NOWADAYS, wireless networks are regulated by fixed

Transcription

1 030 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 5, MAY 203 Scalig Laws of Cogitive Ad Hoc Networks over Geeral Priary Network Models Cheg Wag, Chagju Jiag, Shaojie Tag, ad Xiag-Yag Li, Seior Meber, IEEE Abstract We study the caacity scalig laws for the cogitive etwork that cosists of the riary hybrid etwork (PhN) ad secodary ad hoc etwork (SaN). PhN is further corised of a ad hoc etwork ad a base statio-based (BS-based) etwork. SaN ad PhN are overlaig i the sae deloyet regio, oerate o the sae sectru, but are ideedet with each other i ters of couicatio requireets. The riary users (PUs), i.e., the ad hoc odes i PhN, have the riority to access the sectru. The secodary users (SUs), i.e., the ad hoc odes i SaN, are equied with cogitive radios, ad have the fuctioalities to sese the idle sectru ad obtai the ecessary iforatio of riary odes i PhN. We assue that PhN adots oe out of three classical tyes of strategies, i.e., ure ad hoc strategy, BS-based strategy, ad hybrid strategy. We ai to directly derive ulticast caacity for SaN to uify the uicast ad broadcast caacities uder two basic riciles: ) The throughut for PhN caot be uderied i order sese due to the resece of SaN. 2) The rotocol adoted by PhN does ot alter i the iterest of SaN, ayway. Deedig o which tye of strategy is adoted i PhN, we desig the otial-throughut strategy for SaN. We show that there exists a threshold of the desity of SUs accordig to the desity of PUs beyod which it ca be rove that: ) whe PhN adots the ure ad hoc strategy or hybrid strategy, SaN ca achieve the ulticast caacity of the sae order as it is stad-aloe; 2) whe PhN adots the BS-based strategy, SaN ca asytotically achieve the ulticast caacity of the sae order as if PhN were abset, if soe secific coditios i ters of relatios aog the ubers of SUs, PUs, the destiatios of each ulticast sessio i SaN, ad BSs i PhN hold. Idex Ters Cogitive etworks, hybrid etworks, ulticast caacity, rado etworks, ercolatio theory Ç INTRODUCTION NOWADAYS, wireless etworks are regulated by fixed sectru assiget olicy. The liited available sectru coexists with the iefficiecy i the sectru usage, [], [2]. To coe with this roble, dyaic sectru access with cogitive radio has recetly bee ivestigated, which is a ovel aradig, called cogitive etwork, that iroves the sectru utilizatio by allowig secodary users (SUs) to exloit the existig wireless sectru oortuistically without havig a egative iact o PUs, i.e., licesed users. I this aer, we focus o scalig laws of ulticast caacity for cogitive etworks. We costruct the cogitive etwork as a suerositio of two ideedet etworks, called riary etwork ad secodary etwork, that oerate at the sae tie, sace ad frequecy. The SUs are assued to. C. Wag ad C. Jiag are with the Deartet of Couter Sciece ad Egieerig, Togji Uiversity, ad with the Key Laboratory of Ebedded Syste ad Service Coutig, Miistry of Educatio, Buildig of Electroics ad Iforatio Egieerig, NO. 4800, Caoa Road, Shaghai 20804, Chia. E-ail: 3chegwag@gail.co, cjjiag@togji.edu.c.. S. Tag is with the Deartet of Couter ad Iforatio Sciece, Tele Uiversity, 805 N. Broad Street, Philadelhia, PA 922. E-ail: stag7@iit.edu.. X.-Y. Li is with the Tsighua Natioal Laboratory for Iforatio Sciece ad Techology (TNLIST), the Deartet of Couter Sciece ad Egieerig, Togji Uiversity, ad the Deartet of Couter Sciece, Illiois Istitute of Techology, 0, West 3st Street, Chicago, IL E-ail: xiagyag.li@gail.co. Mauscrit received 2 Ja. 202; revised 20 May 202; acceted July 202; ublished olie 20 July 202. Recoeded for accetace by E. Leoardi. For iforatio o obtaiig rerits of this article, lease sed e-ail to: tds@couter.org, ad referece IEEECS Log Nuber TPDS Digital Object Idetifier o. 0.09/TPDS be equied with cogitive radios ad have the fuctioalities to sese the idle sectru ad obtai the ecessary iforatio of PUs [], [2], [3]. We assue the riary etwork to be a hybrid etwork, deoted by PhN, cosistig of base statios (BSs) ad ad hoc odes (PUs) [4], [5]. We assue the secodary etwork as a ad hoc etwork, deoted by SaN. To atch the reality of sectru cosutio better, we assue that the etwork odel has a Pyraid structure. That is, the uber of PUs, which are licesed to access to the sectru at ay tie, is relatively less tha the uber of SUs, which ca oortuistically access to the sectru. Our odel has three ovel oits relative to the existig works:. Sice ulticast caacity ca be regarded as the geeral result of uicast ad broadcast caacities [6], [7], [8], [9], we directly study the ulticast caacity for cogitive etworks to ehace the geerality of this study. 2. Sice ure ad hoc etworks ad BS-based etworks (static cellular etworks) ca be regarded as the secial case of hybrid etworks i ters of the uber of BSs [5], [0], we cosider the odel where the riary etwork is a hybrid etwork, which further ehaces the geerality of our odel. 3. We use the iroved geeralized hysical odel [], [2] that ca cature the ature of wireless chael better tha other classic iterferece odels, such as rotocol odel, hysical odel [3], ad the geeralized hysical odel [4]. 4. We focus o the odel where PhN is a exteded scalig etwork [4], [5], [6] while SaN is a dese /3/$3.00 ß 203 IEEE Published by the IEEE Couter Society

2 WANG ET AL.: SCALING LAWS OF COGNITIVE AD HOC NETWORKS OVER GENERAL PRIMARY NETWORK MODELS 03 scalig etwork [4], [5], rather tha the odel cosidered i ost existig works where the riary ad secodary etworks are both dese scalig, [7], [8], [9], [20]. The diversity of scalig atters of the riary ad secodary etworks exads the techical challege ad stregthes the theoretical cotributio of this aer. We ited to derive the ulticast caacity for SaN uder two basic riciles: ) The order of throughut for PhN ust ot be uderied by the resece of SaN. 2) The rotocol adoted by PhN will ot alter ayway due to the resece of SaN. These two basic riciles are coicidet with the abstract of ractical techiques of cogitive etworks. We first derive the uer bouds of ulticast caacity for a sigle etwork isoorhic to SaN, called sigle SaN. Obviously, we ca use such uer bouds as those for SaN whatever strategy is adoted by PhN, because PhN ad SaN always have egative ifluece (iterferece) o each other uder the ocooerative couicatio schee as log as they share the sae sectru at the sae tie. To coute such uer bouds, we directly exloit the hoogeeity roerty ad radoess roerty of etwork toology [2]. Our ai work is to desig ulticast strategies for SaN uder two riciles etioed above, by which the ulticast throughut, i.e., the lower bouds of ulticast caacity, for SaN ca be achieved of the otial order atchig the uer bouds. We desig two tyes of ulticast strategies for SaN. I the first tye of strategy, we devise the hierarchical ulticast routig based o the highway syste cosistig of the first-class highways (FHs) ad secod-class highways (SHs) [8], ad we use a hierarchical TDMA schee to schedule those highways. I the other tye, we build the routig based o the highway syste oly corised of SHs, to avoid the bottleeck o the accessig ath ito highways for soe cases [4]. By itegratig these two strategies together, we obtai the achievable ulticast throughut as the lower bouds of ulticast caacity for SaN. Protectig the caacity for PhN fro decreasig i order sese is the recoditio i the desig of ay strategy for SaN. Our solutio is to set a reservatio area (PA) for every ode i PhN. As a iortat characteristic differet fro other related works such as [7], [8], [9], [20], [22], we allow a PA to be dyaic accordig to the state of the corresodig riary ode. Beefittig fro the dyaics of PAs, SUs ca access oortuistically ito the sectru fro both tie doai ad sace doai. While, static PAs used i [7], [8], [9], [20], [22] ake soe SUs ever be served. I our solutio, a ituitive view is that: Whe a lik i PhN is scheduled, the receiver ca receive data at a rate of the sae order as i the sceario where PhN ooolizes the sectru, as log as all active trasitters i SaN are out of a large eough PA of this receiver; siilarly, whe a lik i SaN is scheduled, the receiver ca receive data at the sae rate (i order sese) as that for a sigle SaN, as log as this receiver is out of all PAs of the active trasitters i PhN. Two techical challeges i our desig are listed as follows: What is the otial size of PA with resect to the caacity for both SaN ad PhN? As discussed above, the larger PAs are better for rotectig the throughut for PhN. Meawhile, too large PAs will result i a decrease i throughut for SaN. I other words, there is a tradeoff betwee the throughuts for PhN ad SaN i ters of the size of PAs. Furtherore, it is easy to uderstad that the desig of ulticast strategies for SaN deeds o the secific strategy adoted by PhN. As a hybrid etwork, PhN could geerally adot three broad categories of ulticast strategies, accordig to [4], [5], [0], [23]. The first oe is the classical BS-based strategy uder which couicatios betwee ay users are relayed by soe secific BSs. The secod oe is the ure ad hoc strategy, i.e., the ultiho schee without ay BSsuorted. The third oe is the hybrid strategy, i.e., the ultiho schee with BS-suorted. Accordig to these three strategies adoted by PhN, we defie the aroriate PAs for each PU ad BS, ad call the A-Tye PA ad B- Tye PA, resectively. Secifically, uder ure ad hoc strategy, the B-Tye PAs are ever active; uder BS-based strategy, the B-Tye PAs are always active; ad uder hybrid strategy, both A-Tye PAs ad B-Tye PAs ight be active i a certai tie. How to build the highways, icludig FHs ad SHs? Differet fro the traditioal highways i [8], [4], [6], the costructio of highways i SaN is ore colicated because it is ivolved with the blockig of soe active PAs. For FHs, we desig a detourig schee uder which every FH detours the PAs, ad we ca rove that the roduced FHs have the large eough desity ad caacity to suort the relay of data i SaN. For SHs, we desig a hierarchical TDMA schedulig schee by which sufficiet aout of SHs ca be scheduled i a costat schedulig eriod, ad all SUs have the oortuity to be served via accessig to the SHs, excet whe PhN adots BS-based strategy. As the fial result, cobiig the uer bouds ad lower bouds, we show that: ) Whe PhN adots the ure ad hoc strategy or hybrid strategy, the er-sessio ulticast caacity (PMC) for SaN is of order ðffiffiffiffiffiffiffi d Þ whe! d ¼ O ðlog Þ 3 ; ad is of order ð Þ whe d ¼ ð log Þ, where is the total uber of SUs ad d is the uber of destiatios of each ulticast sessio i SaN. 2) Whe PhN adots the BSbased strategy, a ifiitesial fractio of SUs caot be served. The PMC for SaN is asytotically of the sae order as i Case. The rest of the aer is orgaized as follows: I Sectio 2, we itroduce the syste odel. I Sectio 3, we reset ai results. We roose the uer bouds of ulticast caacity for the secodary etwork i Sectio 4, ad derive the achievable throughut as the lower bouds of ulticast caacity by desigig the secific ulticast strategies i Sectio 5. I Sectio 6, we review the related works. I Sectio 7, we draw soe coclusios ad future ersective. NOTATIONS: I the aer, we adot the followig otatios:. x!deotes that variable x takes value to ifiity.. For a discrete set U, juj reresets its cardiality.. For a cotiuous regio R, let krk deote its area.

3 032 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 5, MAY 203. For a 2-diesio lie seget L ¼ uv, jlj reresets its euclidea legth. For a tree T, deote its total euclidea edge legths by kt k.. For evet E, deote the robability of E by PrðEÞ.. To silify the descritio, let the exressio ðþ : ½ 0 ðþ; ðþš rereset that ðþ ¼ð 0 ðþþ ad ðþ ¼Oð ðþþ. 2 SYSTEM MODEL 2. Network Deloyet The etwork odel has a two-layer structure over a square ffiffiffi regio RðÞ ¼½0; Š 2. The first layer is the PhN cosistig of ðþ riary users (PUs, riary ad hoc odes) ad bðþ BSs. I PhN, PUs are laced accordig to a Poisso oit rocess (...) of uit itesity over regio R; the regio R ffiffiffiffiffiffiffiffiffiffiffiffiffiffi is artitioed ito bðþ square subregios of side legth =bðþ; oe BS is located at the ceter of each subregio. Assue that BSs are coected via the highbadwidth wired liks that are certaily ot the bottleecks throughout the whole routig. The secod layer is the secodary ad hoc etwork (SaN) cosistig of ðþ secodary users (SUs, secodary ad hoc odes). I SaN, SUs are distributed accordig to a... of itesity over the regio R. We radoly choose s (or s ) odes fro all PUs (or SUs) as the sources of ulticast sessios i PhN (or SaN), ad for each PU v (or SU v s ), ick uiforly at rado d PUs (or d SUs) as the destiatios. Fro Chebychev s iequality (Lea B.3 i Aedix B, which ca be foud o the Couter Society Digital Library at htt://doi.ieeecoutersociety.org/0.09/tpds ), we ca assue that the ubers of PUs ad SUs are ad, resectively, as i [4], [24], which does ot chage our results i order sese. The followig are our basic assutios that are coicidet with the abstract of ractical techiques of cogitive etworks. Assutio. PhN oerates as if SaN were abset. That is, PhN does ot alter its rotocol due to SaN ayway. Assutio 2. SaN kows the locatios of odes i PhN ad the rotocols adoted by PhN. 2.2 Network Scalig Model I the research of scalig laws of etwork caacity, i ters of the scalig ethod of etwork, there are two tyical odels: the exteded etworks, i which the total area is fixed ad the desity of odes icreases, ad dese etworks, i which the desity of odes is fixed ad the total area icreases [4], [5], [25]. Ideed, the exteded etworks ad dese etworks are the reresetative cases of exteded scalig odel ad dese scalig odel, resectively, [], [2], [26]. To deterie which tye of etwork scalig odel SaN belogs to, we dig out the ai differece betwee these two scalig odels. Due to liited sace, we ove the detailed aalysis to Aedix B.2, available i the olie suleetal aterial. Accordig to the settig i Sectio 2., PhN obviously belogs to the exteded ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi etwork odel; the critical side legth of SaN is of order ð log Þ. Fro Defiitio B.5 i Aedix B, available i the olie suleetal aterial, whe ¼ oð= log Þ, SaN is dese scalig, which is the case cosidered i this aer. This case ca be forulated ito the followig assutio. Assutio 3. PhN ad SaN are overlaed ito a layered etwork with a Pyraid structure. Secifically, ¼ oð log Þ. The case where SaN is also exteded scalig ay be treated as a future work. 2.3 Couicatio Model We adot a iroved geeralized hysical odel [], [2], which catures the ature of wireless chaels better tha the rotocol odel ad hysical odel [3], ad ca avoid the athological throughut degradatio for dese scalig etworks that haes uder the geeralized hysical odel. Let VðÞ deote the set of trasitters i tie slot. The, durig ay tie slot : v i 2VðÞ, v i ad v j ca couicate via a direct lik, over a chael with badwidth B, of rate Rðv i ;v j ; Þ ¼i R 0 ;Blog þ Sðv i;v j ; Þ ; ðþ N 0 þ Iðv i ;v j ; Þ where R 0 > 0 is a redefied costat, the costat N 0 > 0 is the abiet oise, Sðv i ;v j ; Þ is the stregth of the sigal iitiated by v i at the receiver v j, ad Iðv i ;v j ; Þ is the su iterferece o v j roduced by all odes belogig to the set VðÞ fv i g. The wireless roagatio chael tyically icludes ath loss with distace, shadowig ad fadig effects. I this aer, we assue that the chael gai deeds oly o the distace betwee a trasitter ad a receiver, igore shadowig ad fadig, ad defie the chael ower gai fuctio as ðv i ;v j Þ¼d ij, where d ij ¼ dðv i ;v j Þ¼kv i v j k is the euclidea distace betwee two odes v i ad v j, >2 deotes the ower atteuatio exoet, [], [2]. Based o this, Sðv i ;v j ; Þ ad Iðv i ;v j ; Þ are defied as: Sðv i ;v j ; Þ ¼P ðv i ; Þ ðv i ;v j Þ;Iðv i ;v j ; Þ X ¼ Pðv k ; Þ ðv k ;v j Þ; v k 2VðÞ fv i g where P ðv l ; Þ deotes the trasittig ower of v l 2VðÞ i tie slot. For o itercouicatio occurs betwee the two etworks, we have: For a lik v i! v j i PhN, deote the su of iterferece o v j roduced by all odes i tie slot by I ðv i ;v j ; Þ, the it holds that I ðv i ;v j ; Þ ¼I ðv i ;v j ; ÞþI s ðv i ;v j ; Þ; where I ðv i ;v j ; Þ, or I s ðv i ;v j ; Þ, deotes the su of iterferece o v j roduced by all odes i VðÞ\V fv i g,orivðþ\v s, where V ad V s deote all odes i PhN or SaN, resectively. For a lik v i! v j i SaN, deote the su of iterferece o v j roduced by all odes i tie slot by I s ðv i ;v j ; Þ, the it holds that I s ðv i ;v j ; Þ ¼I s ðv i ;v j ; ÞþI ss ðv i ;v j ; Þ; where I s ðv i ;v j ; Þ, or I ss ðv i ;v j ; Þ, deotes the su of iterferece o v j roduced by all trasitters i VðÞ\V, or i VðÞ\V s fv i g. We assue that all PUs ad BSs trasit with the costat wireless trasissio ower P. This settig is the sae as i [4], [5]. Note that the work ca be further exteded ito the case that BSs with wireless trasissio ower P ðþ, where ðþ ¼ðÞ. For SaN, we assue ð2þ ð3þ

4 WANG ET AL.: SCALING LAWS OF COGNITIVE AD HOC NETWORKS OVER GENERAL PRIMARY NETWORK MODELS 033 that each SU, say v i, i tie slot, trasits with a secific trasissio ower Pðv i ;Þ2ð0;P 0 Š, if it is scheduled i, where P 0 > 0 is the axiu of the trasissio ower. 2.4 Caacity Defiitio Sice the riority of the licesed users, i.e., the PUs, ust be guarateed, there ight be soe olicesed users, i.e., SUs, that caot be served. For exale, whe PhN adots the BS-based strategy as reseted i [5], all BSs oerate siultaeously all the tie. Therefore, i the eriheral of BSs, there will be soe SUs that caot access ito the sectru fro either tie doai or sace doai. Sice we assue that PhN ad SaN oerate o the sae sectru, there is o idle frequecy via which SaN ca access ito the sectru. Hece, we eed to geeralize the foral defiitio of ulticast caacity roosed i [6]. Cosequetly, we defie asytotic ulticast caacity that is siilarly defied i [9]. Please see the defiitios of asytotic aggregated ulticast caacity (Asy-AMC) ad asytotic er-sessio ulticast caacity (Asy-PMC) i Aedix B., available i the olie suleetal aterial. We adot the sae ethod of costructig ulticast sessios as i [9], ad assue that all odes (users) i SaN act as the sources, i.e., s ¼, where is the total uber of odes ad s is the uber of sources. 3 MAIN RESULTS Now, we reset the uer bouds ad lower bouds of ulticast caacity for the SaN; fially, cobiig the lower bouds ad uer bouds, we obtai the ulticast caacity. 3. Uer Bouds of Multicast Caacity for SaN We first derive the uer bouds of ulticast caacity for SaN as if the PhN were abset. Straightforwardly, such results are also the uer bouds for SaN whe PhN works. Theore. The PMC for SaN is at ost of order 8 O ffiffiffiffiffiffiffiffiffiffi d >< P ¼ O d log O >: " # whe d : ; ðlog Þ 2 " # whe d : ðlog Þ 2 ; log whe d : log ; : The aggregated ulticast caacity for SaN is at ost of order P. Here, d deotes the uber of destiatios of each ulticast sessio i SaN. The result i Theore always holds regardless of what strategy PhN adots. 3.2 Lower Bouds of Multicast Caacity for SaN We derive the lower bouds of ulticast caacity by desigig the strategies for SaN corresodig to three classical tyes of strategies adoted i PhN, [4], [5], [0] Whe PhN Adots Pure Ad hoc Strategy I this case, all SUs ivolved i all ulticast sessios ca be served. Theore 2. The achievable PMT for SaN is of order 8 " # ffiffiffiffiffiffiffiffiffiffi whe d : ; d ðlog Þ 3! " # whe P ¼ d ðlog Þ 3 d : 2 ðlog Þ 3 ; >< ðlog Þ 2 " # ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi whe d : d log ðlog Þ 2 ; log >: whe d : log ; : ð4þ The achievable AMT for SAN is of order ð P Þ Whe PhN Adots BS-Based Strategy I this case, soe SUs are covered by the B-Tye PAs that are always active, the they caot be served. Uder our strategy for SaN, we ca esure that there are at least s ðþ ulticast sessios of SaN whose d ðþ d destiatios ca be served, where s ðþ!; d ðþ!, as!. Theore 3. Uder two cases, i.e., ) d ¼ ðlog Þ, or2) d ¼ Oðlog Þ ad bðþ¼oð d log Þ, the asy-achievable PMT for SaN is of order P, where P is defied i (4). The asyachievable AMT for SaN is of order P Whe PhN Adots Hybrid Strategy I this case, we set SaN to be idle whe the dowliks ad uliks ivolved with the BSs are scheduled i PhN, ad we schedule SaN i the other hases. Uder this strategy, we get the throughut for SaN of the sae order as i Theore Multicast Caacity for SaN Cobiig the uer boud o ulticast caacity for SaN, described i Theore, ad the lower boud i Sectio 3.2, we ca obtai Theore 4. Whe PhN adots the ure ad hoc strategy or the hybrid strategy, the PMC for SaN is of order 8 >< ffiffiffiffiffiffiffiffiffiffi C P ¼ d >: " # whe d : ; ðlog Þ 3 whe d : log ; : The aggregated ulticast caacity for SaN is of order C P. Whe PhN adots the BS-based strategy, Asy-PMC ad Asy-AMC for SaN are of order C P ad C P, resectively. Fro Theore 4, we kow that there still exists a ga betwee the uer bouds ad lower bouds for the case that d : ½=ðlog Þ 3 ;=log Š. How to close the ga ay be left for future work. 4 UPPER BOUNDS FOR SAN I this sectio, we coute the uer bouds o ulticast caacity for SaN whe PhN is abset. Ituitively, such uer bouds are ossibly too loose. However, i Sectio 5, ð5þ

5 034 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 5, MAY 203 we show that such bouds ca be achieved ideed. Thus, we focus o the sigle SaN, where odes are distributed ffiffiffi ito the square regio ½0; Š 2 accordig to a... of desity with ¼ oð log Þ, ad we begi to rove Theore. First, we have Lea. The PMC for SaN is at ost of order O ax ffiffiffiffiffiffiffiffiffiffiffi ; log : d Proof. We ca use the hoogeeity roerty of the etwork to rove this result. A useful aalysis tool, called trasissio area [2], is exloited. Please see the detailed roof i Aedix C., available i the olie suleetal aterial. tu Lea 2. The PMC for SaN is at ost of order 8 O whe d ¼ O >< d log O >: whe log d ¼ : log Proof. This result ca be roved accordig to the radoess ositios of the odes i the etwork. Radoess roerty roduces soe relatively isolated clusters of odes. These clusters ca act as a bottleeck o the ulticast caacity. Please see the detailed roof i Aedix C.2, available i the olie suleetal aterial. tu Cobiig Lea ad Lea 2, we obtai Theore. 5 LOWER BOUNDS FOR SAN Geerally, the lower bouds of the caacity ca be obtaied by desigig the secific ulticast strategy. Deote a class of the ulticast strategies by S, cosistig of routig schee S r ad trasissio schedulig S t. The routig schee S r ight have a hierarchical structure cosistig of & hases that corresod to subroutig schees S r ; S r2 ;...; S r&, accordigly, the trasissio schedulig S t cosists of & hases, i.e., S t ; S t 2 ;...; S t &, where & is a costat ad it eas that the routig schee S r is ohierarchical whe & ¼. 5. Overview of Multicast Strategy Due to the overwhelig riority to access the sectru, PhN oerates as if SaN were abset. Deote the rotocol for PhN by S cosistig of routig schee S r ad trasissio schedulig S t. We will desig the ulticast strategy for SaN accordig to the secific strategy adoted by PhN. Assue that every subhase of S t oerates uder a ideedet TDMA schee. Deote the schedulig eriods of those TDMA schees by Kj 2, where K j 3 ad j ¼ ; 2;...;&. We first costruct the secific PAs for each PU ad each BS i PhN, ad call the A-Tye PA ad B-Tye PA, resectively. Please see the illustratio i Figs. a ad b. The, at slot ð j ;jþ, j &, ad j Kj 2, which reresets the j th tie slot i a schedulig eriod i hase j, we set the status of PAs of the odes scheduled i ð j ;jþ Fig.. PhN cosists of PU layer ad BS layer; SaN has oly oe layer, i.e., SU layer. (a) The black sall square is the source of a give ulticast sessio. The bigger shaded squares are A-Tye PAs. (b) The sall black hexagos are the BSs that are laced i the ceter ositios of subregios of area bðþ. The shaded squares aroud BSs are B-Tye PAs. (c) Dashed lies deote the EST of a give ulticast sessio. ffiffiffi as active. Thus, i ay tie slot, the regio RðÞ ¼½0; Š 2 is artitioed ito two regios: the occuied regiooðþ, which is a regio covered by all active PAs i tie slot, ad the vacat regiovðþ, which is the coleet of regio OðÞ. Accordigly, we deote the set of the SUs covered by the occuied regio OðÞ or surrouded by the active PAs i tie slot, as the set PðÞ. Fially, for a give ulticast sessio M S;i with the source ode v S;i, whe v S;i 2 T ð j ;jþ PðÞ, the ulticast sessio M S;i will be igored; otherwise, by usig the algorith i [6], we costruct the euclidea saig tree (EST), deoted by ESTð~U S;i Þ, based o the set ~U S;i, where U S;i is the saig set ad ~U S;i ¼U S;i T ð j;jþ PðÞ. The, like the ulticast routig desiged i [6], our routig for SaN is guided by the saig tree ESTð~U S;i Þ. The couicatio of each lik i ESTð~U S;i Þ is routed via the highway syste siilar to that i [8], if alicable. However, ituitively, the routig aths ight be broke by the active PAs i soe (or eve all) tie slots. Thus, how to deal with such itractability? Is it ossible that the otial throughut for SaN ca be achieved? Here, the called otial order of throughut is the uer bouds of ulticast caacity i the sigle etwork isoorhic to SaN. The, give a secific rotocol i PhN, there are three questios i the desig of ulticast strategies for SaN. Questio. How to costruct ad schedule the FHs ad SHs such that, i ay tie slot whe SaN is scheduled, o lik alog the highways coes across the active PAs? Questio 2. How large is the desity of the highway syste i SaN, icludig the FHs ad SHs, if exists? Questio 3. How to esure our ulticast strategy to serve the SUs (or ulticast sessios) as uch as ossible? Obviously, the status of PAs ad the ethod of costructig the highway syste i SaN are deteried by the strategy adoted by PhN. Thus, all of these three questios should be aswered deedig o the rotocol of PhN. Accordig to the existig works [4], [5], whe the TDMA trasissio schedulig schee is adoted i PhN, the strategy for hybrid etwork ca be classified ito three tyes, i.e., ure ad hoc strategy, BS-based strategy, ad hybrid strategy. Next, we itroduce cocisely these strategies, ad aswer the three questios above accordig to the secific rotocol of PhN. 5.2 Whe PhN Adots Pure Ad Hoc Strategy I PhN, uder the ure ad hoc strategy, sice o BS is used, all B-Tye PAs are always iactive. For a A-Tye PA, its

6 WANG ET AL.: SCALING LAWS OF COGNITIVE AD HOC NETWORKS OVER GENERAL PRIMARY NETWORK MODELS 035 ffiffiffi Fig. 2. The cells are of side legth c ¼ c ffiffiffi. The slab is of side legth l ¼ðlog h h Þ 2 c. The shaded regios are the A-Tye PAs. The sall square odes at the ceter of A-Tye PAs rereset the PUs, ad the sall circle odes rereset the SUs. Those oshaded cells cotaiig at least oe SU are called orotected oe. status (active or iactive) is deteried by the routig ad trasissio schedulig adoted by PhN. To achieve the otial order of throughut for PhN, we assue that the ulticast strategy i [8] is adoted i PhN. The strategy is divided ito two hases. Deote the routig schee i the first hase i PhN by S r ad deote the trasissio schedulig i PhN as S t. I this hase, the strategy is desiged based o the schee lattice (Defiitio B.3 i Aedix B, available i the olie suleetal aterial) ILð ;c; 4Þ, where c>0 is a costat defied i [8]. The routig is costructed based o the FHs cosistig of the short liks of costat legth; ad those short liks are scheduled by a TDMA schee. Assue that the costat schedulig eriod is K 2 ([8, K ¼ 3]). Deote the routig schee i the secod hase i PhN by S r2 ad deote the trasissio schedulig as S t2.i this hase, the strategy ffiffiffiffiffiffiffiffiffiffi is desiged based o the schee lattice ILð ; log ; 0Þ, where is a costat defied i [8, Lea 4] ad ffiffiffiffiffiffiffiffiffiffi is a adjustig costat to esure the value of =ð log Þ to be a iteger; the routig is costructed based o the ffiffiffiffiffiffiffiffiffiffi SHs cosistig of the liks of legth of order ð log Þ; ad those liks are also scheduled by a TDMA schee of costat eriod K2 2 ([8, K 2 ¼ 4]). A iortat ethod is called the arallel trasissio schedulig uder which ðlog Þ liks iitiatig fro each active cell are siultaeously scheduled. As i PhN, the highway syste i SaN also cosists of two levels highways: FHs ad SHs. Next, we first itroduce the fro the situatio where PhN is ot cosidered, ad the exted the to the real situatio i which the riority of PhN is iviolable Highways for SaN Abset of PhN Whe the PhN is igored, the highway syste ca be costructed by the siilar ethod i [8]. The FHs are ideed the highways costructed i [4]. The SHs that are built without usig ercolatio theory [4]. Existece ad desity of FHs. The FHs are costructed ad scheduled based o the schee lattice ILð ;c ; 4 Þ as illustrated i Fig. 2. Sice the distributio of SUs follows a Poisso with ea c 2 (derived by the itesity ties the area of the cell c 2 ), the cell i ILð ;c ; 4Þ has the sae oe robability as the cell of the lattice i [4, Fig. 2], i.e., e c2. Let h ¼ ffiffiffi ffiffi2 c. Fro Lea B. i Aedix B, available i the olie suleetal aterial, by choosig a large eough c, there are, w.h.., ðhþ aths crossig the etwork fro left to right. These aths ca be groued ito disjoit sets cosistig of log h aths, with ffiffiffi each grou crossig ffiffiffiffiffiffiffiffiffi a rectagle slab of size ð log h h Þ 2 c, where >0, is sall eough, ad h is vaishigly sall so that the side legth of each rectagle is a iteger. Therefore, we ca ffiffiffi divide such a rectagle ito log h slices of size $ð; Þ, where $ð; Þ ¼ð Þ. Deote the jth slice i the ith slab by s h ði; jþ, where i h log h h ad j log h. The, we allocate the relay burde of odes i s h ði; jþ to a secific FH, deoted by h h ði; jþ reresetig the jth horizotal FH i the ith slab. Siilarly, for the vertical case, we ca exlai the corresodig s v ði; jþ ad h v ði; jþ, ad defie the aig betwee the. Existece ad desity of SHs. The SHs are costructed ad scheduled based o the schee lattice rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IL ; log ; 0 : For the dese scalig etwork odel, the arallel trasissio schedulig does ot work [8]. The, i SaN, havig o arallel SHs like i PhN [5], there exists oly oe SH i each colu (or row). Deote each colu as s 0 vðiþ, where ffiffiffi i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi log : Based o this, we allocate the relay burde of odes i s 0 v ðiþ to a secific SH, deoted by h0 vðiþ reresetig the SH cotaied i the ith colu. Siilarly, for the vertical case, we ca exlai the corresodig s 0 h ðiþ ad h0 h ðiþ, ad defie the aig betwee the. Reark that we ca use a TDMA with the costat eriod K2 2, to schedule the SHs. We will rovide the detailed aalysis i Sectio Highways for SaN Preset of PhN Cosequetly, we costruct the highway syste for SaN based o ercolatio theory [4], esurig that o highway i SaN crosses the active PAs i ay tie slot. Existece ad costructio of FHs. The FHs i SaN will be scheduled i the first hase i PhN, i.e., S r or S t. I this hase, aroud each PU, we build ffiffiffi its A-Tye PA as a cluster of ie cells of side legth c, as illustrated i Fig. 2. I ay tie slot, a A-Tye PA is active or iactive deedig o whether the cetral PU is scheduled (icludig both trasittig ad receivig) or ot. Recall that the trasissio schedulig of FHs i PhN is a TDMA schee with costat eriod K 2. The, i ay tie slot of the schedulig for FHs i PhN, the uber of scheduled cell will be of the uber of all cells, i.e., K 2 c. That is, i 2 soe tie slots, the uber of scheduled PUs is of order ðþ. Hece, it has o iact o our results whe the dyaic of the status of A-Tye PAs i the first hase is igored, i.e., all A-Tye PAs are always regarded as active i the first hase. Next, we build the FHs i SaN that coexists with PhN. We first odify the defiitio of oe cells [4]. A cell i

7 036 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 5, MAY 203 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. 4. SHs built based o ILð ; log ; 0Þ. (a) Whe PhN adots ure ad hoc strategy, the SHs i SaN eed ot detour the PAs, but wait for their iactive status. (b) Whe PhN adots BS-based strategy, sice the PAs are always active, SHs i SaN have to detour all B-Tye PAs alog the SHs adjacet to the PAs. The bold olylies deote the detourig aths. Fig. 3. Illustratio of schedulig schee. For readability, we describe the case that K ¼ 3 ad K 2 ¼ 4. The schedulig for SaN is divided ito two hases corresodig to the two hases i PhN. I the first (or secod) hase, SaN schedules i sequece K K (or K 2 K 2 ) cells i the ffiffiffiffiffiffiffi schee lattice ILð ;c = ; 4 Þ (or ILð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; log = ; 0Þ) durig oe eriod of 3K 2 (or K4 2 ¼ K2 2 K2 2 ) slots; that is, each cell will be scheduled cotiuously 3 (or K2 2 ) slots. Reark that, durig the cotiuous K2 2 slot for each cell i ILð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; log = ; 0Þ, the cell is really scheduled oly whe it is ot covered by active PAs. ILð ;c ; 4Þ is called orotected oe if it is oety, i.e., it cotais at least oe SU, ad does ot belog to ay A-Tye PAs. Please see the illustratio i Fig. 2. The, we have the followig lea. Lea 3. Whe ¼ oðþ, a cell i ILð ;c ; orotected oe with robability s! as!. 4 Þ is Please see the detailed roof i Aedix C.3, available i the olie suleetal aterial. By Lea 3, we ca rove the existece of FH i SaN, ad obtai the sae desity of FHs i SaN as that i PhN. Thus, we ca use the sae otatios of FHs i the situatio abset of PhN, which will be used i Algorith. Schedulig of FHs i SaN. Let the trasittig ower of ffiffiffi SUs i the first hase be P 0 ðc Þ, where P 0 2ð0;P 0 Š is a costat. Recall that the costat P 0, defied i Sectio 2.3, is the axiu trasittig ower i SaN. Obviously, ffiffiffi P 0 ðc Þ 2ð0;P 0 Š. Because the FHs i SaN detour all PAs, the caacity of FHs i PhN ca be rotected fro icreasig i ters of order, which is roved i Theore 5. As illustrated i Fig. 3, i the first hase i SaN, the schedulig uit is also the cluster of K K cells. Ulike i PhN, each cell i a schedulig uit is scheduled cotiually three slots. By this ethod, it holds that there is at least oe out of these three slots durig which the earest distace betwee the trasitter i PhN to the receiver i SaN is of a costat order. Schedulig of SHs i SaN. As the ew schee lattice rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IL ; log ; 0 is used, we defie the ew A-Tye PA that is a cluster of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ie cells i ILð ; log ; 0Þ cetered at a PU. Please see the illustratio i Fig. 4a. Let the trasittig ower of SUs i the secod hase be rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 0 log ; where P 0 2ð0;P 0 Š is a costat. Obviously, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 0 log 2ð0;P 0 Š: Do the SHs, costructed for SaN abset of PhN, still work ow? The followig Lea 4 will aswer this questio. Lea 4. T K 2 2 2¼ Pðð 2; 2ÞÞ ¼ ;, where Pðð 2 ; 2ÞÞ reresets the set of SUs covered or surrouded by the active PAs i the tie slot ð 2 ; 2Þ, i.e., the 2 th schedulig slot of the secod hase i PhN. Please see the detailed roof i Aedix C.4, available i the olie suleetal aterial. This lea eas that for ay SU, there is at least oe slot out of the schedulig eriod of the secod hase i PhN, i.e., K2 2 tie slots, i which the SU ca be ossibly scheduled. Recall that K2 2 is the costat eriod of TDMA schee used for SHs i SaN abset of PhN. Hece, we ca use a TDMA schee with the eriod of K2 4 ¼ K2 2 K2 2 to schedule the SHs at least oce. Sice K2 2 2ð0; þþ, we ca obtai the sae order of caacity for SHs i SaN regardless of the resece of PhN Multicast Strategy for SaN For a give ulticast sessio M S;i with source v S;i ad the saig set U S;i, we first costruct the EST ESTðU S;i Þ by the ethod i [6]. The, we ca build the ulticast routig tree based o the highway syste ad ESTðU S;k Þ. More secifically, for each couicatio air i ESTðU S;k Þ, i.e., a edge, the ackets will access to the secific FH via the secific SH. The strategy for SaN is divided ito two hases that are sychroous to the two hases i PhN. See the illustratio i Fig. 3a. The detailed ulticast routig schee is reseted i Algorith. To clarify the descritio, we first recall the forulatio of the highway syste:. s h ðx; yþ: The yth horizotal slice i the ffiffiffi xth horizotal slab i the schee lattice ILð ;c ; 4 Þ.

8 WANG ET AL.: SCALING LAWS OF COGNITIVE AD HOC NETWORKS OVER GENERAL PRIMARY NETWORK MODELS 037. s 0 hðzþ: The zth row i rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IL ; log ; 0 :. h h ðx; yþ: The horizotal FH bearig the relay load iitiated fro the odes i the slice s h ðx; yþ.. h 0 hðzþ: The horizotal SH bearig the relay load iitiated fro the odes i the row s 0 h ðzþ. The forulatios for the vertical case are siilarly defied. Algorith. Multicast Routig based o FHs ad SHs Iut: The ulticast sessio M S;k ad ESTðU S;k Þ. Outut: A ulticast routig tree TðU S;k Þ. : for each lik u i! u j of ESTðU S;k Þ do 2: Accordig to the ositio of u i ad u j, deterie the idexes a i, b i, y i ad c j, d j, x j, where u i 2 s h ða i ;b i Þ\s 0 v ðy iþ; u j 2 s v ðc j ;d j Þ\s 0 h ðx jþ. 3: Packets are draied fro u i ito the horizotal FH h h ða i ;b i Þ via the vertical SH h 0 v ðy iþ. 4: Packets are carried alog the horizotal FH h h ða i ;b i Þ. 5: Packets are carried alog the vertical FH h v ðc j ;d j Þ. 6: Packets are delivered fro the vertical FH h v ðc j ;d j Þ to u j alog the horizotal SH h 0 h ðx jþ. 7: ed for 8: Cosiderig the resulted routig grah, we erge the sae edges (hos), ad reove those circles which have o iact o the coectivity of the couicatios for ESTðU S;k Þ. Fially, we obtai the fial ulticast routig tree TðU S;k Þ Aalysis of Multicast Throughut for SaN Without loss of coatibility to ost existig related results, we assue that s ¼ ðþ, i.e., jsj ¼ ðþ. Please see the corresodig detailed roofs i Aedix C, available i the olie suleetal aterial. Above all, we should guaratee the riority of PhN i ters of the throughut. The, we roose the followig theore. Theore 5. By usig Algorith to costruct the ulticast routig for SaN, deoted by S r s, ad the trasissio schee described i Fig. 3, deoted by S t s, to schedule SaN, the caacity of highways i PhN, icludig FHs ad SHs, ca be rotected fro decreasig i order sese due to SaN. Because SaN does ot add the load of ay highway i PhN, by Theore 5, we obtai that the resece of SaN has o iact o the order of throughut for PhN, whe the strategy for SaN is desiged as i Theore 5. Next, we study the throughut for SaN. First, we aswer Questio 3 roosed above. The eaig of S r s ad St s ca be foud i Theore 5, so we have Theore 6. Uder the ulticast routig S r s ad trasissio schee S t s, all ulticast sessios i SaN ca be served. Heceforth, we start to aalyze the ulticast throughut for SaN uder S r s ad S t s, by usig Theore B.2 i Aedix B, available i the olie suleetal aterial. We first coute the caacity of the FHs ad SHs i SaN. Theore 7. Uder the trasissio schedulig S t s, the caacity of FHs ad SHs i SaN ca achieve ðþ. Accordig to Theore 7, we ca obtai Theore 8. Theore 8. Durig the first ad secod hases, whe! d ¼ O ðlog Þ 2 ; the er-sessio ulticast throughuts for SaN are achieved of ðffiffiffiffiffiffiffiþ ad ð d ðlog Þ 3 2 Þ, resectively. d Like i the sigle rado exteded etwork [8], [6], whe the uber of destiatios is beyod soe threshold, to be secific, d ¼ ð Þ, the ulticast throughut ðlog Þ 2 derived by the ulticast routig based o the FHs ad SHs cooeratively is ot otial i order sese. For this case, the ulticast routig based oly o SHs ca derive a larger throughut. Next, we describe such a routig schee i Algorith 2. Algorith 2. Multicast Routig based o Oly SHs Iut: The ulticast sessio M S;k ad ESTðU S;k Þ. Outut: A ulticast routig tree TðU S;k Þ. : for each lik u i! u j of ESTðU S;k Þ do 2: Accordig to the ositio of u i ad u j, deterie the idexes x i ad y j, where u i 2 s 0 h ðx iþ; u j 2 s 0 v ðy jþ. 3: Packets are draied fro u i ito the horizotal SH h 0 h ðx iþ by a sigle ho. 4: Packets are carried alog the horizotal SH h 0 h ðx iþ. 5: Packets are carried alog the vertical SH h 0 v ðy jþ. 6: Packets are delivered fro the vertical SH h 0 v ðy jþ to u j by a sigle ho. 7: ed for 8: Usig the siilar rocedure i Ste 8 of Algorith, we ca obtai the fial ulticast routig tree TðU S;k Þ. Theore 9. By usig the ulticast routig based oly o SHs, i.e., the ulticast routig costructed by Algorith 2, ad schedulig oly for SHs, the er-sessio ulticast throughut for SaN ca be achieved of order 8 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi whe d ¼ O >< d log log >: ð=þ whe d ¼ : log The, accordig to Theore B.2 i Aedix B, available i the olie suleetal aterial, cobiig Theore 8 ad Theore 9, we ca get Theore Whe PhN Adots BS-Based Strategy For this case, PhN adots the classical BS-based strategy ffiffi based o the schee lattice ILð ; b ; 0Þ, where b ¼ bðþ is the uber of BSs i PhN. Uder this strategy, the sources deliver the data to BSs durig the Ulik hase ad BSs deliver the received data to destiatios durig the Dowlik hase. The couicatio betwee ay airs of PUs will be relayed by the BSs. To achieve a better throughut, the BS-based strategy is adoted i PhN oly

9 038 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 5, MAY 203 whe b ¼ ð= log Þ [5]. Because BSs are regularly laced, i.e., each BS locates at the ceter of each cell, all cells ca be siultaeously scheduled durig both Ulik hase ad Dowlik hase, ad ca sustai a rate of ð b Þ 2, [5]. I each cell, all dowliks ad uliks are scheduled i sequece. All B-Tye PAs will be always active. The SUs cotaied i such PAs caot be served. Thus, we study the ulticast caacity for SaN uder the geeral defiitio of ulticast caacity, i.e., asytotic ulticast caacity (Defiitio B.2 i Aedix B, available i the olie suleetal aterial). Obviously, the classic defiitio of caacity [6], [3], ca be regarded as a secial case of the asytotic ulticast caacity. Next, we focus o the Dowlik hase i which SaN is scheduled. Whether or ot SaN is scheduled i Ulik hase has o iact o the ulticast throughut i order sese Highway Syste SaN still refers to adot the ulticast strategy based o the FHs ad SHs as i the case that all BSs are always iactive. Also, the B-Tye PA for each BS i the first hase is a cluster of ie cells i the schee lattice ILð ;c ; 4Þ; ad i the secod hase it is a cluster of ie cells i rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi IL ; log ; 0 : Notice that the effectiveess of such a B-Tye PA relies closely o the fact that SaN is dese scalig while PhN is exteded scalig. Ituitively, such FHs or SHs as i the case abset of active BSs ight be daaged by the PAs as the uber of BSs is icreasig. Fortuately, for b ¼ OðÞ the b ¼ oð log Þ, ad BSs are regularly laced, [5], which guaratees SaN the sae desity, i order sese, of FHs as i the case abset of BSs. The revious SHs i SaN are still reaied. However, whe the ackets carried alog a SH are stoed by a PA, they will detour the PA alog the adjacet SHs. Hece, the load of the SHs aroud the PAs is robably heavier tha that of other SHs. See illustratio i Fig. 4b. Throughout the routig, the bottleeck i the secod hase should be i those SHs with heavy burde. We exloit this fact to aalyze the ulticast throughut for SaN Served Set of SaN Now, we deal with the questio that how ay SUs are ot served at all. Deote the set of all SUs that are ot served by P. Deote the set of all sources i SaN by S. Based o the sets P ad S, we roose a defiitio of the served set of ulticast sessios that ca be divided ito two regies deedig o d, i.e., the uber of destiatios of each ulticast sessio. Defiitio (Served set). The served set, deoted by S,isa ~ subset of S. Defie S ~ :¼ S S\P, whe d ¼!ðlog Þ; ad defie S:¼ ~ fv S;i ju S;i \P¼;g, whe d ¼ Oðlog Þ. For a ulticast sessio M S;i, we defie a subset of U S;i as ~U S;i ¼fv S;i g[d 0 S;i, where D0 S;i ¼D S;i D S;i \P. The, for M S;i, we build its saig tree ESTð~U S;i Þ, ad ake it as the guidelie of ulticast routig, corresodigly actig as ESTðU S;i Þ i the case that PhN adots ure ad hoc strategy. Sigificatly, we have the followig result. Theore 0. As!, it holds that j ~ Sj! jsj; ad for each v S;i 2 ~ S, it holds that uifor w.h.., jd 0 S;i j!jd S;ij. The, accordig to Theore 0, the throughut derived by the strategy based o the served set is asy-achievable Guaratee of Priority of PhN I the first hase of SaN, the su iterferece roduced by SaN at a receivig PU i PhN, deoted by I s, is of order OðÞ, due to the settig of PAs. The, I s ¼ OðN 0 Þ, where the costat N 0 > 0 is the abiet oise. Hece, the resece of SaN does ot chage the order of the caacity of FHs i PhN. Siilarly, we ca rove that SaN does ot iair the caacity of SHs i PhN Asy-Achievable Multicast Throughut for SaN We first cosider the caacity of FHs ad SHs i SaN. Lea 5. I the first hase, the su iterferece roduced by PhN at a receivig SU is of order OðÞ. Please see the detailed roof i Aedix C.0, available i the olie suleetal aterial. Accordig to the roof of Theore 7 i Aedix C.7, available i the olie suleetal aterial, durig ay tie slot i the first hase whe a lik v i! v j is scheduled, the iterferece o v j roduced by SaN itself is bouded by I ss ðv i ;v j ; Þ¼OðÞ. Cobiig with Lea 5, we get that the caacity of FHs i SaN does ot decrease due to PhN, ad it is still of order ðþ. Usig the siilar rocedure, we rove that the caacity of SHs i SaN is still of order ðþ. Next, we should aalyze the load of FHs ad SHs, resectively. The forer is obviously the sae as the load of FHs i SaN whe BSs are abset. For the latter, the schee of detourig icreases the load of the SHs adjacet to PAs; but it ca be roved that the icreet does ot chage the order of the load of those SHs oadjacet to the PAs. Accordig to the techical Theore B. i Aedix B, available i the olie suleetal aterial, such questios coe dow to boudig the area of sufficiet regios (Defiitio B.6 i Aedix B, available i the olie suleetal aterial). Thus, it is the tie to rove Theore 3, as oe of our ai results. Please see the roof i Aedix C., available i the olie suleetal aterial. 5.4 Whe PhN Adots Hybrid Strategy I this case, the strategy for PhN ca be divided ito four hases: FHs hase, SHs hase, Dowlik hase, ad Ulik hase [4], [5]. The, we ca use a sile iterissio ethod to reserve the order of throughut ad caacity for SaN as i the case that o BS is used, described i Theore 2 ad Theore 4. That is, let SaN be idle durig Dowlik hase ad Ulik hase, ad schedule SaN i FHs hase ad SHs hase as i the case that PhN adots ure ad hoc strategy. Please see the illustratio i Fig LITERATURE REVIEW I this sectio, we aily review the literature about caacity scalig laws for cogitive etworks. I [3], the riary source destiatio ad cogitive S-D airs are odeled as a iterferece chael with

10 WANG ET AL.: SCALING LAWS OF COGNITIVE AD HOC NETWORKS OVER GENERAL PRIMARY NETWORK MODELS 039 Fig. 5. Schedulig schee i SaN whe PhN adots the hybrid strategy. asyetric side iforatio. I [27], the couicatio oortuities are odeled as a two-switch chael. Note that both [3] ad [27] oly cosidered the sigle-user case i which a sigle riary ad a sigle cogitive S-D airs share the sae sectru. Recetly, a sigle-ho cogitive etwork was cosidered i [28], where ultile secodary S-D airs trasit i the resece of a sigle riary S-D air. They showed that a liear scalig law of the sigle-ho secodary etwork is obtaied whe its oeratio is costraied to guaratee a articular outage costrait for the riary S-D air. For ultiho ad ultile users case, Jeo et al. [7], [8] first studied the achievable uicast throughut for cogitive etworks. I their cogitive odel, the riary etwork is a rado dese SANET or a dese BS-based etwork [5], ad the secodary etwork is always a rado dese SANET; two etworks oerate o the sae sace ad sectru. Followig the odel of [7], [8], Wag et al. [20] studied the ulticast throughut for the riary ad secodary etworks. To esure the riority of PUs i eaigs of the throughut, they defied a ew etric called throughut decreet ratio (TDR) to easure the ratio of the throughut of PaN i resece of SaN to that of PaN i absece of SaN. Edowig PaN with the right to deterie the threshold of the TDR, they [20] devised the ulticast strategies for SaN. Both the uicast routig i [7], [8] ad ulticast routig i [20] are built based o the backboes siilar to the SHs i [8], which suggests that the derived throughuts are ot otial uder the Gaussia Chael odel for ost cases. By itroducig ercolatio-based routig [8], [4], Wag et al. [9] iroved the ulticast throughut for the sae cogitive etwork odel as i [7], [8], [20]; they showed that uder soe coditios, there exist the corresodig strategies to esure both etworks to achieve asytotically the uer bouds of the caacity as they are stad-aloe. Oe of the coo characteristics i [7], [8], [9], [20] is that the riary ad secodary etworks i all three odels are dese scalig. More iortatly, the coo roble of three works is that all the strategies i [7], [8], [9], [20] shield the tie doai, which akes the routig ath always detour the PAs (or reservatio regios), although they are soeties iactive. Uder those strategies, there are ossibly soe SUs that ca ever be served. As a iortat characteristic differet fro existig related works, our strategies allow a PA to be dyaic accordig to the state of the corresodig riary ode. Thaks to the dyaics of PAs, SUs ca access oortuistically ito the sectru fro both tie doai ad sace doai. Huag ad Wag [29] studied the throughut ad delay scalig of geeral cogitive etworks. They roosed a hybrid rotocol odel for secodary odes to idetify trasissio oortuities. Based o it, they showed that secodary etworks ca obtai the sae otial erforace as stad-aloe etworks whe riary etworks are soe classical wireless etworks. This work reseted a fudaetal isight o the architectural desig of cogitive etworks. Recetly, Li et al. [30] studied the caacity ad delay scalig laws for cogitive radio etwork with static PUs ad heterogeeous obile SUs coexist i the uit regio. Liu et al. [3] ivestigated the scalig behavior of trasissio delay i large scale ad hoc cogitive etworks by aalyzig the ratio of delay to distace as the distace goes to ifiite. 7 CONCLUSION AND FUTURE WORK We study the ulticast caacity for cogitive etworks that oerate uder TDMA schee. The etwork odel cosists of a PhN ad a SaN. We devise the dyaic PA for each riary ode accordig to the strategy adoted i PhN. Based o PAs, we desig the ulticast strategy for SaN uder which the highway syste acts as the ulticast backboe. Uder the recoditio that SaN should have o egative iact o the order of the throughut for PhN, our strategy has the followig erits: ) By our strategy, the otial throughut for SaN ca be (asytotically) achieved for soe cases. 2) Uder our strategy, ulike ost related works, secodary odes ca access oortuistically ito the sectru fro both tie doai ad sace doai. 3) Uder our strategy, all SUs ca be served excet for the case that PhN adots BS-based strategy. There are soe future directios to be cosidered:. The case where PhN ad SaN are both exteded scalig ay be studied. 2. A iterestig ad sigificat issue is to study the etwork odel i which the secodary etwork is a obile ad hoc etwork. 3. As for ulticast caacity of stad-aloe wireless ad hoc etworks, the ost challegig issue is to close the reaiig gas betwee the lower ad uer bouds of ulticast caacity i soe regies by establishig ossibly tighter uer bouds or creatig ore effective strategies to irove the lower bouds. 4. I our syste odel, whe SHs are scheduled, A- Tye PAs are dyaic, the their statuses, i.e., the statuses of those cetered riary odes, are ecessary for the corresodig secodary odes. However, i a ractical cogitive etwork, it is difficult for the secodary odes to kow the locatios of riary receivig odes. A ore reasoable assutio is that SUs ca locate the riary trasitters, [30], [32]. I soe existig literatures, such as [32], soe rotocols, which oly costruct the reservatio regios aroud the riary trasitters, are desiged by alifyig the sizes of reservatio regios. The effectiveess of this tricky techique deeds o the fact that the legth of each ho uder the rotocol i the riary etwork is liited to a certai order, thus, the distace betwee a secodary ode ad a riary

11 040 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, VOL. 24, NO. 5, MAY 203 receiver ca be liited by restrictig the distace betwee this secodary ode ad the seder of this riary receiver. Ituitively, the logic is based o the triagle iequality. Our syste odel will be ossibly iroved by itroducig this techique ito our strategies. 5. I this aer, all our strategies are desiged for the odel where the riary ad secodary etworks are both of hoogeeous ad hoc ode desity, ad the results are derived based o the ercolatio theory for Poisso-distributed etworks. Whe etwork odels of ihoogeeous ode desity, such as those i [], [2], are cosidered, the clusterig behavior of users will affect the distributio of PAs ad ossibly ivalidate the ulticast strategies for the hoogeeous odel. Sice the satial ihoogeeity aears to be a quite ubiquitous feature of real etwork odels, it is a sigificat future work to exted our work to the case where either the riary etwork or secodary etwork is of ihoogeeous ode desity. ACKNOWLEDGMENTS The authors would like to thak the aoyous reviewers for their costructive coets. The research of authors is artially suorted by the Natioal Natural Sciece Foudatio of Chia (NSFC) uder grat No , the Natioal Basic Research Progra of Chia (973 Progra) uder grat No. 200CB3280, the Progra for New Cetury Excellet Talets i Uiversity (NCET) uder grat No. NCET-2-044, the Natural Sciece Foudatio of Shaghai uder grat No. 2ZR45200, the Natioal Sciece Foudatio for Postdoctoral Scietists of Chia uder grat No. 202M508, the Research Fud for the Doctoral Progra of Higher Educatio of Chia (RFDP) uder grat No , the Progra for Chagjiag Scholars ad Iovative Research Tea i Uiversity, the Shaghai Key Basic Research Project uder grat No. 0DJ400300, ad the US Natioal Sciece Foudatio (NSF) CNS REFERENCES [] I.F. Akyildiz, W.-Y. Lee, M.C. Vura, ad S. Mohaty, Next Geeratio/Dyaic Sectru Access/Cogitive Radio Wireless Networks: A Survey, Couter Networks, vol. 50, , [2] C. Fortua ad M. Mohorcic, Treds i the Develoet of Couicatio Networks: Cogitive Networks, Couter Networks, vol. 53, o. 9, , [3] N. Devroye, P. Mitra, ad V. Tarokh, Achievable Rates i Cogitive Radio Chaels, IEEE Tras. Iforatio Theory, vol. 52, o. 5, , May [4] B. Liu, P. Thira, ad D. Towsley, Caacity of a Wireless Ad Hoc Network with Ifrastructure, Proc. ACM MobiHoc, [5] C. Wag, X.-Y. Li, C. Jiag, S. Tag, ad Y. Liu, Multicast Throughut of Hybrid Wireless Networks Uder Gaussia Chael Model, IEEE Tras. Mobile Coutig, vol. 0, o. 6, , Jue 20. [6] X.-Y. Li, Multicast Caacity of Wireless Ad Hoc Networks, IEEE/ACM Tras. Networkig, vol. 7, o. 3, , Jue [7] A. Keshavarz-Haddad ad R. Riedi, Multicast Caacity of Large Hoogeeous Multiho Wireless Networks, Proc. IEEE Sixth It l Sy. Modelig ad Otiizatio i Mobile, Ad Hoc, ad Wireless Networks ad Workshos (WiOt), [8] C. Wag, C. Jiag, X.-Y. Li, S. Tag, Y. He, X. Mao, ad Y. Liu, Scalig Laws of Multicast Caacity for Power-Costraied Wireless Networks Uder Gaussia Chael Model, IEEE Tras. Couters, vol. 6, o. 5, , May 202. [9] C. Hu, X. Wag, ad F. Wu, Motiocast: O the Caacity ad Delay Tradeoffs, Proc. ACM MobiHoc, [0] X. Mao, X.-Y. Li, ad S. Tag, Multicast Caacity for Hybrid Wireless Networks, Proc. ACM MobiHoc, [] G. Alfao, M. Garetto, ad E. Leoardi, Caacity Scalig of Wireless Networks with Ihoogeeous Node Desity: Uer Bouds, IEEE J. Selected Areas i Co., vol. 27, o. 7, , Set [2] G. Alfao, M. Garetto, E. Leoardi, ad V. Martia, Caacity Scalig of Wireless Networks with Ihoogeeous Node Desity: Lower Bouds, IEEE/ACM Tras. Networkig, vol. 8, o. 5, , Oct [3] P. Guta ad P.R. Kuar, The Caacity of Wireless Networks, IEEE Tras. Iforatio Theory, vol. 46, o. 2, , Mar [4] M. Fraceschetti, O. Dousse, D. Tse, ad P. Thira, Closig the Ga i the Caacity of Wireless Networks via Percolatio Theory, IEEE Tras. Iforatio Theory, vol. 53, o. 3, , Mar [5] A. ÖzgÜr, O. L Ev^Eque, ad D. Tse, Hierarchical Cooeratio Achieves Otial Caacity Scalig i Ad Hoc Networks, IEEE Tras. Iforatio Theory, vol. 53, o. 0, , Oct [6] S. Li, Y. Liu, ad X.-Y. Li, Caacity of Large Scale Wireless Networks uder Gaussia Chael Model, Proc. ACM MobiCo, [7] S.-W. Jeo N. Devroye, M. Vu, S.-Y. Chug, ad V. Tarokh, Cogitive Networks Achieve Throughut Scalig of a Hoogeeous Network, Proc. IEEE Seveth It l Sy. Modelig ad Otiizatio i Mobile, Ad Hoc, ad Wireless Networks (WiOt), [8] S.-W. Jeo, N. Devroye, M. Vu, S.-Y. Chug, ad V. Tarokh, Cogitive Networks Achieve Throughut Scalig of a Hoogeeous Network, IEEE Tras. Iforatio Theory, vol. 57, o. 8, , Aug. 20. [9] C. Wag, S. Tag, X.-Y. Li, ad C. Jiag, Multicast Caacity Scalig Laws for Multiho Cogitive Networks, IEEE Tras. Mobile Coutig, vol., o., , Nov [20] C. Wag, C. Jiag, X.-Y. Li, ad Y. Liu, Multicast Throughut for Large Scale Cogitive Networks, ACM/Sriger Wireless Networks, vol. 6, o. 7, , 200. [2] A. Keshavarz-Haddad ad R. Riedi, Bouds for the Caacity of Wireless Multiho Networks Iosed by Toology ad Dead, Proc. ACM MobiHoc, [22] M. Vu, N. Devroye, ad V. Tarokh, O the Priary Exclusive Regios i Cogitive Networks, IEEE Tras. Wireless Co., vol. 8, o. 7, , July [23] A. Agarwal ad P. Kuar, Caacity Bouds for Ad Hoc ad Hybrid Wireless Networks, ACM SIGCOMM Couter Co. Rev., vol. 34, o. 3,. 7-8, [24] R. Zheg, Iforatio Disseiatio i Power-Costraied Wireless Networks, Proc. IEEE INFOCOM, [25] O. Dousse ad P. Thira, Coectivity vs Caacity i Dese Ad Hoc Networks, Proc. IEEE INFOCOM, [26] A. ÖzgÜr, R. Johari, D. Tse, ad O. Lévque, Iforatio- Theoretic Oeratig Regies of Large Wireless Networks, IEEE Tras. Iforatio Theory, vol. 56, o., , Ja [27] S. Jafar ad S. Sriivasa, Caacity Liits of Cogitive Radio with Distributed ad Dyaic Sectral Activity, IEEE Tras. Couters, vol. 25, o. 5, , Ar [28] M. Vu ad V. Tarokh, Scalig Laws of Sigle-Ho Cogitive Networks, IEEE Tras. Wireless Co., vol. 8, o. 8, , Aug [29] W. Huag ad X. Wag, Throughut ad Delay Scalig of Geeral Cogitive Networks, Proc. IEEE INFOCOM, 20. [30] Y. Li, X. Wag, X. Tia, ad X. Liu, Scalig Laws for Cogitive Radio Network with Heterogeeous Mobile Secodary Users, Proc. IEEE INFOCOM, 202. [3] Z. Liu, X. Wag, W. Lua, ad S. Lu, Trasissio Delay i Large Scale Ad Hoc Cogitive Radio Networks, Proc. ACM MobiHoc, 202. [32] C. Yi, L. Gao, ad S. Cui, Scalig Laws for Overlaid Wireless Networks: A Cogitive Radio Network versus a Priary Network, IEEE/ACM Tras. Networkig, vol. 8, o. 4, , Aug. 200.

12 WANG ET AL.: SCALING LAWS OF COGNITIVE AD HOC NETWORKS OVER GENERAL PRIMARY NETWORK MODELS 04 Cheg Wag received the PhD degree i the Deartet of Couter Sciece at Togji Uiversity, i 20. Curretly, he is a associate research rofessor of couter sciece at Togji Uiversity. His research iterests iclude wireless couicatios ad etworkig, obile social etworks, ad obile cloud coutig. Shaojie Tag received the BS degree i radio egieerig fro Southeast Uiversity, Chia, i 2006, ad has bee workig toward the PhD degree i the Couter Sciece Deartet at the Illiois Istitute of Techology sice His curret research iterests iclude algorith desig ad aalysis for wireless ad hoc etworks, wireless sesor etworks, ad olie social etworks. Chagju Jiag received the PhD degree fro the Istitute of Autoatio, Chiese Acadey of Scieces, Beijig, Chia, i 995. Curretly, he is a rofessor with the Deartet of Couter Sciece ad Egieerig, Togji Uiversity, Shaghai. He is also a coucil eber of Chia Autoatio Federatio ad Artificial Itelligece Federatio, the director of Professioal Coittee of Petri Net of Chia Couter Federatio, the vice director of Professioal Coittee of Maageet Systes of Chia Autoatio Federatio, ad a iforatio area secialist of Shaghai Muicial Goveret. His curret areas of research are cocurret theory, Petri et, ad foral verificatio of software, wireless etworks, cocurrecy rocessig, ad itelliget trasortatio systes. Xiag-Yag Li (M 99, SM 08) received the bachelor s degree at the Deartet of Couter Sciece ad the bachelor s degree at the Deartet of Busiess Maageet fro Tsighua Uiversity, Chia, both i 995, ad the MS ad PhD degrees i 2000 ad 200 at the Deartet of Couter Sciece fro the Uiversity of Illiois at Urbaa-Chaaig. He is a rofessor of couter sciece at the Illiois Istitute of Techology. He serves as a editor of several jourals, icludig IEEE TPDS ad Networks. He also serves i Advisory Board of Ad Hoc & Sesor Wireless Networks fro 2005, ad IEEE CN fro 20. He is a seior eber of the IEEE.. For ore iforatio o this or ay other coutig toic, lease visit our Digital Library at