Throughput and Delay Scaling of Cognitive Radio Networks with Heterogeneous Mobile Users

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1 Trougput and Delay Scalng of Cogntve Rado Network wt Heterogeneou Moble Uer Pengyuan Du, Maro Gerla, and Xnbng Wang 2 Department of Computer Scence, Unverty of Calforna, Lo Angele, USA {pengyuandu, gerla}@c.ucla.edu 2 Scool of Electronc, Info. & Electrcal Engneerng, Sanga Jao Tong Unverty, Sanga, Cna Abtract. We tudy te trougput and delay calng law of cogntve rado network (CRN) wt prmary and econdary uer operatng at te ame tme, pace and arng te pectrum. Bot te prmary and econdary uer are ntally randomly and unformly dtrbuted, ten move accordng to a General Heterogeneou Speed-retrcted Moblty (GHSM) model. However, te prmary uer ave ger prorty to acce te pectrum wle te econdary uer ould acce opportuntcally. In GHSM model, we defne ( + ) = eterogeneou movng pattern ung a unveral et 0 T = {T, A = n χ 0 }, were A determne te movng area of eac pattern. Te et of prmary (econdary) movng pattern T (p) (T () ) a ubet of T wc randomly and ndependently elected. We agn n (n β, were β > ) prmary (econdary) node to eac movng pattern, and ter ntal poton are ubect to a poon pont proce. Addtonally, n GHSM model wt T (p) T () = Θ 3 () = Θ(). By propong a cooperatve routng trategy, we fully utlze te moblty eterogenety of prmary and econdary uer to aceve near-optmal trougput and delay performance of order Θ(poly ) wen χ 0 β. In oter cae, our tranmon ceme ow advantage over [7] n delay performance of te prmary network and over [3] n delay of te econdary network. Keyword: Cogntve Rado Network, Trougput Capacty, Delay, Heterogeneou Moblty, Cooperatve Routng Introducton Recently, tere a been more tre over te already-crowded rado pectrum nce wrele applcaton demand ever more bandwdt. Drven by uc demand, many reearce ave been conducted on cogntve rado network (CRN) becaue of t effcent uage of pectrum. Intated by Gupta and Kumar work [5], te fundamental performance calng law of CRN raed great nteret n te networkng reearc communty. Vu et al. condered te trougput calng law for a ngle-op cogntve rado network and obtaned a lnear calng law for econdary network n [0] and []. In [6], Jeon et al. tuded te trougput calng of a cogntve network under general envronment and owed tat bot prmary and econdary network can aceve te ame trougput calng law a a tand-alone wrele network wle te econdary network may uffer from a fnte outage probablty. Yn et al. n [4] developed te trougput calng law under a mlar aumpton and adopted tranmon protocol tat could guarantee zero outage probablty for econdary network. In contrat to te above-mentoned tatc cenaro, capacty and delay performance n wrele moble network yet anoter topc tat a been explored. In [4], te autor owed tat te moble network could aceve te optmal trougput of Θ() under te 2-op relay algortm at te cot of Θ(n) delay per packet. Oter moblty model ave alo been tuded ubequently, ncludng te..d. moblty model [8], random way-pont moblty model [9], random walk moblty model [5], and retrcted moblty model [3][2]. Moreover, recent reearc owed tat moblty n CRN could brng even more beneft. In partcular, te movement econdary uer would facltate poble cooperaton between te two coextng network 3 Te followng aymptotc notaton are ued trougout t paper. Gven non-negatve functon f(n) and g(n):. f(n) = ω(g(n)) mean tat lm n g(n)/f(n) = f(n) = o(g(n)) mean tat g(n) = ω(f(n)). 3. f(n) = O(g(n)) mean tat tere ext a contant c and nteger N uc tat f(n) c g(n) for n > N. 4. f(n) = Θ(g(n)) mean tat for two contant 0 < c 2 < c 3, c g(n) f(n) c 2g(n) for uffcently large n. 5. f(n) g(n) mean tat lm n f(n)/g(n) =.

2 2 P. Du, M. Gerla, X. Wang and mprove ter performance calng law. In [2], Gao et al. propoed a upportve two-ter network were econdary uer are wllng to relay packet for te prmary uer to mprove trougput and delay calng of te prmary network. Ten, Wang et al. [3] derved a cooperaton ceme wc aceve near-optmal capacty and delay calng for te tatc prmary network, but wt le upportve moble econdary uer. T aceved by dvdng te econdary uer nto dfferent layer and te moble econdary uer of dfferent layer are aocated wt dfferent movng area. However, n order to regulate te movng area of dfferent layer, t network model requre a trct cell partton ceme wc arbtrarly parttoned te wole network nto dfferent layer cell. L et al. utlze a mlar approac to mprove capacty calng for econdary network n a more general and flexble erarccal moblty model [7]. Baed on tee work, we go one tep furter by extendng te network model to a more general tuaton were bot prmary and econdary uer poe dfferent movng ablty. T motvated by te fact tat even toug te prmary uer ave prorty to acce te pectrum reource, ter movng pattern ould not be any dfferent compared to te econdary uer. Terefore we frt defne a unveral et T, wc nclude ( + ) type of movng pattern. For eac movng pattern, a movng area of n χ determned were χ a random varable followng te dcrete unform dtrbuton wt + dfferent value, rangng from 0 to χ 0. Ten we contruct te et of prmary movng pattern T (p) by coong randomly and ndependently from te unveral et wt probablty 2. Te left element n T form et T (). We agn n (n β, β > ) node for eac prmary(econdary) movng pattern. Te key queton we want to explore under t General Heterogeneou Speed-retrcted Moblty (GHSM) model nclude: () wt more prmary node n te network, ow to explot te moblty of prmary and econdary uer to upport ter tranmon? (2) weter or not we can tll aceve near-optmal capacty and delay performance? (3) we dvde all movng pattern nto two et, o tere may ext movng ablty gap among prmary or econdary uer. Wat mpact doe uc penomenon ave on te trougput and delay calng law? Our man contrbuton are a follow: We preent a GHSM model wt bot moble prmary uer and moble econdary uer, and movng ablte of all uer are determned n a general and repreentatve way. A cooperatve routng ceme propoed for prmary tranmon wc guarantee te prmary uer better delay performance compared wt te ceme n [7]. A for te econdary network, we ow tat te movng ablty gap among econdary uer wll not degrade trougput and delay performance n order ene,.e., te performance of econdary network no wore tan te performance n [7] [3]. Te ret of t paper organzed a follow. In Secton 2 we ntroduce te network model and defnton. In Secton 3 we preent te routng and cedulng ceme. In Secton 4 and 5, we analyze te trougput capacty and delay of prmary and econdary network, repectvely. In Capter 6, we dcu te ngt of our reult. Fnally we conclude t paper n Secton 7. 2 Sytem Model Trougout t paper we denote te probablty of an event E a P(E) and we manly deal wt event wc take place wt g probablty (w..p.), or wt probablty a te node denty tend to nfnty. 2. Network Geometry We conder te network extenon to be a unt toru were prmary node and econdary node coext. In te moble cogntve network, bot te prmary and econdary network cont of moble uer. Te prmary node are randomly dtrbuted accordng to Poon Pont Proce (P.P.P.) of denty N = ( p + )n, wle te econdary node are of denty M = ( + )m = ( + )n β, wt p + = = Θ(). Tee node move accordng to te eterogeneou peed-retrcted moblty model wc wll be ntroduced later. All node are randomly grouped nto ource-detnaton (S-D) par. 2.2 Communcaton Model For te wrele cannel n t work, we aume te cannel gan depend only on te dtance between te tranmtter and t recever. Terefore te normalzed cannel power gan g(d) gven by g(d) = d δ, ()

3 Trougput and Delay Scalng of CRN wt Heterogeneou Moble Uer 3 Table. Defnton of ymbol related to acevable rate for eac prmary and econdary tranmt par Pp P N 0 Xp,tx Xp,rx X,tx X,rx Ip Ip I Ip Rp R Tranmt power of te -t prmary par Tranmt power of te -t econdary par Termal noe power Tx locaton of te -t prmary par Rx locaton of te -t prmary par Tx locaton of te -t econdary par Rx locaton of te -t econdary par Interference power from te prmary Tx to te Rx of te -t prmary par Interference power from te econdary Tx to te Rx of te -t prmary par Interference power from te econdary Tx to te Rx of te -t econdary par Interference power from te prmary Tx to te Rx of te -t econdary par Rate of te -t prmary par Rate of te -t econdary par were d denote te dtance between te tranmtter (Tx) and t recever (Rx) and δ > 2 te pat lo exponent. To determne te tranmon rate of eac network, we aume tat eac tranmon deploy a ceme tat can aceve te addtve wte Gauan noe (AWGN) cannel capacty. For a gven gnal to nterference and noe rato (SINR), t capacty gven by te well known formula R = log(+sinr) bp/hz aumng te addtve nterference alo wte, Gauan, and ndependent from te noe and gnal. We now caracterze te rate aceved by te prmary and econdary tranmt par. Suppoe tat N p prmary par and N econdary par communcate multaneouly. Before proceedng wt a detaled decrpton, let u defne te notaton ued n te paper, gven by Table. Tu te -t prmary Tx-Rx par can communcate at a rate of Rp = log( + P pg( X p,tx Xp,rx ) N 0 + Ip + Ip ), (2) were denote te Eucldean norm of a vector. Smlarly, te -t econdary par can communcate at a rate of P R g( X,tx X,rx ) = log( + N 0 + I + Ip ). (3) 2.3 Moblty Model Te prmary and econdary uer would move under a GHSM model modfed baed on te HSRM model n [7]. At te begnnng, all uer are unformly and randomly dtrbuted over te network. Ten tey would move wtn ter own crcular area centered at te ntal poton, accordng to a..d. moblty model. Te radu R denote te retrcted peed of dfferent uer, and te node poton would be totally reuffled n eac movng area from one tme lot to anoter. Te movng area of prmary and econdary moble uer et to be n χ, were χ a varable followng te dcrete unform dtrbuton of + dfferent value: χ = 0, χ0 ( )χ0,...,, χ 0. Here, 2χ0 χ 0 a random potve value and = Θ(). Eac movng area A = n χ 0 caracterze an -t type of movng pattern T, and T = {T 0 }. Dfferent from te HSRM n [7], we ave bot prmary and econdary moble uer. Specfcally, we randomly cooe p type of movng pattern from T wc form T (p) and agn n prmary node for eac movng pattern T T (p). Te oter movng pattern form T () and we agn n β econdary node for eac T T (). Smlar to HSRM n [7], larger χ 0 lead to larger dfference between mobng area of dfferent uer, and larger correpond to larger T. Tu, χ 0 and determne te moblty eterogenety of our moblty model.

4 4 P. Du, M. Gerla, X. Wang Moreover, we denote te q-t prmary uer of T T (p) a P q and t ntal poton a X q p,, were 0 and q n. Te r-t econdary uer of T T () a S r and t ntal poton a Xr,, were 0 and r n. Under our HSRM, P q Xq p, A R, were R = π = Θ(n χ 0 2 ), and S r Xr, R. 2.4 Capacty Defnton Trougout te paper, te acevable per-node trougput of te prmary and econdary network defned te average data rate tat eac ource node can tranmt to detnaton w..p. under a partcular cedulng and routng ceme. 2.5 Flud Model and Delay In t paper, we ue a flud model [] to tudy te delay performance for te prmary and econdary network. Specfcally, we dvde eac tme lot nto everal packet lot, and te ze of te packet wll be caled down to arbtrarly mall wt repect to te node denty n (or m) n te network. 3 Network protocol In t capter, we frt preent te routng and cedulng trategy n our moble cogntve network wt GHSM. Ten we analyze te trougput and delay performance of te prmary and econdary network. At lat we wll dcu ow moblty eterogenety of prmary and econdary uer can affect te trougput and delay calng of moble cogntve network. 3. Prmary Network Routng Sceme In our ceme, bot te prmary and econdary uer are movng under te GHSM, o te prmary packet would not need to be relayed only by econdary uer. Intead, prmary packet would be relayed progrevely to reac te detnaton by bot prmary and econdary uer. We tll dvde te unt area nto prmary and econdary cell wt a p = 2 n and a = 2 log m m, ten te prmary packet would be routed n tee two network grd. Snce T (p) and T () are randomly contructed, t clear tat te can of relay node for a partcular prmary S-D par cont of two nterlaced prmary and econdary relay can. We frt derve te Crtcal Relay Type a defned n [7] for te prmary node, nce wen te movng pattern a large type, t poble tat te movng area of a prmary node tend to be wtn a prmary cell wc wll not elp explot te advantage of moblty. Defnton. Te crtcal relay type of prmary node p denoted by: = max{ T T (p), R 2 2 a p }. (4) p From te above defnton, we can calculate te crtcal relay type a follow:, f χ 0 <, p = ( log + log 6π ), f χ 0. χ 0 χ 0 p (5) Here x = max{n Z n x}, and x p = max{n Z Tn T (p), n x}. It can be een tat p = Θ(). Now we ntroduce te man dea of our prmary routng ceme. For a partcular ource node P k, t would frt tranmt packet to a T 0 type relay node nce t relay node a acce to any prmary cell wtn te unt area and tu can delver te packet to te detnaton P k. Ten te packet would be forwarded to te next type of relay node woe movng area maller but cover te movng area of. So on fort, te packet would be delvered to a relay node of te ame type of movng pattern a P k P k, and wen tey meet te packet would approac te detnaton. We note tat nce T (p) and T () are ut part of T, mot packet are propagated troug bot prmary and econdary network and tere are four type of delvery cae:

5 Trougput and Delay Scalng of CRN wt Heterogeneou Moble Uer 5 P p p (prmary to prmary relay). It may appen wen prmary ource node P k tranmt to a T 0 relay node, or wen relay node P u tranmt to detnaton P k, or between two relay node (P u k k, P u k+ k+ ) woe type of movng pattern T k, T k+ T (p). We take te trd tuaton a an example to ow te operaton rule for P p p. Te ntal poton of detnaton P k and relay node P u k k, P u k+ k+ atfe X u k p,k Xu p,k+ < R k and X u k+ p,k+ Xk p, < R k+ 2a p. P u k k tranmt to P u k+ k+ wt power of P p a δ/2 p. P p (prmary to econdary relay). It may appen wen prmary ource node P k tranmt to a T 0 relay node, or between two relay node P u k k, Su k+ k+ woe type of movng pattern T k T (p), T k+ T (). We take te econd tuaton a an example to ow te operaton rule for P p. Te ntal poton of detnaton P k and relay node P u k k, Su k+ k+ atfe X u k p,k Xu,k+ < R k and X u k+,k+ Xk p, < R k+ 2a p. P u k k tranmt to S u k+ k+ wt power of P p a δ/2 p. We note tat to mplfy te analy, te prmary tranmtter of P p would broadcat t packet to all econdary uer redng n t prmary cell. P (econdary to econdary relay). It appen between two econdary relay node (S u k k, Su k+ k+ ) woe movng pattern T, T + T (). Te operaton rule are: te ntal poton of detnaton P k and relay node S u k k, Su k+ X k+ atfe u k,k Xu k+,k+ < R k and X u k+,k+ Xk p, < R k+ 2a p. S u k k tranmt to S u k+ k+ wt power of P a δ/2. P p ( econdary to prmary relay). It appen between relay node (S u k k, P u k+ k+ ) woe movng pattern T T (), T + T (p). Te operaton rule are: te ntal poton of detnaton P k and relay node S u k k, P u k+ k+ atfe X u k,k Xu k+ p,k+ tranmt to P u k+ k+ wt power of P a δ/2. < R k and X u k+ p,k+ Xk p, < R k+ 2a p. S u k k Baed on te ntroducng of te crtcal relay type, te prmary S-D par (P k relay node from T 0 to T p, P k ) would utlze te wen > p, tu te number of relay tep would be p + 2. Wen p te packet would be relayed to T. So we denote p = mn (, p), ten te relay proce would take up to p + 2 tep. Te algortm own n Algortm. Algortm Relay Algortm for Prmary Packet B p Input: Te prmary ource node P k and detnaton node P k Output: Te p + ntermedate prmary and econdary relay node : P k move wtn t movng area untl t meet a node of T 0 n te ame prmary cell. 2: P k execute P p p or P p contngent on weter T 0 T (p) or T 0 T (). 3: for k = 0 to ( p ) do 4: Te relay node of T k move wtn t movng area untl t meet a node of T k+ n te ame cell ( prmary cell f T k+ T (p), econdary cell f T k+ T () ). 5: Execute one of te four relay procee baed on te correpondng rule. 6: end for 7: P u p move wtn t movng area untl t meet P k n te ame prmary cell. p 8: Execute P p p to delver Bp to te detnaton. To verfy te feablty of eac tep n Algortm, we preent te followng lemma. Lemma. Wt te eterogenety factor χ 0 and = Θ(), te number of elgble relay n eac tep of Algortm larger tan, w..p.. Proof. We denote C r (x) a te crcle wt radu r centerng at x. We conder te wore condton a R p p, nce te node denty of econdary uer for eac movng pattern larger tan te prmary uer.

6 6 P. Du, M. Gerla, X. Wang Fg.. Illutraton of te relay proce ung S 0. From Algortm, we can ee tat elgble relay for T k+ ould rede n te area of C Rk (X u k p,k ) C Rk+ 2a p (X k p, ) > π 3 (R k+ 2a p ) 2. Ung properte of Poon dtrbuton, we ave P (te number of elgble relay of T (k+) 0) = e λ λ π 3 R2 k+ n e n p χ 0 / { e n χ 0 f χ 0 <, 0 e nlog / f χ 0, 3.2 Secondary Network Routng Sceme Dfferent from te prmary network routng ceme, te econdary network would only utlze econdary uer a relay node to propagate econdary packet. We note tat T T () are a ere of dcrete number, o te relay proce for econdary packet may go troug dcrete type of relay node. We organze T T () accordng to ter value, and T, denote te -t type of movng pattern n T (). Lke Defnton, we defne te crtcal relay type of econdary network. Defnton 2. Te crtcal relay type of econdary node denoted by: = max{ T T (), R 2 2 a }. (6) It can be ealy derved tat, f χ 0 < β, = ( β log + log 6πβ ), f χ 0. χ 0 χ 0 Ten we ave = Θ(). If we denote = mn(, ), te relay proce of B p would take up to K + 2 tep, for T = T,K (te mappng from T to T () own n Fgure 2 ). We note tat a dfference n te econdary relay algortm tat, wen T 0 / T (), T,0 node are ued to relay a econdary packet toward t detnaton. And we notce tat te movng area of T,0 n t cae only cover part of te unt toru, o we keep te packet tranmttng among T,0 node wt power of P a δ/2 untl a pecfc relay node of T,0 cover te ntal poton of te detnaton. We denote eac tep of t relay proce a S 0. For oter tep of te wole relay proce, te operaton rule are mlar to P, and we denote tem a S. Te econdary relay algortm own n Algortm 2. Smlar to Lemma, we can verfy te feablty of eac tep n te econdary relay algortm. 3.3 Prmary and Secondary Scedulng Sceme After ntroducng our routng ceme, we wll ow te cedulng ceme for te prmary and econdary network. Smlar to te protocol n Capter??, we ue te 64-TDMA cedulng ceme for prmary and econdary network, and te duraton of a econdary frame equal to te duraton of a prmary lot. To lmt te nterference between two network, we alo adopt preervaton regon. From te knowledge of prevou capter, we know tat t wll not caue te decreae of tranmon opportunte for econdary uer n order ene. (7)

7 * * * Trougput and Delay Scalng of CRN wt Heterogeneou Moble Uer 7 T0 T T3 T4 T5 T6 T7... * * * T 2 T + TD... T-... T ( ) adfd K 0 adfd K 4 adfd K7 2 T,0 T, T,2... TS, K - T, S TS, K +... K SDFASDF K * K * 2 T,d K Fg. 2. Mappng from T to T (). Algortm 2 Relay Algortm for Secondary Packet B Input: Te econdary ource node S k and detnaton node S k Output: Te K + ntermedate econdary relay node : S k move wtn t movng area untl t meet a node of T,0 n te ame econdary cell. 2: wle B eld by a T,0 econdary node woe movng area doe not cover X k, do 3: Execute S 0. 4: end wle 5: for k = 0 to (K ) do 6: Te relay node of T,k move wtn t movng area untl t meet a node of T,k+ n te ame econdary cell. 7: Execute S baed on te correpondng rule. 8: end for 9: S u move wtn t movng area untl t meet S k n te ame econdary cell. 0: Execute S to delver B to te detnaton. Prmary Scedulng Sceme For te prmary network, we alo organze T k T (p) and T p,k denote te k-t type of movng pattern n T (p). Ten te cedulng ceme nclude K p + 2 pae, for = T p,k p, and eac pae cot one prmary frame. T p Pae 0 Durng te actve lot of eac prmary cell, randomly cooe a ource node P k Bp to a random node of T 0 troug P p p or P p. For k =, 2,..., K p, and relay packet Pae k Durng te actve lot of eac cell, two type of tranmon wtn t cell could appen n t pae: () tranmon between prmary node of P u ˆKk ˆK k and feable relay node of T ˆKk + (T ˆKk = T p,k ) for prmary packet B p ( ˆK k p); (2) tranmon between prmary node of T p,k and P ˆKk for a packet B ˆK k p to reac te detnaton. One of uc tranmon would be elected ˆK k randomly to perform. Oterwe te cell tay dle. Pae K p + Durng te actve lot of eac cell, all par of node (P u p p, P k ) ( p ) wt n te cell are elgble for tranmon to relay B p to te detnaton. One of uc par would be elected randomly to tranmt. Oterwe te cell tay dle. Secondary Scedulng Sceme Accordng to te routng ceme, te econdary network are requred to relay bot prmary and econdary packet. Tu we propoe a cedulng ceme cont of more pae to guarantee tranmon opportunty for bot knd of packet. Specfcally, we preent a K p +K pae ceme. For te frt K p pae, econdary node erve a relay for prmary packet; for te oter pae, econdary node relay ter own packet to te detnaton. For k =, 2,..., K p, Pae k Durng te actve lot of eac econdary cell, randomly cooe a node S u ˆKk to tranmt to a ˆK k feable relay node of T ˆKk + for prmary packet B p ( ˆK k p). If uc par doe not ext, tay dle.

8 8 P. Du, M. Gerla, X. Wang Pae K p + Durng te actve lot of eac econdary cell, randomly cooe a econdary ource node S k and relay packet B to a randomly elected S u ˆK0 troug S ˆK 0 0. For k =, 2,..., K, Pae K p + + k Durng te actve lot of eac econdary cell, two type of tranmon wtn t cell could appen: () tranmon between S u ˆKk ˆK k packet B ( ˆK k ); (2) tranmon between S u ˆKk ˆK k and a feable relay node S u ˆKk ˆK k and S ˆKk ˆK k for a packet B ˆK k for econdary to reac te detnaton. One of uc tranmon would be elected randomly to perform. Oterwe te cell tay dle. Pae K p + K + 2 Durng te actve lot of eac econdary cell, all par of node (S u p, S k ) ( p ) wt n te cell are elgble for tranmon to relay B to te detnaton. One of uc par would be elected randomly to tranmt. Oterwe te cell tay dle. 4 Trougput and Delay Scalng for te Prmary Network In t ecton, we wll tudy te trougput and delay performance for te prmary network baed on our predefned protocol. 4. Trougput Performance Before we proceed to te dervaton of trougput performance of te prmary network, we provde te followng lemma. Lemma 2. [7]Lemma 6 At any moment, te number of P k (for T T (p), 0 ) node n a prmary cell Θ(), w..p.; at any moment, te number of S k (for T T (), 0 ) node n a econdary cell Θ(), w..p.. We frt conder te data rate of eac tranmon n prmary and econdary network ung te followng lemma. Lemma 3. Durng te routng proce of prmary packet, eac delvery type P p p, P p, P, P p can aceve a contant data rate n eac cell. Proof. Baed on analy n prevou ecton, we know tat for our TDMA cedulng ceme wt frame tructure n Fgure?? and te defned preervaton regon, eac prmary and econdary cell can aceve a contant data rate w..p.. Under te ame cedulng ceme, we can alo upport eac tep of te routng proce of prmary packet wt a contant data rate. Ten we can derve te per-node trougput of te prmary network wt te followng teorem. Teorem. Wt te propoed prmary cedulng and routng ceme, te prmary network can aceve te per-node trougput w..p.: λ(n) = Θ( 2 ). (8) Proof. Our proof follow a mlar logc a [7]. We dvde te routng proce nto tree part: nput, relay and output. In te nput proce, te prmary packet are ntated from all actve prmary cell to a T 0 relay node troug P p p or P p, tu te aggregated trougput over te prmary network n t proce Snce eac nput proce conume K p Λ nput (N) = Θ( n ) = Θ( a p ). fracton of te cedulng cycle, we ave λ nput (N) = Λ nput (N) ( pn) K p Λ nput(n) n = Θ( 2 ).

9 Trougput and Delay Scalng of CRN wt Heterogeneou Moble Uer 9 In te relay proce, we ave four dfferent delvery tuaton. For P p p and P p, te aggregated trougput te ame a Λ nput (N). However tee two tuaton would lat for a contant fracton of K te prmary cedulng cycle p K p, o we ave λ relay (N) = Λ nput(n) ( K p pn) K p Λ nput(n) n = Θ( ). For P and P p, te delvery executed n te econdary network. So we ave and te proce conume K p K p +K Λ relay = Θ( a ) = Θ( nβ ), fracton of econdary cedulng cycle, we ave K λ relay (N) = Λ relay (N) ( p pn) K +K p Λ relay (N) nβ n = Θ( ). Te trougput of te output proce mlar to te tuaton P p p of te relay proce, except tat t conume K p fracton of te prmary cedulng cycle. Terefore we ave λ output(n) = Θ( 2 ). Conequently, te prmary network can aceve a per-node trougput of λ(n) = mn(θ( 2 ), Θ( nβ ), Θ( )) = Θ( 2 ). 4.2 Delay Performance We ave te followng teorem to count te delay for te prmary network. Teorem 2. Wt te propoed prmary relay algortm, te prmary network could aceve te followng delay performance w..p.: D (N) = Θ( n χ 0 log 2 n + 2 log 2 n + 2 n ( χ 0 p ) ), (9) were denote te type of prmary detnaton node, and p = mn( p, ) a we defned prevouly. Proof. We dvde te wole routng proce nto nput, relay and output proce. To derve te delay performance of te prmary network, we would evaluate te average number of frame durng eac of te tree procee for a prmary packet B p to reac t detnaton. Input Proce Now we conder te delay performance of nput proce, were prmary ource n- ode P k wll tranmt to a relay node of T 0. Snce for eac type T T (p) tere are Θ() prmary node wtn a prmary cell, te probablty tat P k coen to tranmt durng te actve tme lot of te cell P(P k coen te prmary cell actve) = Θ( p ). It obvou tat P(te prmary cell actve) = K p = Θ( ), ten we can obtan te probablty for a ucceful tranmon for P k n te nput proce P nput = Θ( 2 ). Delay performance n t proce D nput (N) = Θ( 2 ). Relay Proce Now we conder te delay performance of relay proce. Durng te relay proce, te delvery of packet Bp propagated troug P p p, P p, P, P p. We frt conder te delay of P p p. In t proce, P p p appen between P u and P u, for p. It a been proved n [7] tat te probablty of a ucceful P p p Θ(n χ 0 ). Conderng tat te prmary cell actve for K p = Θ( ) fracton of te prmary cedulng cycle, and tere are Θ() node wtn te cell for T, T, we obtan P P p p = Θ( χ n 0 log 2 n ), and D P p p = n χ 0 log 2 n. Next we conder te delay of P p. It appen between P u and S u, for < p. In t cae, te probablty of a ucceful P p alo Θ(n χ 0 ) becaue te node denty of econdary uer larger tan te prmary uer wc wll not lead to te decreae of t probablty. Ung te ame metod we can obtan D P p = n χ 0 log 2 n.

10 0 P. Du, M. Gerla, X. Wang For P, te tuaton mlar to P p p except tat te tranmon appen K p +K +c fracton of te econdary cedulng cycle, and t wll not nfluence te delay n order ene. At lat, we conder te tuaton of P p. Recall tat for P p durng te relay proce, eac prmary uer broadcat t packet Bp wt power P p a δ/2 p to all econdary uer redng n te econdary cell. It clear tat tere are at leat Θ(n β ) econdary uer n eac prmary cell,.e., at leat Θ(n β ) cope of packet Bp are propagated n te econdary network untl tey are tranmtted back to te prmary network troug P p. Baed on te proof n [7], we can obtan tat te probablty tat an elgble prmary relay node P u rede n te ame econdary cell a S u n a actve tme lot Θ(n χ0/ ), ten at leat one copy n β of Bp could be forwarded uccefully, becaue P(Bp χ0/ can be forwarded) = ( Θ(n )) nβ n β χ0/ mn(θ(), Θ(n n β )) Θ(), n β for n χ0/ = Θ(). Tu we obtan D P p = log 2 n. To elmnate te effect of copyng Bp, once a copy uccefully execute P p and forward Bp back to te prmary network, oter cope are outdated and would be reected by any P u. Suc aumpton to guarantee tat no bottleneck effect would appen regardng te trougput of te prmary network. Output Proce Durng te output proce, P p p appen between P u p p Te probablty of P u p p meetng P k n te ame prmary cell and P k, for p = mn (, p). P P p p = Q P (P u p p Q)P (P k Q) Q ( ap πr 2 p ) 2 R2 p a p ( ap πr 2 p ) 2 = Θ( ), n p χ 0 / were Q denote te et of cell nde C R p (X u p p, p ) C R (X k p, ). For te wort cae were p, te probablty tat par (P u p, P k p ) coen to tranmt Θ( log 2 n ), tu we can obtan D output(n) = 2 n p χ0/. Combnng all te delay for eac tep of te wole relay proce, we can obtan te delay performance of prmary packet: D (N) = Θ( n χ0/ log 2 n + 2 log 2 n + 2 n ( p χ0/) ), (0) were we ue te fact tat P p p, P p, P, P p eac conume Θ() tep to relay te prmary packet. 5 Trougput and Delay Scalng for te Secondary Network In t ecton we tudy te trougput and delay performance for te econdary network. Te econdary network dfferent from te prmary network for tey can only acce te pectrum opportuntcally, and we ue preervaton regon to guarantee uc aumpton. Accordng to analy n prevou ecton, we know tat te adopton of preervaton regon wll not caue te decreae of trougput and delay performance n order ene. Terefore we proceed wtout dcung ome pecfc ue regardng preervaton regon. 5. Trougput Performance Smlar to Lemma 3, we ave te followng lemma. Lemma 4. Durng te routng proce of econdary packet, eac delvery type S 0 and S can aceve a contant data rate n eac cell. Ten we can derve te per-node trougput of te econdary network wt te followng teorem. Teorem 3. Wt te propoed econdary cedulng and routng ceme, te econdary network can aceve te per-node trougput w..p.: λ(m) = Θ( 2 ). ()

11 Trougput and Delay Scalng of CRN wt Heterogeneou Moble Uer Proof. We tll dvde te routng proce nto tree part: nput, relay and output. In te nput proce te econdary packet are ntated from all actve econdary cell to a T,0 type relay node troug S 0, tu λ nput (M) = Λ nput (M) ( n β ) nβ K +K p = Θ( (n β ) ) = Θ( 2 ). In te relay proce, econdary packet are delvered troug S, t can upport an average per-node trougput of λ relay (M) = Λ S (M) ( n β ) K + nβ K +K p = Θ( (n β ) ) = Θ( ). In te output proce, te tuaton mlar to tat of te nput proce, we ave λ output (M) = Θ( 2 ). Conequently, te econdary network can aceve a per-node trougput of λ(m) = mn(θ( 2 ), Θ( )) = Θ( 2 ). 5.2 Delay Performance We ave te followng teorem to count te delay for te prmary network. Teorem 4. Wt te propoed econdary relay algortm, te econdary network could aceve te followng delay performance w..p.: D (M) = Θ( n χ 0 log 2 n + 2 n (β χ 0 ) ), (2) were denote te type of prmary detnaton node, and p = mn( p, ) a we defned prevouly. Proof. We tll dvde te wole routng proce nto nput, relay and output proce. We evaluate te average number of econdary frame durng eac of te tree procee for a econdary packet B to reac t detnaton. Input Proce Frt, we conder te delay performance of nput proce, were econdary ource node S k wll tranmt to a relay node of T,0 troug S 0. Snce for eac type T T () tere are Θ() econdary node wtn a econdary cell, te probablty tat S k coen to tranmt durng te actve tme lot of te cell P(S k coen te econdary cell actve) = Θ( ). It obvou tat P(te econdary cell actve) = K +K p = Θ( ), ten we can obtan te probablty for a ucceful tranmon for S k n te nput proce P nput = Θ( 2 ). Delay performance n t proce D nput (M) = Θ( 2 ). Relay Proce Now we move to te delay performance of relay proce. Durng te relay proce, te delvery of packet B cont of two part: relay proce troug S 0 and relay proce troug S. We frt tudy te cae wt S 0. Recall tat n te econdary routng ceme, we keep packet B propagated among relay node of T,0 X untl we fnd one uc relay node woe movng area cover te ntal poton X k,,.e., u ˆK0, ˆK X k 0 p, < R ˆK0 2a. We now ow tat uc an elgble relay node of T,0 could be found troug a contant number τ of S 0. Snce eac element n T () randomly elected from T wt probablty 2, f T,0 not 0-t type, ten T,0 Θ()-t type, w..p., becaue ( 2 )ω() 0. We denote T,0 τ 0 -t type. It obvou tat te probablty tat X k, locate wtn te movng area of Su ˆK0 πr ˆK 2ˆK0 = πn τ0χ0/. Ten te 0 probablty tat t cot ω() number of S0 to fnd a relay node of T,0 ( πn τ0χ0/ ) ω() 0, for n τ0χ0/ Θ().

12 2 P. Du, M. Gerla, X. Wang Conderng tat te econdary cell actve for K p +K fracton of te econdary cedulng cycle, and tere are Θ() node of T,0 redng n te cell, we obtan D S = τ = Θ( ). 0 Next we conder te delay of oter relay tep troug S. We denote T,k a τ k -t type of movng pattern, and = max( k 0 < k K ) for k = τ k+ τ k. It clear tat = Θ(). Ten we ave tat te probablty of a ucceful S at leat Θ(n χ 0 ). Terefore we obtan D S = n χ 0 log 2 n. Output Proce Durng te output proce, S appen between S u Te probablty of S u meetng S k cae were, te probablty tat par (S u n te ame prmary cell P S output = Θ( and S k, for = mn (, )., S k ) coen to tranmt Θ( log 2 n ). For te wort n β χ 0 / ), tu we can obtan D output (M) = 2 n β χ0/. Combnng all te delay for eac tep of te wole relay proce, we can obtan te delay performance of econdary packet: D (M) = Θ( n χ 0 log 2 n + 2 n (β χ 0 ) ), (3) were we ue te fact tat t conume Θ() tep to relay te econdary packet troug S. 6 Dcuon on te Delay and Trougput Performance In t ecton, we wll dcu ow moblty eterogenety of prmary and econdary uer can affect te delay and trougput performance of cogntve network. We frt evaluate te optmal performance of prmary and econdary network under our routng and cedulng ceme. Specfcally, we compare our reult wt [7] and ow te advantage of our ceme. Ten we dcu te performance n dfferent type of movng pattern. At lat, we wll ow ome poble extenon to our work. 6. Optmal Trougput and Delay Performance We ave proved tat te prmary network can aceve λ(n) = Θ( 2 ), w..p.. Snce = Θ(), te acevable per-node trougput λ(n) = Θ( log 3 n ). A for te optmal delay performance for prmary network, we ave Θ(n χ0 log 3 log n), 0 χ 0 <, D optmal (N) = Θ(log 4 n), χ 0 log. We can ee tat for te prmary network, te near-optmal trougput and delay, a well a te delay-trougput tradeoff Θ(poly log(n)) can be aceved wen = Θ() and χ 0 >. For te econdary network, te acevable per-node trougput λ(m) = Θ( ). Te optmal log 3 n delay performance can be caracterzed a follow, Θ(n β χ0 log 3 log n), 0 χ 0 < β D optmal (M) = (5) Θ(log 4 log n), χ 0 β. We can ee tat for te econdary network, te near-optmal trougput and delay, a well a te delay-trougput tradeoff Θ(poly log(n)) can be aceved wen = Θ() and χ 0 > β. Now we compare our reult wt [7] to ow te advantage of our routng and cedulng ceme. In [7], te autor propoed a tranmon ceme wt an acevable trougput λ(n) = Θ( ) and delay 3 log Θ(n β χ0 ), 0 χ 0 < β D optmal (n) = (6) Θ(log 4 3 log n), χ 0 β, for n prmary uer. Tu bot our ceme and te ceme n [7] can aceve near-optmal trougput of Θ(poly ). However our ceme can guarantee more prmary node (Θ(n)) wt better delay performance a own n Fgure 3. For te econdary network, our ceme can aceve mlar performance wt repect to trougput and delay to ter ceme. (4)

13 Trougput and Delay Scalng of CRN wt Heterogeneou Moble Uer 3 D optmal n 3 nlog n 0 n log 3 n n 0 Y. L ceme Our ceme Fg. 3. Comparon between our ceme and te ceme n [7]. 6.2 Performance of Dfferent Movng Type Under our propoed routng and cedulng ceme, te prmary and econdary uer can aceve nearoptmal trougput regardle of ter dfferent movng type. However te delay performance gly related to te movng pattern eac node poee, for D (N) = Θ( n χ 0 D (M) = Θ( n χ 0 log 2 n + 2 log 2 n + 2 n ( χ 0 p ) ), log 2 n + 2 n (β χ 0 ) ), were p = mn(, p T T (p) ), = mn(, T T () ). Obvouly, D (N) wll decreae wen movng type of prmary node ncreae. For 0-t type 4 prmary node, D 0 (N) = Θ(n log 3 n) wt λ 0 (N) = Θ( ), and our reult ndcate a mlar penomenon log 3 n a decrbed n [4],.e., D(n) = nλ(n), for λ(n) =. Wen become larger, te delay performance D (N) mproved wt te elp of relay node of lower movng type, wc ndcate tat eterogeneou moblty ntroduce tranmon dverty nto te network to reduce tranmon delay and guarantee near-optmal trougput performance. We alo notce tat te optmal performance occur to te prmary node wt mnmal movng area. Wen 0 χ 0 <, te prmary node of T p,p aceve optmal delay D ˆKp (N) = Θ(n χ0 log 3 n) ; wen χ 0, te prmary node woe movng pattern are at leat ( p log Θ())-t type can aceve optmal delay of D ˆK (N) = Θ(log 4 n). p For te econdary network, te delay performance mlar to tat of te prmary network. It alo preented tat better performance can be aceved for econdary node of larger movng type. However, wen uer tend to ave trong movng ablty, te overall performance of econdary network wore tan te prmary network by an order of Θ(n β ) due to larger node denty. Oterwe bot network can aceve te ame near-optmal performance. 7 Concluon T paper tude te trougput and delay calng law of GHSM cogntve rado network. In te GHSM model, we tudy te calng law of two ter of moble node wt eterogeneou movng pattern. For prmary tranmon, we utlze bot prmary and econdary node to forward prmary packet progrevely to te detnaton troug a can of relay node wt contnuou movng pattern. By employng uc a cooperatve routng trategy, we guarantee te prmary uer a better optmal performance, and te delay-trougput tradeoff aceve a near-optmal order of Θ(poly ) wen eterogenety ncreaed uffcently. For te econdary tranmon, only econdary node are ued to relay packet due to te prorty of prmary uer. We prove tat te econdary network can tll aceve te ame performance a own n [7] even toug econdary relay proce may uffer from movng ablty gap among node wt dfferent movng pattern. Te mpact of dfferent movng ablty on trougput and delay performance alo dcued and poble extenon are provded for future work. 4 Even f T 0 / T (p), we ave D ˆK0 (N) = Θ(n log 3 n) for T p,0 Θ()-t type w..p. n t cae. (7)

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