R ]) μ(e[ ] β), where λ and μ are he Lagrange muliplier. hen an opimal opping rule can be derived a decribed in he following propoiion. Propoiion 3.4: he opimal opping rule for he cheduling under double conrain can be decribed a N =min{n 1:R φ or R r φ r }, where φ and φ n are given by { φ = x +λα+μ μβ, φ r = x + λα + μ. he proof can be found in Appendix C. he calculaion of opimal hrehold require he knowledge of (x,λ,μ), which can be deermined wih following equaion: (x + λα + μ) = P ()E[(R φ ) + ]+P r E[(R r φ r ) + ], (5) φ r (na//hz) α= Fig β=100 imu. calc φ (na//hz) α=0.35, β= Opimal hrehold for ecure link and regular link. and { λ(e[r ] αe[ ]) = 0, (6) μ(e[ ] β) =0. Eq. (5) i derived from he maximum expeced profi equaion a Eq. (4) and Eq. (6) i from KK condiion. Similar wih ingle conrain cenario, here exi an area where boh hroughpu requiremen α and delay requiremen β are effecive. o characerize hi area, we have he following propoiion. Propoiion 3.5: For a given delay requiremen β, he effecive range for α i bounded by [ rdf φ L (r) β(1 F (φ L )), φ H ] rdf (r) β(1 F (φ H )), where { φ L = F 1 (1 +Pr φ H = F 1 (1 he proof can be found in Appendix D. P ), (β ) P ). (β ) IV. PERFORMANCE EVALUAION In hi ecion, he new cheduling cheme i evaluaed by everal experimen. In our imulaion, we conider a nework coniing of five node, and each of hem mainain i own regular link and ecure link o oher node. he ranmiion rae on a link i aumed o be equal o he channel capaciy given by R = log(1 + ρ H 2 ) na//hz, where ρ i normalized SNR for he link and H denoe he random channel gain which follow a complex Gauian diribuion wih variance equal o 1. For any ecure link, i normalized SNR i e o 1; for any regular link, i normalized SNR i given by 5. Alo, he ranmiion ime i equal o 30 lo ime. Moreover, o reflec raffic load of he nework, channel occupancy raio i inroduced and defined a P =1 (1 p,m p r,m ). m o examine he opimum of hrehold derived from equaion in Secion III, we compare he derived opimal hrehold pair wih one obained from enumeraion mehod under hree cenario: 1) wih hroughpu conrain (α =0.35); 2) wih delay conrain (β = 100); 3) wih boh conrain. o obain he opimal hrehold by enumeraion, we ake each elemen 4527

5 Globecom Wirele Neworking Sympoium Overall hroughpu (na//hz) Random acce DOS QDOS wih α=0.35 QDOS wih β=100 hroughpu of ecure link (na//hz) Random acce DOS QDOS wih α=0.35 QDOS wih β=100 Delay of ecure link (lo ime) Random acce DOS QDOS wih α=0.35 QDOS wih β= Channel occupancy raio Channel occupancy raio Channel occupancy raio (a) Overall hroughpu (b) hroughpu of ecure link (c) Delay of ecure link Fig. 3. hroughpu and delay in differen cheme. ABLE I HROUGHPU AND DELAY WIH DOUBLE CONSRAINS P θ oal (na//hz) θ (na//hz) σ (lo ime) in {( x 200 ) x, y 800 and x, y N} a he hrehold pair and run he imulaion for 10 6 lo ime. According o imulaion reul, hrehold pair ha do no guaranee he QoS conrain are removed and among remaining one he pair wih maximum overall hroughpu i eleced. o minimize he influence of randomne in he imulaion, he enumeraion procedure menioned previouly i repeaed by 40 ime for each cenario and he opimal hrehold pair obained in each run i recorded. he comparion reul are provided in Fig. 2. In hi figure, he hrehold pair marked by red cro are calculaed from equaion preened in Secion III, while he pair labeled wih green plu are colleced from imulaion baed on enumeraion mehod. he comparion reul how ha all opimal hrehold pair obained from imulaion are locaed around he derived one, which demonrae he opimum of derived reul baed on equaion in Secion III. Performance reul of our cheduling cheme (QDOS) wih differen QoS conrain are hown in Fig. 3. For comparion, performance of anoher wo acce cheme i provided: 1) random acce cheme, namely he cheme wih boh φ and φ r e o zero; 2) diribued opporuniic cheme (DOS) propoed in [6]. From Fig. 3(a), we know ha he overall yem hroughpu of our cheme i ignificanly higher han ha of random acce cheme. Alo, when he channel occupancy raio i greaer han 0.2, he hroughpu performance lo of our cheme a compared o DOS cheme i wihin 10%. hi reul indicae ha he overall hroughpu of our cheme i no ignificanly compromied. Fig. 3(b) and Fig. 3(c) how he QoS of ecure link in differen cheme. he reul indicae ha here i no QoS guaranee on ecure link in DOS cheme. In conra, our cheme wih he hroughpu conrain can effecively guaranee he hroughpu performance of ecure 200, y link, while our cheme wih he delay conrain uccefully conrol he delay on ecure link o an accepable level. Moreover, noe ha he hroughpu requiremen α = 0.35 i no aified in low channel occupancy raio range when only he delay conrain i impoed in our cheme. Alo, he delay requiremen β = 100 i lighly violaed in high channel occupancy range if our cheme only e he hroughpu conrain. herefore if boh hroughpu QoS and delay QoS are required, our cheduling cheme wih double conrain need o be applied. In able I, he overall hroughpu (θ oal ) and QoS of ecure link for our cheme wih double conrain are ummarized. he reul how ha boh delay (σ ) and hroughpu (θ ) requiremen of ecure link are aified a any channel occupancy raio (P ). Alo, he overall hroughpu i cloe o ha of he DOS cheme hown in Fig. 3(a). V. CONCLUSIONS A QoS-oriened diribued opporuniic cheduling wa developed o balance hroughpu and QoS guaranee. Performance reul howed ha QoS of a pecific link ype could be guaraneed wihou ignificanly compromiing he overall hroughpu of all link. Alhough our reearch conidered a hybrid of ecure link and regular link, he new cheduling cheme i compleely applicable o oher cenario of hybrid link. APPENDIX A PROOF OF PROPOSIION 3.1 For convenience, we define he profi of cheduling cheme a R N + λr x N λα. hu L, defined a max N ζ1 E[R N ] x E[ N ] λ(αe[ ] E[R ]), can be inerpreed a he maximum expeced profi of he yem. Furhermore, according o heorem 6.1 in [7] and KK condiion, he value of L i zero. he opporuniic cheduling cheme follow a baic rule: given a node ha ucceed in conenion, if he profi from aking curren ranmiion opporuniy i greaer han ha from kipping hi opporuniy, he node ar ranmiion. If a ecure link uccefully conend he channel, he profi from aking he ranmiion opporuniy can be quanified a R x λ(α R ) (x + λα) con, 4528

6 Globecom Wirele Neworking Sympoium where con i he conenion period before hi ucceful channel conenion. In addiion, according o ime-invarian characeriic of he yem, he maximum expeced profi from kipping hi opporuniy i given by L (x +λα) con. Hence, if (1 + λ)r x λα L, namely R x +λα, he ecure packe i ranmied. Similarly, if a regular link win he channel conenion, he profi from kipping he ranmiion i L (x + λα) con, while he profi from aking he opporuniy i given by R r x λα( + E[ ]) + λe[r ] (x + λα) con. Baed on KK condiion, λ(e[r ] αe[ ]) = 0. hu if R r x λα L, namely R r x + λα, he regular packe i ranmied. APPENDIX B PROOF OF PROPOSIION 3.2 he hroughpu of ecure link can be expreed a φ M rdf,m(r) 1 F θ =,m(φ ) +. i m P,i(1 F,i(φ)) + i Pr,i(1 Fr,i(φr)) m=1 P,m(1 F.m(φ )) + Previou equaion can be implified a P φ θ = rdf (r) + P (1 F (φ )) + P r (1 F r (φ r )). (7) For lower bound, if he hroughpu requiremen α i le han θ L, hen he opimal hrehold φ (for boh ecure link and regular link) in he unconrained cae i locaed in he feaible domain, which mean ha hi hrehold i alo he opimal oluion for he conrained problem. In hi cae, he hroughpu conrain i inacive. For upper bound, he hroughpu of ecure link can achieve he maximum when φ r approache. In hi cae, here i no hroughpu on regular link and he overall hroughpu i equal o he hroughpu of ecure link. Hence, in hi cenario, x = θ U. hu, we have φ = x + λθ U = θ U. herefore, baed on Eq. (7), he maximum hroughpu of ecure link can be deermined by P rdf θ U θ = U (r) + P (1 F (θ U )). APPENDIX C PROOF OF PROPOSIION 3.4 Similar wih Appendix A, we can define he profi of cheduling cheme and L denoe he maximum expeced profi of he yem. Following imilar framework propoed in Appendix A, we can derive ha if (1 + λ)r x (λα + μ) + μβ L, namely R x +(λα + μ) βμ, he ecure packe i ranmied. Similarly, if a regular link uccefully conend he channel, he packe i ranmied when R n + λe[r ] x (λα + μ)( + E[ ]) + μβ L, namely R r x + λα + μ. Propoiion 3.4 i proved. APPENDIX D PROOF OF PROPOSIION 3.5 he average delay for ecure link i given by β = E[ ]= + P r(1 F r (φ r )) +. P (1 F (φ )) hu we have he following inequaliie P (1 F (φ )) + β + P r P (1 F (φ )) +. I follow ha P (β ) (1 F (φ )) + P r P (β ). Since F i a rae diribuion funcion, i value increae wih φ. hu, i can be hown ha F 1 (1 + P r P (β ) ) φ F 1 (1 P (β ) ). For convenience, he lower bound and he upper bound in above inequaliie are denoed by φ L and φ H, repecively. Alo, baed on E[R ] αe[ ]=0and E[ ] = β, wehave E[R ]=αβ. In addiion, E[R ] i a funcion increaing wih φ, hu i can be hown ha rdf φ L (r) β(1 F (φ L )) α rdf φ H (r) β(1 F (φ H )). hi prove Propoiion 3.5. REFERENCES [1] I. Cizar and J. Korner, Broadca channel wih confidenial meage, IEEE ran. Inf. heory, vol. 24, no. 3, [2] A. D. Wyner, he wire-ap channel, Bell Sy. ech. J., vol. 54, no. 8, pp , [3] R. Liu and W. rappe, Securing Wirele Communicaion a he Phyical Layer, Springer Publihing Company, [4] X. Qin and R. Berry, Exploiing muliuer dieveriy for medium acce conrol in wirele nework, in Proc of IEEE INFOCOM, [5] X. Liu, E.K. Chong, and N.B. Shroff, A framework for opporuniic cheduling in wirele nework, Compuer Nework, vol. 41, no. 4, [6] D. Zheng, W. Ge, and J. Zhang, Diribued opporuniic cheduling for ad-hoc nework wih random acce: an opimal opping approach, IEEE ran. Inf. heory, vol. 55, no. 1, [7]. Ferguon, Opimal Sopping and Applicaion [Online]. Available: hp:// om/sopping/conen.hml [8] Jabir S. Arora, Inroducion o Opimum Deign, Elevier Academic Pre,