A Distributed Spectrum Sharing Algorithm in Cognitive Radio Networks

Transcription

1 A Dstrbuted Spectrum Sharng Algorthm n Cogntve Rado Networks We Sun, Tong Lu, Yanmn Zhu, Bo L Shangha Jao Tong Unversty Hong Kong Unversty of Scence and Technology Emal: {sunwt,lutong ,yzhu}@sjtu.edu.cn, bl@cse.ust.hk Abstract In ths paper we study a socal welfare maxmzaton problem for spectrum sharng n cogntve rado networks. To fully use the spectrum resource, the lcensed spectrum owned by the prmary user PU) can be leased to secondary users SUs) for transmttng data. We frst formulate the socal welfare of a cogntve rado network, consderng the cost for the prmary user sharng spectrum and the utlty ganed for secondary users transmttng data. The socal welfare maxmzaton s a convex optmzaton, whch can be solved by standard methods n a centralzed manner. However, the utlty functon of each secondary user always contans the prvate nformaton, whch leads to the centralzed methods dsabled. To overcome ths challenge, we propose an teratve dstrbuted algorthm based on a prcng-based decomposton framework. It s theoretcally proved that our proposed algorthm converges to the optmal soluton. Numercal smulaton results are presented to show that our proposed algorthm acheves optmal socal welfare and fast convergence speed. Index Terms Decomposton, socal welfare maxmzaton, optmzaton, cogntve rado network. I. INTRODUCTION Wth the ncreasng development of wreless communcatons, spectrum becomes a more and more lmted resource. However, many studes [1] [2] [3] show that spectrum s under utlzed n realty. The technology of cogntve rado CR) [4] provdes a promsng mechansm for flexble usage of spectrum. The utlzaton of spectrum can be mproved by sharng vacant spectrum wth unlcensed users. We consder a cogntve rado network, whch conssts of a prmary user PU), the lcensed owner of spectrum, and several secondary users wthout lcences. Each secondary user conssts of a par of nodes, a sender and a recever, whch have the demand of transmttng data. To fully utlze spectrum, the prmary user can lease ts vacant spectrum to secondary users wth monetary rewards. We consder the problem of spectrum sharng among secondary users n a cogntve rado network. A cost s ncurred and pad by the prmary user for allocatng spectrum to secondary users. Each secondary user obtans an amount of utlty for transmttng data, whch contans ts prvate nformaton. We frst formulate the cost and the utltes, respectvely. Then, our objectve of such a cogntve rado network s to maxmze the socal welfare, whch s defned as the sum of the utlty obtaned by each secondary user mnus the cost pad by the prmary user. Fg. 1. PU Allocated spectrum SU1 SU SU 2 3 The llustraton of spectrum allocaton n a cogntve rado network. A number of exstng works [5] [6] have been done to solve the problem of spectrum sharng n cogntve rado networks. Some of these works [7] [8] [9] take advantage of the models n game theory to characterze cogntve rado networks, whch cannot acheve socal welfare maxmzaton. These works always assume that secondary users are ratonal and selfsh. Some other works [1] [11] [12] can really maxmze the network utlty for spectrum sharng. However, the algorthms desgned n these works are executed n a centralzed manner. Therefore, the prvacy of secondary users s not protected. Dfferent wth these works, we focus on desgnng a dstrbuted algorthm to maxmze the socal welfare. Such an optmal spectrum sharng problem s sgnfcantly challengng because of unque characterstcs of cogntve rado networks. Frst, the utlty functon of each secondary user contans the user-specfc nformaton, whch s ther prvacy. It means that the problem cannot be solved n a centralzed manner by the standard methods such as Newton method. A dstrbuted algorthm s necessary by decouplng the problem nto several subproblems, each of whch can be ndependently solved by a secondary user. Second, the objectve n the problem, maxmzng socal welfare, s coupled wth the cost functon of the prmary user and the utlty functon of each secondary user. In other words, the objectve functon cannot be drectly decoupled nto several ndependent subproblems. In response to the challenges mentoned above, we propose a fully dstrbuted algorthm for spectrum sharng n cogntve

2 2 rado networks. As the shared spectrum s prced by the prmary user, we fnd that the optmal socal welfare s obtaned when optmal soluton of an equvalent prcng problem s acheved. Then, we propose a prcng-based decomposton framework, whch decouples the problem nto several subproblems. The prmary user and each secondary user s responsble for a subproblem, whch only contans ts own prvacy. Based on the decomposton framework, a dstrbuted algorthm s provded, whch s executed teratvely. In each teraton, each secondary user nforms the prmary user ts demanded spectrum, accordng to the gven prce. The prmary user updates the prce based on the collected nformaton. We also propose a prce updatng rule, under whch our teratve algorthm s rgorously proved to converge to the optmal soluton n fnte steps. Extensve smulatons have been conducted to evaluate the socal welfare and convergence speed acheved by our dstrbuted algorthm. We have made three techncal contrbutons n our paper. We model the cost pad by the prmary user for spectrum sharng and the utlty obtaned by secondary users. The concept of socal welfare s formally defned as the objectve of a cogntve rado network to pursue. To maxmze the socal welfare n a dstrbuted manner, we propose a novel prcng-based decomposton framework. A dstrbuted algorthm s desgned based on the decomposton framework, where teratve operatons are executed by the prmary user and the secondary users n turn. Rgorous theoretcal analyss s conducted to demonstrate that our teratve algorthm converges to the optmal soluton n fnte steps. Extensve smulaton results show that our dstrbuted algorthm can acheve the optmal socal welfare, and converge quckly. The remander of ths paper s organzed as follows. We ntroduce the network model and problem formulaton n Secton II. The optmal dstrbuted approach s descrbed n Secton III. In Secton IV, smulaton results are reported to show the performance of our algorthm and compared algorthms. We revew related work n Secton V and conclude the paper n Secton VI. II. NETWORK MODEL AND PROBLEM FORMULATION In ths secton, we frst descrbe the model of cogntve rado networks, and then formulate the problem solved n ths paper. A. Network Model We consder a cogntve rado network whch conssts a prmary user and n secondary users. The set of secondary users s denoted by {SU 1, SU 2,, SU n }. Each secondary user actually conssts of a sender and a recever. The prmary user, the owner of the spectrum, allocates ts unused spectrum to secondary users. Each secondary user s allocated wth a porton of spectrum, namely bands. Each band s assocated wth a bandwdth representng a dvded frequency range. The bandwdth allocated to secondary user SU s noted by p, 1 n. We assume that the prmary user can communcate wth each secondary user drectly. Moreover, the frequency allocated to dfferent secondary user s orthogonal to avod nterference. The network model s llustrated n Fg. 1. B. Problem Formulaton In ths subsecton, we formulate the socal welfare of a cogntve rado network, ncludng two folds: the cost pad by the prmary user for spectrum allocaton and the utlty obtaned by the secondary users for data transmsson. Cost for spectrum allocaton. The spectrum sharng wth secondary users may mpact the data transmsson of the prmary user. Ths leads to a hghly ncreasng cost assocated wth the allocated bandwdth, whch s formulated as an exponental functon. We defne the cost functon as follows, Defnton 1. The cost functon of the prmary user s formulated as p C p) = βe 1 1) where p = p 1, p 2,, p m ) and β s a system parameter. The prmary user expects a payment n return from secondary users for usng spectrum. The prmary user provdes a prce q for the spectrum spent by the SU. Consderng ths payment, we formulate the proft of the prmary user n Defnton 2. Defnton 2. The prmary user obtans a proft for sharng spectrum as Φ p, q) = pq T C p). 2) Utlty for data transmsson. It s ntutve that each secondary user obtans a utlty for transmttng data. Accordng to the bandwdth p allocated to the secondary user, we defne the utlty functon n the followng, Defnton 3. The utlty functon of secondary user SU s defned as u p ) = a log 1 + p ), 3) where a s a user-specfc parameter. Accordng to the law of dmnshng margnal utlty n economcs [13], the log functon mples that the ncreasng rato of utlty gan obtaned by secondary users degrades as the allocated bandwdth ncreases. Note that the parameter a s the prvacy nformaton of secondary user SU. whch s unknown to the prmary user and other secondary users. Thus, the payoff of SU s formulated as follows, Defnton 4. The payoff of secondary user SU s equal to the utlty obtaned mnus the payment pad to the prmary user, P q, p ) = u p ) q p. 4) In ths paper, we consder secondary users are ratonal, tryng to maxmze ther own payoffs gven a prce q by the secondary user. Socal welfare. We ntroduce the concept of socal welfare, whch s equal to the utltes ganed by secondary users

3 3 mnus the cost pad by the prmary user, s formulated n the followng defnton. Defnton 5. The socal welfare of a cogntve rado network s defned as S p) = u p ) C p). 5) In ths paper, we am to maxmze the socal welfare of a cogntve rado network. The problem of socal welfare maxmzaton s defned n the followng. Defnton 6. The problem of maxmzng the socal welfare of a cogntve rado network s defned as max u p ) Cp) 6) s.t. p [, δ ], [1, n], 7) Varable SU n p q β a Cp) Φp, q) u p ) P q, p ) Sp) t q t) p t) ε TABLE I MAJOR ADOPTED NOTATIONS. Descrpton The -th secondary user The number of secondary users The bandwdth of spectrum allocated to the -th secondary user The prce gven to the -th secondary user The system parameter n the cost functon of the prmary user The user-specfc parameter n the utlty of the -th secondary user The cost functon of the prmary user The proft functon of the prmary user The utlty functon of the -th secondary user The payoff functon of the -th secondary user The socal welfare of a cogntve rado network The current number of teratons The prce updated for the -th secondary user n the t-th teraton The value of spectrum computed by the -th secondary user n the t-th teraton The system varable n the prce updatng rule where δ s the upper bound of p, determned by secondary user SU. Note that the problem defned n Defnton 6 s a convex optmzaton problem. Accordng to [14], the problem must have an optmal soluton p = [p 1, p 2,, p m]. The problem of maxmzng the socal welfare n a cogntve rado network should be solved n a dstrbuted manner. Frst, f the problem s centrally solved n one pont, t ncurs a huge computaton load on the sngle pont. Second, the utlty functon of each secondary user s prvacy, whch leads to unknowng the exact formulaton of socal welfare. A. Overvew III. OPTIMAL DISTRIBUTED APPROACH In response to the dffcultes descrbed above, we desgn a dstrbuted algorthm based on a prcng-based decomposton method. The algorthm can acheve the maxmal socal welfare and protect the prvate nformaton for each secondary user at the same tme. We frst convert the orgnal problem n 6) nto an e- quvalent prcng problem. The optmal spectrum allocaton can be obtaned when the optmal prces are acheved. Then, we propose a decomposton framework, whch decouples the optmzaton problem nto several subproblems. In the framework, each secondary user maxmzes ts own payoff based on the prce gven by the prmary user. The prmary user updates the prce after collectng the returned spectrum demand from secondary users. The above operatons are executed teratvely untl the spectrum allocaton converges to the optmal soluton. A prce updatng rule s also provded, gven the nsght of the condton that the optmal prces should satsfy. At last, we theoretcally prove that our teratve algorthm converges to the optmum n fnte steps. The notatons appeared n ths paper are summarzed n Table I. Algorthm 1 Optmal Dstrbuted Algorthm for Spectrum Sharng Input: A cogntve rado network consstng of many secondary users, denoted by {SU 1, SU 2,, SU n }. The cost functon of the prmary user s Cp) and the utlty functon of each secondary user s u p ), {1,, n}. Output: The optmal spectrum allocaton p for each secondary user. 1: t = // t counts the number of teratons. 2: The prmary user sets the same ntal value q ) = θ sent to all secondary users, where θ s a lttle postve real value. 3: Repeat for t =, 1,. 4: For each secondary user, the value of p t+1) computed accordng to p t+1) q ) = arg max u p ) q p. p [,δ ] 5: For the prmary user, accordng to the returned p t+1), the value of q t+1) s updated as q t+1) = 1 ε) q t) + ε C p) p p= p t+1). 6: Untl f p t+1) p t) ζ, where ζ s a tunable lttle real number. 7: p = p t+1), q = q t+1). B. Equvalent optmal prcng problem As mentoned above, the prmary user gves a prce q to each secondary user for leasng spectrum. Accordng to the prce, each secondary user decdes the bandwdth of the spectrum p t demands. Based on the prcng mechansm, the socal welfare can be rewrtten as follows, s

4 4 Fg. 2. SU 1 q 1 p 1 q 2 PU SU 2 p C q 1 q, p q 3 p 2 p 3 SU 3 p q arg maxu p qp p, p p The prcng-based decomposton framework context. The alternate operatons are executed teratvely by the prmary user and secondary users n turn untl the optmal soluton q and p s acheved. Intutvely, the prce updatng rule s sgnfcant to guarantee the teratve operatons eventually converge to the optmal soluton. Here, we propose a prcng updatng rule accordng to the the Karuth-Kuhn-Tucker KKT) condton. Accordng to the convex optmzaton theory [15], the optmal soluton q = [q1,, qn] of 1) must satsfy the KKT condton as shown n the followng proposton. [ Proposton 1. An optmal prce vector q = q 1, n], q should satsfy q = C p) p,, 11) p= p S p) = = = u p ) C p) u p ) p q ) + pq T C p) P q, p ) + Φ p, q) 8) Consderng secondary users are ratonal, they wll choose the bandwdth of allocated spectrum by maxmzng ther own payoffs gven a spectrum prce, p q ) = arg max u p ) q p. 9) p [,δ ] We denote p q ) as p for short n the next context. Therefore, the socal welfare maxmzaton problem 6) can be transformed nto an equvalent optmal prcng problem, whch determnes the optmal prce vector q = [ q 1,, q n] by maxmzng the socal welfare as follows, max q n Φ p, q) + P q, p ) 1) We llustrate why the prcng problem s equvalent to the orgnal problem brefly. Assume that the optmal soluton of problem 1) s q. The secondary user chooses p q ) dependng on the optmal prce q by maxmzng the payoff 9). After knowng p q ) and q, the socal welfare s maxmzed dependng on 8). Therefore, the optmal soluton of p can be obtaned when fndng the optmal prce vector q = [q 1,, q n]. C. Prcng-based decomposton framework In ths subsecton, we ntroduce a prcng-based decomposton framework, whch s shown n Fg.2. Each secondary user computes the value of p as shown n 9) locally and sends the result to the prmary user. Accordng to the collected value of p, the prmary user updates the prce q accordng to a updatng rule, whch s gven n the comng where p = [ p 1,, p n ] T wth p = p q ) gven n 9). Proof: Frst, accordng to the frst order condton, there exsts P p, q )/ p = for 2). Therefore, we obtan q = u p ). 12) Moreover, applyng the KKT condton S p)/ q = to ) 5), there s u p ) Cp) p p q p= p = snce p q <, we have u C p) p ) p =. 13) p = p The proposton s proved by combnng the 12) and 13). Gven the nsght of Proposton 1, we propose a prce updatng rule as follows, q = 1 ε) q +ε C p) p, 14) p= p, where ε, 1] s a tunable parameter. Note that n our decomposton framework, updatng prce s unnecessary to be executed after all p, {1,, n} are returned. The prmary user can update q for secondary user SU arbtrary tmes before q j j ) s updated. It means that our decomposton framework can be realzed n an asynchronous way so that the number of teratons can be largely degraded as dfferent secondary users have dfferent communcaton delay wth the prmary user. D. Dstrbuted Algorthm In ths subsecton, we propose a dstrbuted algorthm to maxmze the socal welfare of a cogntve rado network. In the dstrbuted algorthm, the prmary user and all secondary users should partcpate as follows, For each secondary user SU, t maxmzes the payoff P p, q ) = u p ) q p locally to obtan the value of p based on the prce q gven by the prmary user. Each secondary user only needs to know ts own utlty functon. For the prmary user, t updates the value of q based on p returned from each secondary user. The prce updatng rule s gven n 14).

5 5 The prmary user generates a new prce based on an orgnal prce and partal dervatve Cp) p, whch pushes the prce p= p q towards the optmal value q = Cp ) p. The detals of the dstrbuted algorthm are descrbed n algorthm 1. E. Convergence Analyss Contracton Mappng. Many teratve algorthms can be expressed as x l + 1) = φx l)), l =, 1,, where x ) X and l denotes the number of teratons. Mappng φ s called a contracton f φ x) φ y) κ x y, x, y X, 15) where s a norm and κ [, 1) s called a modulus of φ. Moreover, the mappng φ s called a pseudo-contracton f there exsts a fxed pont x X means x φ x )) and φ x) x x x, x X. 16) The convergence property of contracton or pseudocontractons s gven n Theorem 1. Theorem 1 Geometrc Convergence). Supposng the mappng or the pseudo-contracton and the modulus of φ s κ [, 1). { Then, φ has an} unque fxed pont x and a sequence xl), l =, 1, generated by x l + 1) = φ x l)) satsfes x l) x κ l x ) x, l, 17) for every choce of ntal x ) X. In partcular, x l) converges to x geometrcally. In ths secton, we analyze the convergence of our algorthm under the condton of ε = 1 and ε < 1 n Proposton 2 and Proposton 3, respectvely. For smplcty, the second order partal dervatves of Cp) s denoted by pjp C p) = Cp) p j p. Convergence of Algorthm 1 wth ε = 1. We frst defne a notaton [λ] + to denote the projecton of λ R onto the range [, τ ], [λ] + = arg max z=[, τ ] z λ. Brefly, a soluton to 9) s equvalent to p q ) = [arg max p u p ) q p ] + = [ u 1 q ) ] +. Therefore, when ε = 1, the prce updatng rule turns to q = c p). Substtutng t nto p q ) obtans p = φ p) = [ u 1 c p)) ] +. 18) Algorthm 1 appled wth ε = 1 eventually acheves convergence under a certan condton whch s gven n the followng proposton. Proposton 2. Supposng ε = 1, f we have pj p C p) < mn u p ), p p j=1 [, δ ], 19) )} { p q t) generated by Algorthm 1 converges geometrcally [16] to the optmal soluton p of 6), gvng an ntal value q ). Proof: For each, a functon g s defned as follows. g r) = u 1 c z r))) = u 1 c rx + 1 r) y)), where g r) s dfferentable and r [, 1]. We have φ x) φ y) = [ u 1 c x)) ] + [ u 1 c y)) ] + 1 g 1) g ) = dg r) dr 1 dg r) dr max dg r) r [,1] dr, where the frst nequalty s because [x ] + [y ] + x y for all x y R. Furthermore, applyng a chan rule, we have dg r) dr = n j=1 jc 1 c rx + 1 r) y) x j y j )) u 1 ) c z r))) n j=1 jc z r)) x j y j. If condton 19) holds, we have j=1 j c x) < mn u x )) u x u 1 c x)) ) for all x [,τ ]. Therefore, there exsts a number κ < 1 whch enables that dg r) dr κ max x j y j = κ x y j, r [, 1] Therefore, we have φ x) φ y) κ x y, x, y [, τ ], whch shows φ s a contracton wth modulus κ respect to a maxmum norm. Therefore, a proposton s proved. Convergence of Algorthm 1 wth ε < 1. We relax a value of ε by consderng a convergence condton wth ε < 1. In ths stuaton, we suppose that p q l)) generated wth each teraton l s always wthn a range [,τ ]. Therefore, we can rewrte a soluton to 9) as p q ) = u 1 q ), whch has a lttle dfference from ε = 1 stuaton. Combnng t wth q = 1 ε) q + εc p), we can obtan p = φ p) = u 1 1 ε) u p ) + εc p)) 2) Appled wth 2), the Algorthm 1 can acheve optmal under a convergence condton n the followng proposton. Proposton 3. Set p q ) ) to be for all or δ for all. If we have ) 1 ε 1 + p j p Cp) c p ) j p j p Cp), p [, δ ], 21) )} { p q t) generated by our algorthm 1 converges geometrcally [16] to the optmal soluton p of 6).

6 6 Proof: For each a functon g s defned as follows. g r) = u 1 1 ε) u z r))) + εc z r)),, where z r) = rx + 1 r) y. Supposng x < x, we wll show x < φ x) x. We have φ x) φ x ) = g 1) g ) = where dg r) dr s gven by 22). 1 dg r) dr, dr dg r) dr = u 1 ) 1 ε) u z r)) + εc z r)) ) 1 ε) u z r)) x x ) + ε j c z r)) x ) j x j j = u φ z r)))) 1 1 ε) u z r)) + ε c z r))) x x ) ) +ε j jc z r)) x ) j x j 22) Applyng h 1 ) to both sdes yelds x < φ x). Therefore, there exsts a number κ < 1 enables that φ x) x κ x x, whch s equvalent to φ x) x κ x x, x [, x ]. In other words, φ s a pseudo-contracton n [, x ]. Smlarly, [ f x ] > x, we can get φ, whch s a pseudo-contracton n x, τ. Therefore, the proposton s proved. IV. PERFORMANCE EVALUATION A. Methodology and Smulaton Setup We perform smulatons to evaluate the performance of our optmal dstrbuted algorthm, compared wth a baselne algorthm as follows. Dstrbuted weghted allocaton scheme DWAS). Frst, the prmary user provdes an ntal value q for all secondary users. Each secondary user SU returns the value of p q ) by maxmzng P q, p ). Next, the prmary user decdes the fnal value of q, {1,, n} by maxmzng Φq, p), assumng that u p ) s a lnear functon of p,.e., u p ) = b p. b represents the weght of secondary user SU, whch s computed as p q )/q. After recevng q, each secondary user computes ts optmal p q ). In addton to the DWAS, we also compare our algorthm wth the optmal value. The optmal value s obtaned by maxmzng the socal welfare drectly. The evaluaton of our algorthm s performed from two aspects, ncludng the network performance and the convergence speed. The metrcs of network performance are the socal welfare and the total allocated spectrum. We perform the smulatons wth varyng two factors: the number of secondary users and the value of β. The metrc of convergence speed s the number of teratons. The smulatons are performed under dfferent numbers of secondary users and values of ε. The ntal values of q, {1, 2,, n} n our proposed algorthm and the DWAS are set to 1. The default values of β and ε are.2 and.1, respectvely. The values of a are standard unformly dstrbuted n the open nterval.5, 1). Fg. 3. Socal welfare Algorthm 1ε =.1) Algorthm 1ε =.15) Number of teratons Socal welfare acheved wth the number of teratons B. Impacts of the number of secondary users We frst study the socal welfare and allocated spectrum of our algorthm and the DWAS under the condton of dfferent numbers of secondary users, compared wth the optmum. The results are shown n Fg. 4 and Fg. 7. Fg. 4 shows that the socal welfare obtaned by our algorthm performs as well as the optmum, and much better than the DWAS. The socal welfare obtaned by our algorthm and the DWAS ncreases wth the ncreasng number of secondary users. When there are 2 secondary users, the socal welfare obtaned by our algorthm s 5.34% hgher than the DWAS. In Fg. 7, the spectrum allocated by our algorthm s the same as the optmum, and more than the DWAS. The spectrum obtaned by our algorthm and the DWAS ncreases wth the ncreasng number of secondary users. However, the spectrum allocated by DWAS s always less than the spectrum allocated by our algorthm. When there are 2 secondary users, the allocated spectrum obtaned by our algorthm s 96.23% more than the DWAS. C. Effects of the value of β We next study the performance of our algorthm and the DWAS under the condton of dfferent values of β, compared wth the optmum. The results are shown n Fg. 5 and Fg. 8. Fg. 5 shows that the socal welfare acheved by our algorthm performs as well as the optmum, and more than the DWAS. The socal welfare obtaned by our algorthm and the DWAS decreases wth the ncreasng value of β. When there are 2 secondary users, the socal welfare obtaned by our algorthm s 4% more than the DWAS. In Fg. 8, the spectrum allocated by our algorthm s the same as the optmum, and more than the DWAS. For our algorthm and the DWAS, the allocated spectrum decreases, when the value of β ncreases. When there are 2 secondary users, the allocated spectrum obtaned by our algorthm s 97.43% more than the DWAS. D. Convergence Speed In ths subsecton, we study the convergence speed of our algorthm under dfferent numbers of secondary users and values of ε. The results are shown n Fg. 3, Fg. 6 and Fg. 9.

7 7 Socal welfare Optmum Algorthm 1 DWAS Socal welfare Optmum Algorthm 1 DWAS Number of teratons n = 2 n = 4 n = 6 n = 8 n = Number of secondary users Value of β varepslon =.1 varepslon =.15 Number of secondary users Fg. 4. Socal welfare comparson of dfferent Fg. 5. Socal welfare comparson of dfferent Fg. 6. Convergence of Algorthm 1 wth dfferent algorthms wth varyng number of secondary users algorthms wth varyng β number of secondary users Allocated spectrum Algorthm 1 Optmum DWAS Allocated spectrum Optmum Algorthm 1 DWAS Number of teratons Algorthm 1 wth 6 secondary users Algorthm 1 wth 1 secondary users Number of secondary users Value of β Value of ε Fg. 7. Allocated spectrum comparson of dfferent Fg. 8. Allocated spectrum comparson of dfferent Fg. 9. Convergence of Algorthm 1 wth dfferent algorthms wth varyng number of secondary users algorthms wth varyng β value of parameter ε Fg. 3 plots the socal welfare acheved by our algorthm n each teraton, when the value of system parameter ε s.1 and.15, respectvely. Under the two settngs, we can see that 99.9% of the optmal socal welfare s acheved n the frst ten teratons, whch mples our algorthm converges quckly. In Fg. 6, We study smulatons of our algorthm when the number of secondary users s 2, 4, 6, 8 and 1, respectvely. The value of parameter ε s.1 and.15. The numbers of teratons under dfferent numbers of secondary users and dfferent values of ε are close, whch llustrates our algorthm s adapted to large-scale cogntve rado networks. Fg. 9 shows that the convergence speed of our algorthm under the condton of the value of ε varyng from.1 to.15. The number of secondary users s 6 and 1. The number of teratons under dfferent values of ε s less than 32. The hgher value of system parameter ε, the faster the convergence speed. Our algorthm converges fast under varyng values of ε. V. RELATED WORK Increasng works focus on the problems n cogntve rado networks. Some works focus on estmaton of nterference temperature of the rado envronment and detecton of spectrum holes. such as [17] [18]. Works n [19] [2] [21] focus on estmaton of channel-state nformaton and predcton of channel capacty for usng by the transmtter. Some other works [5] [6] focus on transmt-power control and dynamc spectrum management. The spectrum allocaton n cogntve rado networks has been extensvely studed. By takng advantage of the models n game theory, some of exstng works solve the spectrum sharng problem by achevng Nash Equlbrum. Nyato et al. [7] formulate the problem of spectrum sharng as an olgopoly market competton and use a Cournot game to obtan the spectrum allocaton. However, the statc Cournot game only adapts to some specal envronment. Han et al. [9] propose a correlated equlbrum concept for users to have the dstrbutve opportunstc spectrum access. Wang et al. [8] propose a compettve spectrum-sharng scheme accordng to aucton theorem wthout achevng socal welfare maxmzaton. Some centralzed algorthms n some other works can actually acheve the socal welfare maxmzaton. However, these algorthms could not protect the prvacy of each user. Iler et al. [12] propose a framework under the regulaton of a spectrum polcy server for spectrum sharng. Peng et al. [1] propose a framework that defnes the spectrum sharng problem for some defntons of overall system utlty. The central sever s desgned for allocaton assgnments. Raman et al. [11] propose a centralzed spectrum server that coordnates users to sharng a common spectrum.

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