Joint Pricing and Load Balancing for Cognitive Spectrum Access: Non-Cooperation vs Cooperation

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1 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications Joint Picing and Load Baancing fo Cognitive Spectum Access: Non-Coopeation vs Coopeation Nguyen H. Tan, Membe, IEEE, Long Bao Le, Senio Membe, IEEE, Shaoei Ren, Membe, IEEE, Zhu Han, Feow, IEEE, and Choong Seon Hong, Senio Membe, IEEE, Abstact In the dynamic spectum access DSA, picing is an efficient appoach poviding economic incentives fo opeatos, wheeas oad baancing yieds congestionavoidance incentives fo seconday uses SUs. Despite compexities of i the coupings among picing, oad baancing and SUs spectum access decision, and ii the heteogeneity of pimay uses taffic and SUs casses/types, we tacke the joint oad baancing and picing pobem to opeatos evenue in two cognitive adio makets: monopoy and duopoy. Fo the monopoy maket, we fist show thee exists a unique SUs uiibium aiva ate to the monopoist s channes. We then show that the joint pobem can be soved efficienty by expoiting its convex stuctue. Fo the duopoy maket, we fist chaacteize a unique SUs uiibium aiva ate to two opeatos empoying diffeent DSA appoaches. When two opeatos ae noncoopeative, we show that thee exists a unique Nash uiibium fo each opeato s evenue. When they ae coopeative, we show that the socia evenue optimization can achieve a unique optima soution. Using the Nash bagaining famewok, we aso pesent a shaing contact that detemines the optima faction of the socia evenue fo each opeato. In both makets, we popose two agoithms that can find the agest SU cass suppotabe by the opeatos. Index Tems Picing, Load Baancing, Nash Equiibium, Dynamic Spectum Access, Cognitive Radio. I. INTRODUCTION Dynamic spectum access DSA has been intoduced to efficienty utiize scace wieess spectum that is conventionay contoed via static icensing. Vaious Manuscipt eceived Jan 4, 204; evised May 8, 204 and Ju 5, 204. This eseach was suppoted by Basic Science Reseach Pogam though Nationa Reseach Foundation of Koea NRF funded by the Ministy of EducationNRF-204RA2A2A D. C. S. Hong is the coesponding autho. N. H. Tan and C. S. Hong ae with the Depatment of Compute Engineeing, Kyung Hee Univesity, Koea emai: {nguyenth, cshong}@khu.ac.k. L. B. Le is with INRS-EMT, Univesity of Quebec, Montea, Quebec, Canada emai: ong.e@emt.ins.ca. S. Ren is with Schoo of Computing and Infomation Sciences, Foida Intenationa Univesity, Foida, USA sen@cis.fiu.edu. Z. Han is with the Eectica and Compute Engineeing Depatment, Univesity of Houston, Houston, Texas, USA emai: zhan2@uh.edu. DSA appoaches, incuding two popua dynamic shaeduse and excusive-use paadigms, have been poposed to enabe seconday uses SUs to fexiby access undeutiized egacy spectum that is used spoadicay by pimay uses PUs [] [3]. The shaed use aows SUs to oppotunisticay access the seconday opeatos inteuptibe spectum without haming the PUs activities, wheeas the excusive use aows opeatos to ease pats of a tempoaiy unused spectum i.e. no PUs opeations fo SUs sevice povisioning. In these two paadigms, picing is one effective maket-based method to distibute spectum fom opeatos to SUs since it not ony povides economics incentives fo opeatos, but aso has the ow-ovehead opeation [4] [6]. Howeve, fo a given opeato s pice, vaious SU appications casses with distinguished physica conditions types wi have diffeent spectum access decisions. Theefoe, if the heteogeneity of SUs casses and types is consideed, how to design an efficient picing mechanism to achieve the optima evenue fo opeatos is one of the maket-based chaenges. Whie picing povides an economics incentive fo opeatos, oad baancing, which distibutes the SUs taffic oads to ight channes, povides a congestionavoidance incentive fo SUs [7] [9]. Nevetheess, SUs congestion is infuenced by sevice times of the opeato s channes, which ae affected by vaying PUs taffic pattens. Hence, if the heteogeneity of PU taffic is consideed, how to design a ow-compexity mechanism that can distibute SUs taffic eveny on a channes is one of the oad baancing chaenges. In this wok, by incopoating the heteogeneity of PUs taffic and SUs types and conditions, we study a joint picing and oad baancing spectum access conto in muti-channe cognitive adio netwoks CRNs. This joint pobem can be iustated as a two-eve stuctue in Fig.. At the opeato eve, in evey beginning peiod of a stationay statistics, the opeato wi decide the coesponding pices and oad baancing infomation to its evenue. Based on the infomation, at the SU eve, an aiving SU with a specific appication and physica condition wi decide whethe o not to join the c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

2 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 2 opeato to its expected utiity. We see that how SUs make joining decisions depends on both pice and oad baancing infomation of the opeato, and how the opeato sets its pices and oad baancing depends on SUs joining poicy. Hence, thee ae cetain coupings not ony between picing and oad baancing, but aso between these infomation and SUs joining poicy. Sepaatey consideing ony picing o oad baancing in muti-channe CRNs wi ead to suboptima soutions of the opeato s evenue optimization. Theefoe, by tacking this joint pobem, ou contibutions can be summaized as foows. The fist netwok scenaio is a monopoy maket with one opeato empoying shaed-use DSA. Given the opeato s oad baancing and pices, we fist chaacteize the SUs joining poicy and show that thee exists a unique SUs uiibium aiva ate to a paae M/G/ queueing system modeing the opeato s channes with PU taffic. By integating this uiibium constaint into the opeato s evenue maximization pobem, we then show that this pobem can be soved efficienty by a suentia optimization method, which eveas its convex stuctue. We aso popose an agoithm that can find the agest SU cass suppotabe by the opeato. The second netwok scenaio is a duopoy maket with two opeatos empoying shaed-use and excusive-use DSAs, espectivey. We fist chaacteize the SUs joining poicy and show the existence of a unique SUs uiibium aiva ate to both opeatos. We then investigate thei inteactions though two behavios: non-coopeation and coopeation. In the noncoopeative case, we show that thee exists a unique Nash uiibium. In the coopeative case, we show that the socia evenue optimization pobem can be spitted into two convex pobems that can be soved by each opeato to achieve a unique optima soution. Befoe jointy optimizing the socia evenue, both opeatos can agee on a shaing contact that detemines a faction of the tota evenue fo each opeato. Using the Nash bagaining famewok, we show that thee exists a unique soution of this shaing faction. Finay, we popose an agoithm that can find the agest SUs cass suppotabe by both opeatos. The est of this pape is oganized as foows. Section II pesents eated wok. We anayze the monopoy and duopoy makets in Section III and Section IV, espectivey. Section V povides numeica esuts and Section VI concudes ou wok. The opeato makes its picing and oad baancing decision to its evenue SUs makes thei spectum access and channe seection decision Monopoy Opeato-eve SU-eve Two opeatos make thei picing and oad baancing decisions to thei evenues SUs choose which opeato to join fo spectum access decision Duopoy Fig.. Two-eve stuctue between opeatos and SUs. Due to space imitations, a missing poofs can be found in the technica epot avaiabe onine [0]. II. RELATED WORKS In the iteatue, picing and oad baancing ae two DSA eseach diections. On the one hand, thee ae many inteests in oad baancing spectum conto in muti-channe CRNs [7] [9]. Using the non-peemptive pioity M/G/ queueing mode, [9] poposed a dynamic eaning scheme to detemine a oad baancing stategy that can convege to a Nash uiibium, which is not necessaiy a goba optima point. In contast, with a peemptive esume pioity M/G/ queueing mode, [7] tied to minimize the system time by poviding the optima channe seection soution which, howeve, eies on the numeica optimization that uses a high-compexity exhaustive seach agoithm. Using the M/M/ queueing mode, the ecent wok [8] suggested a ow-compexity optima oad baancing agoithm based on convex optimization theoy; howeve, its channes wee esticted to exponentia distibutions. On the othe hand, picing methods, which addess the DSA economic aspect, have ecenty eceived temendous attention. One of the main inteests incudes easing and picing mechanisms in a thee-tie maket: spectum ownes, opeatos and SUs [] [3]. The othe key diection focuses on the picing schemes in the twotie maket between opeatos and SUs, which eithe chaacteizes the competition between mutipe opeatos [4] [6], mutipe SUs [4], o inteactions between opeatos and SUs [5], [6]. Howeve, most of these papes chaacteize SUs esponses via thei demand functions, such as bandwidth uiement. Thee ae few papes that conside the picing impact on SUs uiibium behavios in a stategic queueing system as compaed with ou wok in which SUs make thei joining decisions stategicay based on thei peceived queueing deay. Picing in the stategic queueing system, oiginated fom [7] see [8] fo the suvey, can be categoized into the obsevabe [9] and unobsevabe queueing systems c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

3 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 3 [20] [22], whee the atte mode is adopted in this wok due to its pactica meaning in the CR context. Resoting on M/M/ anaysis, whie [20] poposed a picing scheme to a monopoist opeato s evenue, [2] accounted fo a sociay-maximizing picing mechanism. Howeve, both conside homogeneous SUs with the same cass and type, which is an ovesimpified mode. The ecent wok [22] investigated not ony evenue but aso sociay-optima picing schemes; howeve, its assumptions ae imited to a singe-channe case and the same type fo a SUs. By taking heteogeneous SUs casses and types into account, we ovecome many pevious simpified assumptions to study the jointy optima picing and oad baancing with ow-compexity mechanisms. This pobem is impotant to muti-channe CRNs, since it can optimay chaacteize both the opeatos economic issue and the SUs spectum access behavios due to thei mutua inteactions. This mutua dependence wi ead to suboptima soutions of the evenue optimization if we conside ony eithe picing o oad baancing issue. III. MONOPOLY In this section, we fist pesent the system mode, how SUs make thei joining poicy, and SUs uiibium with the given pice and oad baancing of the opeato. Based on these infomation, how the opeato s its evenue is studied ate. A. System Mode In this monopoy maket, we conside a netwok that consists of one opeato with mutipe shaed-use channes. A suence of SUs jobs ae assumed to aive at the netwok and each SU wi make a decision as to eithe join this opeato o bak fo its job cf. Fig.2. The mode in this section can be descibed quantitativey as foows. Shaed-use Monopoist Opeato: We assume that the opeato has a set of channes denoted by L = {,..., L} that ae icensed to egacy PUs. Taffic pattens of PUs can be modeed as an ON-OFF enewa pocess atenating between ON busy and OFF ide peiods. On each channe L, the sojoun times of the ON and OFF peiods ae modeed by i.i.d. andom vaiabes.v. Y and Z, with pobabiity density functions pdf f Y y and f Z z, espectivey. ON and OFF peiods ae assumed to be independent with SUs aiva pocess and sevice time. This ON-OFF pocess can be consideed a channe mode fo the SU sevices. This mode captues the ide time peiod in which the SUs can utiize the channe without causing hamfu intefeence Λ+...+ΛK s, p s2, p2 sl, pl Bak Bak Bak X X X YZ Y2Z2 Fig. 2. Cass-k SUs aives with ate Λ k at a monopoy system with L shaed-used channes with PU taffic as ON/OFF pocess and admission pice and oad baancing vectos, p and s, espectivey. and SUs sevice can be inteupted due to incoming PUs taffic with highe pioity. The shaed-use opeato aows SUs to shae the PUs channes oppotunisticay to gain evenue. Befoe each peiod of a stationay distibutions, the opeato boadcasts a oad baancing vecto s = {s } L and an admission pice vecto p = {p } L to a potentia SUs. Whie s with = s = pobabiisticay guides the SUs in seecting channes, p heps SUs decide whethe o not to join the netwok. 2 SUs: We assume that thee is a set of casses of SUs denoted by K = {,..., K} in the netwok whee cass-k SUs aive at the netwok accoding to a Poisson pocess with a potentia ate Λ k, k K. Each cassk SU caies a distinct job e.g. a packet, session, o connection upon aiva and its job is associated with a specific deay-sensitive appication chaacteized by a vaue θ k. Fo exampe, mutimedia appications with stingent deay uiements wi have high vaues of θ k. Without oss of geneaity, we assume 0 < θ <... < θ K. The uested time to compete a SU s job is epesented by a.v. X with pdf f X x. This.v. is assumed to be independent of the aiva pocess. We denote the expectation of any.v. X by X. We futhe assume that each SU beonging to a type α has a monetay evauation vaue α about the opeato s sevice, whee is the opeato s intinsic quaity e.g. the coveage aea and/o the aggegated channes capacity and α is a andom vaiabe chaacteizing the heteogeneity of SUs unde the same opeato s quaity which depends on independent SUs physica conditions such as fast fading, ocation, andom moving, etc. The common technique that can be used to estimate α is conjoint anaysis [23]. We assume that α foows a If two casses m and n have θ m = θ n, we can mege them into one cass with potentia aiva ate Λ m + Λ n. ON OFF c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

4 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 4 unifom distibution on [0, ], which is widey used in the iteatue [24] [26], with thei cumuative distibution function F. One of the main easons to assume α with unifom distibution is fo the anaysis tactabiity. When a potentia SU aives at the netwok, it makes a decision: eithe join the netwok o bak. The utiity of any baking SU is set to zeo. Fo given s and p fom the opeato, the utiity of a cass-k type-α SU that joins channe is modeed by U k, α, s, p = α θ k d s p, k,. This utiity function is peceived by a cass-k type-α SU as the diffeence between net benefit α p and a deay disutiity θ k d s epesenting the deay cost of the SU s job with deay d s in channe and a deay cost pe unit time θ k. We note that this utiity is geneaized to captue the heteogeneity of both use types i.e. α and casses i.e. θ k. In the iteatue, most utiity functions eithe have the same vaue [22] o θ k [27] fo a uses. A vaiant fom is αq p [24], [28], whee the QoS function q pesents an invese effect of ou congestion function d. 3 Steady-State Queueing Deay: Since many SUs may attempt to shae the same icensed channe L, congestion can occu, which wi affect the deay d s of each SU job. An aiving SU at this channe wi be infomed of its job s deay in a queue containing othe SU jobs that aso wish to use that icensed channe. Theefoe, the opeato is assumed to maintain a paae queueing system of L M/G/ queues cf. Fig. 2 whose sevice time of each queue, denoted by a andom vaiabe χ, has a genea distibution dictated by f X x, f Y y and f Z z. We denote T λ the mean steadystate queueing deay i.e. waiting time + sevice time induced by an effective aiva ate λ. Denoting the fist and second moments of channe s sevice time by χ and χ 2, espectivey, we have the extended-vaue mean queueing deay defined as foows accoding to the Poaczek-Khinchin fomua [29] 0 < χ <... < χ 2 L. We assume that a SU can use its spectum sensing and handoff capabiities to detect and potect the PUs. Spectum sensing is used to infom the SU whethe the channe is busy o ide. When the channe is sensed to be ide, the SU job can be in sevice. When the channe is sensed to be busy, the spectum handoff inteupts the cuent SU s sevice, etuns the channe to the PUs, and esumes the SU s sevice when the PU eaves. Using the enewa theoy to hande mutipe inteuptions due to spectum handoffs, we have deived χ and χ 2 in [22] as foows χ = X + Y Z χ 2 = X Y Z, 3 + Y 2 X + gx, 4 Z whee the Lapace tansfom of gx X = x is g s = 2 fz s s 2 Z fz s, 5 and f Z s is the Lapace tansfom of f Z z. Let λ k denote the effective aiva ate of cass-k SUs into the system. Due to the oad baancing conto s, the effective aiva ate into channe is λ s = s Kk= λ k. Then, the utiity function in can be ewitten U k, α, s, p = α θ k T λ s p, k,. 6 B. SUs Decision Poicy and Equibium We assume that the SUs ae ationa decision-makes in that they ony join the netwok when thei utiities ae positive Individua Rationaity. Theefoe, we have: Definition. A cass-k type-α SU in channe with its utiity U k, α, s, p wi foow a joining decision poicy such that it joins the channe if U k, α, s, p > 0, which uies α > α k, λ s, whee T λ = λ χ 2 2 λ χ + χ, if λ < /χ ;, othewise. 2 α k, λ s := θ kt λ s + p ; 7 it baks, othewise. This definition can eiminate the expicit condition λ < /χ in ou aguments heeafte. It tuns out that by deiving χ and χ 2, we can compete the queueing deay mode. Without oss of geneaity, we aso assume 2 If two channes m and n have the same fist moment vaue, we can ode them accoding to the second moment vaue. If they have the same fist and second moments, we can mege them into a vitua channe v with χ v = χ m = χ n such that fo given s v and p v fom the opeato, each of the channes m and n wi have uay s v/2 and p v/ c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

5 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 5 Then, the effective aiva ate of cass-k SU into channe, defined by λ k, := s λ k, is as foows [ ] λ k, = s Λ k P U k, α, s, p > 0 = s Λ k P [α > α k, λ s ] K = s Λ k F α k, λ k= k,. 8 Poposition. Fo given p and s of a shaed-use monopoist opeato, thee exists a unique uiibium aiva ate k, of the cass-k SU into channe such that if p θ k χ, then k, = 0, 9 2 if p < θ k χ, then k, = s K Λ k α k, k= λ k, > 0. 0 C. Opeato s Revenue Maximization In this subsection, we fomuate the evenue maximization pobem and pesent a suentia optimization method, based on which we can achieve the optima soution and agoithm. Pobem Fomuation: At this stage, the opeato tempoaiy assumes that thee exists a unique SUs uiibium k, > 0, k,. Based on this knowedge, the opeato s objective is to its evenue, which can be fomuated as the foowing optimization pobem s, p subject to p = k, = s Λ k s = =, K α k, k= λ k,, k,, 0 s, p 0,. The fist constaint is the SUs uiibium knowedge fom Poposition, wheeas the second constaint is the oad baancing constaint and the thid constaint is the opeationa space of s and p. This pobem is a nonconvex optimization pobem, which is difficut to sove. Fom the fist constaint of, the uiibium aiva ate into channe can be obtained as = K k= λ k, = s Λ ΩT λ + Λp, 2 whee Λ := K k= Λ k and Ω := K k= Λ k θ k. Fom 2, we have p s, = Λ s Ω Λ T λ. 3 Eiminating the fist constaint of pobem by substituting 3 into the objective function, we obtain an uivaent optimization pobem as foows s, ={ } L subject to = λ = s =, Λ 2 Ω s Λ λ T λ 0 s,. 4 We obseve that pobem 4 eveas a stuctue that can be soved efficienty by using a suentia optimization technique as foows. 2 Suentia Optimization: Fist, by fixing, pobem 4 is uivaent to s subject to 2 = Λ s s = =,, 0 s,. 5 It can be seen that 5 is a convex pobem. Using the necessay and sufficient KKT condition, the soution of 5 can be obtained as foows s = =,. 6 Substituting 6 back into 4 and intoducing an auxiiay vaiabe tot = =, we have an uivaent pobem of 4 as foows subject to vaiabes tot Λ = λ tot, 0. λ 2 Ω tot Λ = λ T λ = tot, 7 Lemma. Pobem 7 is a convex optimization pobem. The Lagangian of this pobem is L tot,, µ = L tot λ tot, µ + = L λ, µ, whee L tot λ tot, µ = tot λ 2 Λ tot + µλ tot, 8 L λ, µ = Ω Λ λ T λ µλ,. 9 It is easy to see that L tot λ tot, µ is a sticty concave function of tot fo a given µ, and we aso have L λ, µ is a sticty concave function of,, fo a given µ fom Lemma. Using the fist-ode condition, we can obtain the unique optima soutions fo a given c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

6 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 6 µ as foows tot µ = whee [.] + := max {0,.} and Λ + µ, 20 2 µ = [Φ µ] +,, 2 2 µλ + Ωχ Φ µ :=, 22 ζ µ Ωχ 2 ζ µ with ζ µ = Ω χ 2 2χ 2 foowing popety. 2Λχ µ. We have the Lemma 2. Φ µ is continuous, sticty deceasing, positive on, Ωχ Λ, and non-positive on [ Ωχ Λ, Ω χ Λ χ2,. 2χ 3 Optima Soutions: We can achieve the optima soution tot µ and µ,, by finding the optima dua vaiabe µ that satisfies the fist constaint = µ = tot µ of pobem 7. Thus, we have the foowing esut. Lemma 3. If τ mo := > Ω Λ χ, 23 τ mo, Ωχ Λ thee exist a unique soution µ of = [Φ µ] + = tot µ and a coesponding channe index L L such that Φ µ > 0, L, and Φ µ = 0, > L. If τ mo Ω Λ χ, [Φ µ] + = tot µ = 0,. We futhe iustate this emma numeicay in Fig. 6 in Section V. With this unique µ, we obtain the unique soution tot µ and µ,, as pe 20 and 2, which is aso the goba unique optima soution of pobem 7 since µ, tot µ and µ satisfy the necessay and sufficient KKT condition [30]. Substituting these vaues into 6 and 3, we can achieve s and p,. Poposition 2. With µ and L fom Lemma 3, the oad baancing and picing optima soutions of opeato s evenue maximization pobem ae unique as foows s = Φ µ tot µ, L, 24 p = [p s, Φ µ ] + fom 3, L, 25 s = 0 and p = 0, > L Agoithm: The optima soution of the opeato s evenue maximization pobem povided by Poposition 2 is based on the assumption that k, > 0, k,, which is not aways tue. Theefoe, we popose Agoithm to seach fo a cass K K such that the optima soutions in Poposition 2 coesponds to k, > 0, L, k K. We can conside K the agest cass that can be suppotabe by the opeato. We have the foowing popety on which Agoithm eies. Lemma 4. Defining Λk := k j= Λ j and Ωk := kj= Λ j θ j, Ωk Λk is inceasing in k K. Poof: Since Ωk + = Ωk + Λ k+ θ k+, Λk + = Λk + Λ k+, we have Ωk + Λk + Ωk Λk = Λ k+ Λkθk+ Ωk Λk > 0. Λk + θ k+ Agoithm Optima Picing and Load-Baancing in the Monopoy Maket : The opeato coects χ, χ { 2, } and Λ k, θ k, k 2: i ag max k K > Ωk Λk χ 3: update p i, s i and L i by Poposition 2 with Λi, Ωi 4: whie p i θ i χ fo some L i do 5: i i 6: epeat step 3 7: end whie 8: K i, p p K, s s K, 9: Opeato boadcasts p, s and T λ µ, and a SUs join the netwok by Definition. Poposition 3. If > θ χ, 27 Agoithm aways etuns a cass K. Poof: We pove by contadiction. Assuming that Agoithm cannot etun any optima cass, which means in the wost case with a singe cass k =, we have p θ χ > 0 fo some L i with condition 27, which means, = 0 accoding to 9, impying s = 0 by 6, eading to p = 0 fo theses s accoding to 26, which is a contadiction. Remak. i Since θ χ is the smaest cost that can be expeienced by a potentia SU without queueing deay and with zeo pice, condition 27 pecudes a tivia scenaio whee no SU has any incentive to join the opeato. ii In ine of Agoithm, a paametes can be estimated by the existing method [3] c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

7 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 7 and though some feedback mechanisms fom the SUs. The agoithm then detemines the agest suppotabe cass i ine 2, whee we can aways find such a cass i with condition 27. Accoding to Lemma 4, condition 23 is aways satisfied with Ωj Λj, j i. Thus, by Poposition 2, we aways obtain p j, s j and L j, j i in ine 3 of Agoithm. Hence, the agoithm keeps oweing this agest cass unti, if it is possibe, thee is a cass K and the coesponding L satisfying condition p < θ k χ, L, k K ines 3 to 8 of Agoithm, which induces k, > 0, L, k K accoding to Poposition. iii In ine 3, Agoithm needs to compute µ satisfying Lemma 3. This µ can be found using a bisection method with a constant compexity 3. Hence, Agoithm has a compexity OK. vi The system can opeate on suentia time sots whee Agoithm uns epeatedy in each sot with fixed channe distibutions such that a incoming SUs can eceive boadcast infomation ine 9 fom the opeato in any time sot. IV. DUOPOLY In this section, we fist pesent the system mode and how SUs choose which opeato to join and thei uiibium with the given pice and oad baancing of two opeatos. Based on these infomation, how these two opeatos noncoopeativey and coopeativey, espectivey, thei evenues ae investigated ate. A. System Mode We assume that thee ae two wieess netwok opeatos poviding diffeent DSA modes. The fist opeato, denoted by O, uses the shaed-use mode, wheeas the second opeato, denoted by O 2, empoys the excusiveuse mode. A suence of SUs jobs is assumed to aive at the netwok and each SU wi make a decision as to which opeato to join fo its job cf. Fig. 3. The mode in this section can be descibed quantitativey as foows. Shaed-use Opeato O : The mode of this opeato is simia to Section III-A. Howeve, given s and p fom O, SUs wi see the sevice of O s channes in the aveage sense: the aveage deay = s T λ, and aveage pice = s p. Theefoe, O can simpify the picing stuctue by ony setting a singe pice p instead of p to epace fo = s p. Hence, the utiity of a cass-k type-α SU with O is U,k α, s, p = α θ k = s T λ, p, 28 whee is the intinsic quaity of O s channes and λ, is the effective aiva ate into channe of O. 3 It depends on chosen stating points and a toeance vaue [32]. O s sl s2 p Λ ΛK Bak Fig. 3. A duopoy between shaed-use and excusive-use opeatos. 2 Excusive-use Opeato O 2 : The opeato O 2 is assumed to obtain i.e. via easing the pat of the spectum which is tempoaiy unused by the spectum owne. This spectum chunk is divided into mutipe bands that have the same bandwidth as that of O s channes. Since thee is no PU taffic on these bands, SU sevices ae not inteupted in this case. Wheneve an aiving SU decides to join O 2, the opeato aocates a dedicated channe fo the SU. We assume that O 2 aways has enough dedicated channes to seve the SUs 4. Theefoe, we can conside O 2 to be a M/G/ queueing system whee queueing deays of a SUs ae ua to X. Fom, the utiity of a cass-k type-α SU with O 2 is p2 O2 U 2,k α, p 2 = α 2 θ k X p 2, k, 29 whee 2 is the intinsic quaity of O 2 channes. Since O with inteuptibe sevice aways has highe deay cost than O 2 s dedicated channes, O needs to have bette intinsic quaity in ode to suvive in the maket. Hence, we assume that > 2. An exampe is that O 2 is an incumbent, wheeas O is an entant with a wide coveage aea. B. SUs Decision Poicy and Equibium We denote the type of cass-k citica uses of O and O 2 by α,k and α 2,k such that U,k α,k, s, p = 0 and U 2,k α 2,k, p 2 = 0, espectivey. Since λ, = s Kk= λ,k, we have α,k λ,k = θ k = s T s Kk= λ,k + p, 30 α 2,k = θ kx + p 2, 3 2 whee λ,k = {λ,k } k K is the vecto of effective aiva ate into O of K casses of SUs. We aso denote 4 We can eax this assumption by boowing/easing moe channes fom othe homogeneous opeatos when O 2 acks the dedicated channes [2], [33] c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

8 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 8 the type of cass-k indiffeent use by α k such that U,k α k, s, p = U 2,k α k, p 2. Then we have α,k λ,k 2 α 2,k α k λ,k = Since SUs ae ationa decision-makes, they choose to join O i if thei utiities with O i ae not ony positive Individua Rationaity but aso highe than those of the othe opeato Incentive Compatibiity. Hence, we have Definition 2. A cass-k type-α SU that has U,k α, s, p with O and U 2,k α, p 2 with O 2 wi foow a joining decision poicy such that it joins O if U,k α, s, p > 0 and U,k α, s, p > U 2,k α, p 2, which uies α > α,k λ,k and α > αk λ,k, 33 it joins O 2 if U 2,k α, p 2 > 0 and U 2,k α, p 2 U,k α, s, p, which uies α > α 2,k and α < α k λ,k, 34 it baks if U,k α, s, p 0 and U 2,k α, p 2 0, which uies α α,k λ,k and α α2,k. 35 With SUs joining poicy defined as above, the effective aiva ate of cass-k SUs into O and O 2, espectivey, ae as foows: λ,k = Λ k P [α ] > α,k λ,k and α > αk λ,k { max α k λ,k, α,k [α ] 2,k < α < α k λ,k αk λ,k = Λ k = Λ k λ 2,k = Λ k P λ,k } df α, 36 α 2,k df α. 37 Based on 36 and 37, we have the foowing esut. Poposition 4. Fo a given s, p of O and p 2 of O 2 in a duopoy maket, thee exists a unique pai of uiibium aiva ates,k and λ 2,k of the cass-k SUs k K into O and O 2, espectivey, such that if p 2 β up k s, p, then,k = Λ k F α,k λ,k, 38 2,k = 0, 39 2 if βk o s, p < p 2 < β up k s, p, then,k = Λ k αk λ,k > 0, 40 2,k = Λ k αk λ,k α2,k > 0, 4 3 if p 2 β o k s, p, then whee β up,k = 0, 42 2,k = F α 2,k, 43 k s, p = θ k 2 = s χ X β o k s, p = θ k = s χ X + 2 p, p. We see that thee is an inteaction between O and O 2 s decisions on s, p and p 2, espectivey. If p 2 is geate than a vaue β up k s, p, O becomes a monopoist as case. In contast, if p 2 is ess than a vaue βk o s, p, O 2 becomes a monopoist as case 3. Theefoe, we can conside βk o s, p and β up k s, p the owe and uppe theshods, espectivey, fo the opeationa ange of p 2 in ode to have duopoy coexistence between O and O 2 in case 2. Cases and 3 coespond to the shaed-use and excusive-use monopoist anayzed in Section III and [22], espectivey. Hencefoth, we wi focus ony on the duopoy maket of case 2. If the condition of case 2 is satisfied k, the tota uiibium aiva ates to O and O 2 ae, espectivey, as foows: = K k= λ,k = Λ α, 44 2 = K k= λ 2,k = α α2, 45 whee α = K k= Λ k α,k, α 2 = K k= Λ k α 2,k, and α = Kk= Λ k α k. C. Duopoy: NonCoopeative Opeatos In this subsection, we study the noncoopeative case as a one-shot game and chaacteize the Nash uiibium of this game. Game Fomuation: Based on the SUs uiibium aiva ates in 44 and 45, the opeatos wi compete with each othe to thei evenues, which can be modeed as the foowing one-shot game: Payes: O and O 2, Stategies: O detemines s and p ; O 2 detemines p 2, Payoff functions: π s, p ; p 2 = p and π 2 p 2 ; s, p = 2 p 2. In ode to find the Nash uiibia of this game, we investigate the opeatos best esponses fist c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

9 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 9 2 O s best esponse: The foowing optimization pobem captues O s best esponse s, p subject to p = Λ α, = s =, 0 s,. 46 The fist constaint is uivaent to p λ, s; p 2 = p2 2 Λ λ Ω X Λ s = T s. 47 Eiminating the fist constaint by substituting 47 into the objective function of pobem 46 and intoducing the new vaiabes, := s,, 48 we obtain the foowing uivaent optimization pobem ΩX max. Λ + p λ 2 Λ Ω Λ = λ, T, s.t. = λ, = λ,, 0,, va.,, = {λ, } L. 49 We see that the pobem 49 has the same convex stuctue as pobem 7. Simiay, using the fist-ode necessay and sufficient condition, we have p 2; ν = XΩ + Λν + p , ν = [Φ ν] +,, 5 whee ν is a dua vaiabe associated with the fist constaint. Then, O s best esponse with a given ν is defined as foows whee BR p 2 ; ν := { s p 2 ; ν, p p 2 ; ν}, 52 s p 2 ; ν = Φ ν p,, and 53 2; ν p p 2 ; ν := p p 2; ν, s p 2 ; ν 54 fom 48 and 47, espectivey. Then we can find ν, which is the soution of the foowing uation = [Φ ν] + = p 2; ν, 55 such that p 2; ν and, ν = [Φ ν ] +, ae the unique optima soutions of pobem 49. Theefoe, O s best esponse, which is the optima soution of 46, is BR p 2 ; ν. 3 O 2 s best esponse: The foowing optimization pobem captues O 2 s best esponse p 2 0 subject to 2 p 2 2 = α α2. 56 Eiminating the fist constaint by substituting into the objective function and using the fist-ode condition, we obtain the best esponse of O 2 BR 2 s, p := p 2 s, p = Ω 2 L 2Λ s = T s X + 2 p Nash Equiibium: Based on the best esponses of O and O 2, we can find the Nash uiibia of this game, denoted by s na, p na and p na 2, though the intesections of two best esponses 52 and 57. Specificay, fo a given ν, any pai of s na, p na and p na 2 must satisfy s na, p na = BR p na 2 ; ν, 58 p na 2 = BR 2 s na, p na. 59 Substituting 58 into 59, we have p na 2 = BR 2 BR p na 2 ; ν, 60 Substituting 52 into 57 to sove 60, we see that thee exists a p na 2 fo a given ν as foows p na 2 ν = 2ν Ω X Λ Fom 6, we see that p na 2 ony depends on dua vaiabe ν of pobem 49. Theefoe, if we can find a condition such that a unique ν na exists, then we woud have a coesponding unique p na 2 νna. By substituting 6 into 50, we have p na 2 ν; ν = Λ2 2 ν + XΩ + 2Λ Denoting ν = λ p na 2 ν; ν, we see that ν must satisfy the constaint 55. We have the foowing esut, which can be poved simiay as Lemma 3. Lemma 5. If τ na := XΩ + 2Λ 2 > Ω Λ2 2 Λ χ, 63 thee exists a unique soution ν na τ na, Ωχ Λ of = [Φ ν] + = ν and a coesponding channe c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

10 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 0 index L L such that Φ ν na > 0, L, and Φ ν na = 0, > L. If τ na Ωχ Λ, [Φ ν] + = ν = 0,. Then, we can obtain a unique p na 2 = [pna 2 νna ] + fom 6. Substituting ν na and p na 2 into 58, we can obtain a unique s na, p na. Theefoe, we have the foowing esut. Poposition 5. With ν na and L fom Lemma 5, thee exists a unique Nash Equiibium in a noncoopeative duopoy maket as foows p na 2 = [p na 2 ν na ] + fom 6, 64 s na = Φ ν na, ν na L and s na = 0, > L, 65 p na = [p ν na ; p na 2 ] + fom D. Duopoy: Coopeative Opeatos In this subsection, we study the coopeative case whee both shaed-use and excusive-use opeatos jointy the socia evenue based on a shaing contact ageement. Socia Revenue Maximization: Based on the knowedge of SUs uiibium in the duopoy coexistence case, the socia evenue can be defined as the tota evenue π s, p, p 2 := p + 2 p 2 that both opeatos can achieve with a given setting s, p and p 2. Theefoe, the socia evenue optimization can be fomuated as foows s, p, p 2 subject to p + 2 p 2 = Λ α, 2 = α = s =, 0 s,. α2, 67 Pobem 67 is a non-convex pobem, which is difficut to sove efficienty. Fotunatey, we can spit it into sepaate subpobems which can be soved efficienty as foows. 2 Sepaate Subpobems: The fist constaint of 67, which can be expessed accoding to 47, is eiminated by substituting 47 into p of the objective function. The second constaint can aso be eiminated by substituting it into 2 of the objective function. When a new vaiabe, = s is futhe intoduced, the oigina pobem 67 can decomposed into two sepaate optimization pobems. The fist pobem is λ,, subject to Ω Λ X λ 2 Λ = λ, T Ω Λ = λ, = λ,, 0,, 68 and the second pobem is p 2 0, Λ 2 ΩX 2 p 2 Λ 2 p Pobem 69 is a singe-vaiabe quadatic optimization, which is easy to sove. Pobem 68 has the same convex stuctue as pobem 7, which can be soved simiay. Denoting a given dua vaiabe of this pobem by ξ, we have the foowing esut due to the fist-ode condition ξ = Λξ + XΩ + Λ 2, , ξ = [Φ ξ] +,. 7 We have the foowing esut, which can be poved simiay as Lemma 3. Lemma 6. If τ co := Ω Λ X + 2 > Ω Λ χ, 72 thee exists a unique soution ξ τ co, Ωχ Λ of = [Φ ξ] + = ξ and a coesponding channe index L L such that Φ ξ > 0, L, and Φ ξ = 0, > L. If τ co Ωχ Λ, [Φ ξ] + = ξ = 0,. 3 Optima Soutions: It is easy to obtain the optima soution of 69, denoted by p co 2, by the fist-ode condition. The optima soution of 68 can aso be obtained simiay as that of 7 though ξ in Lemma 6. By substituting these optima soutions of 68 and 69 into 47, we obtain the O s optima pice, denoted by. Hence, we have the foowing esut. p co Poposition 6. With ξ and L fom Lemma 6, thee exists a unique optima soution of the socia evenue maximizing 67, denoted by s co, p co ; pco 2, in the coopeation duopoy maket as foows [ p co 2 2 = 2 Ω ] X +, Λ 2 73 s co = Φ ξ ξ, L and s co = 0, > L, 74 p co = [p ξ ; p co 2 ] + fom c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

11 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 4 Shaing Contact: Afte jointy achieving the optima socia evenue vaue π co := π s co, p co, pco 2, two opeatos wi decide the faction of π co that each of them wi eceive based on a shaing contact γ. Suppose that O eceives its evenue π co = γπ co, then O 2 eceives its shae π2 co = γπco. The pobem becomes how to find a vaue γ that satisfies both opeatos. Based on the Nash bagaining theoy [34], a pope γ can be a soution of the foowing optimization pobem γπ co π na w γπ co π na w2 2 γ [0,] subject to γπ co π na, γπ co π2 na, 76 whee w and w 2 ae weight vaues epesenting the bagaining powe of O and O 2, espectivey, and π na, πna 2, the evenues of O and O 2 at the Nash uiibium, is a disageement point. The fist and second constaints captue the coopeation incentive in that coopeative shaing evenues must be at east ua to the evenues obtained in the noncoopeative scenaio. Denoting the set of a possibe evenues that two opeatos can achieve by S, we have S = {π co, π co 2 π co + π co 2 = π co, π co 0, π co 2 0}, 77 which is a convex set. In addition, the feasibiity of pobem 76 guaantees a foowing unique soution. Poposition 7. Pobem 76 has a unique optima soution such that a if π co = π na + πna 2, then γ = πna π, co 2 if π co > π na + πna 2, then γ = πna π co + w π co w + w 2 E. Duopoy: Agoithms π na + πna 2 π co. 78 The optima soutions povided by Popositions 5 and 6 ae based on the assumption,k, 2,k λ > 0, k K, which is not aways tue. Theefoe, simia to the monopoy case, we popose Agoithm 2 fo the duopoy maket to seach fo the agest cass K K that can be suppotabe by the opeatos such that the optima soutions in Popositions 5 and 6 coespond to λ,k, 2,k λ > 0, k K. Poposition 8. If θ χ > 2 θ X 79 { 2 X and > max, X }, 80 χ /2θ χ Agoithm 2 Optima Picing and Load-Baancing in the Duopoy Maket : Opeatos coect χ, χ { 2, and Λ k, θ k, k } 2: i ag max k K 2 > Ωk Λk χ X 3: Update s co i, p co i and pco 2 i by Pop. 6 with Λi, Ωi 4: whie p co 2 i βo i s co i, p co i do 5: i i 6: epeat step 3 7: end whie 8: K i, p co pco K, s co s co K and p co 2 p co 2 K 9: Opeatos boadcasts p co, = s co T s co ξ, p co 2 and a SUs join the netwok by Definition 2. 0: % Steps to 9 ae appied to coopeative duopoy; fo the noncoopeative duopoy, a of steps ae the same except epacing evey p co, s co, ξ, p co 2 by p na, s na, ν, p na 2 and Pop. 6 by Pop. 5 in ine 3. % Agoithm 2 aways etuns a cass K such that,k, λ 2,k > 0 fo both coopeation and k K noncoopeation. Remak 2. It is cea that if 79 is vioated, no SUs has any incentive to join O. Theefoe, whie condition 79 says that must be age than 2 an amount at east θ χ X fo O to have the maket shae instead of being eiminated by O 2, 80 povides an uppe bound on to sufficienty guaantee fo both noncoopeative and coopeative duopoy coexistence. We expess the intuition of Agoithm 2 in the coopeative case since the othe case can foow the same ines of agument. In ine 2, with condition 79 we can aways find such a cass i. Accoding to Lemma 4, condition 72 is aways satisfied with Ωj Λj, j i. Thus, by Poposition 6 we can aways obtain s co j, p co j and pco 2 j, j i in ine 3. Based on the obsevation that p co 2 i is deceasing by Lemma 4 and Poposition 6, the agoithm keeps oweing this agest cass unti, if it is possibe, thee is a cass K satisfying condition p co 2 K > βk o s co K, p co K ines 3 to 8 in Agoithm 2, which guaantees,k, 2,k λ > 0, k K by Poposition 8. Simia to Agoithm, Agoithm 2 has the compexity OK. V. NUMERICAL RESULTS In this section, we appy the anaytica esuts to numeicay iustate the opeatos optima soutions by Agoithms and 2 coesponding to the monopoy and duopoy scenaios, espectivey c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

12 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 2 s p Ch. Ch. 2 Ch. 3 Ch. 4 Ch s a p π mo and U mo mo mo π mo U mo b Fig. 4. a The optima oad baancing and picing soutions of the monopoist in the setting ExpE eft coumn pots and UniExp ight coumn pots, b The monopoy evenue and the aggegate SUs utiities with the setting ExpE eft pot and UniExp ight pot. Fist, we conside 5 casses of SUs, epesented by θ, θ 2,..., θ 5 = 0.2, 0.4, Futhemoe, Λ k foows a unifom distibution on [0, 3] fo k =,..., 5. Second, we conside a shaed-use opeato with five channes. In the fist setting temed ExpE, X has the exponentia distibution with f X x = µ X e µxx, wheeas Y and Z have the Eang distibutions with f Y y = µ 2 onye µony and f Z z = µ 2 off ze µoffz,, espectivey. We set µ X to i.e. X = and the PU activities fom channe to 5 ae set to µ on, µ off =.5, 0.5,.2, 0.8,.0,.0, 0.8,.2 and 0.5,.5. In the second setting temed UniExp, X is unifomy distibuted on [0.,.9] i.e. X =, wheeas Y and Z have exponentia distibutions with f Y y = µ on e µony and f Z z = µ off e µoffz,, espectivey, whee the PU activities fom channe to 5 ae µ on, µ off =.4, 0.6,.3, 0.7,.0,.0, 0.7,.3 and 0.4,.6. The PU channes in both settings mode the inceasing PU occupancy, i.e., ight to heavy PU taffic. Fom 3 and 4, χ and χ 2 of ExpE ae.25,.54, 2.0, 2.85, 5.0 and 3.52, 5.73, 0.33, 22.37, 72.38, espectivey; and those of UniExp ae.33,.66, 2.0, 2.5, 4.0 and 2.7, 4.64, 7.08,.69, 32.32, espectivey, fo =,..., 5. We can see that fo a channes, the second moments of case ExpE ae geate than those of UniExp. Finay, fo the excusive-use opeato, we simpy set X = to iustate the same bandwidth fo a channes of O and O 2. A. Monopoy Fig. 4a shows a sampe of the optima oad baancing and picing soutions of the monopoist in both settings ExpE and UniExp when is vaied. We can see that when is sma, ony those channes s with ow vaue θk ExpE UniExp Fig. 5. The agest cass vaue θ K that can be suppotabe by O using Agoithm. 5 = [Φ µ] + and totµ [Φ µ] + =,...,5 5 = [Φµ]+ totµ µ Fig. 6. Numeica iustation of Lemma 3 in the setting ExpE with = 2. The ines of [Φ µ] + ae continuous and deceasing to 0 at Ωχ Λ, =,..., 5, espectivey. The functions 5 [Φ µ] + and totµ intesect at a unique µ that coesponds to L = 2. χ and χ 2 ae activated with s > 0, p > 0; and the setting ExpE, with geate channes vaiabiity second moments, has ess activated channes than those of UniExp e.g. when = 2, ony channes and 2 of µ c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

13 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 3 O2 evenue π na, πna 2 π co, πco 2 i π co, πco 2 ii π co, πco 2 iii O evenue a O2 pice p na, p na 2 p co, p co O pice Fig. 7. Duopoy pefomance with 5 SUs casses: a Non-coopeative evenues π na, π2 na and coopeative evenues π co, π2 co with thee diffeent weight settings w, w 2: i ξ p co, 2 ξ p co 2, ii p co L + = sna T s na λ, 2 p, and iii co p co 2 +X L = sna T s na λ, co p2, 2 X b Optima Picing. b 20 5 non-coopeative coopeative 20 5 non-coopeative coopeative θk 0 θk a b Fig. 8. The agest cass K of Agoithm 2 with 40 SUs casses: a is fixed at 50, 2 vaies fom 4 to 47, b 2 is fixed at 45, vaies fom 48 to 54. ExpE and channes, 2 and 3 of UniExp ae active. Fo these activated channes, it is cea that the channe with ow vaue χ and χ 2 wi have high vaue s. When inceases, the numbe of activated channes and the optima pices aso incease in both settings. Futhemoe, we obseve that the oad baancing soution conveges to a fixed distibution when keeps inceasing. In Fig. 4b, we futhe examine the eationship between the monopoy evenue, π mo, and the coesponding aggegate utiity of α k, U k, αdα, via the opeato s SUs, U mo := k quaity. When inceases, we obseve that whie π mo inceases with a sighty inceasing sope, U mo fist inceases with a shapy inceasing sope up to a citica vaue of = 26, then changes to a ineay inceasing state. It can be expained that: Fist, not ony K inceases i.e. moe casses ae suppoted, but aso the utiity of each cass k K inceases when is inceased to the citica vaue 26, whee the maximum numbe of casses K = 5 is achieved. Second, when is inceased past this citica vaue, whie the majoity of the utiities of ow casses i.e. sma k continue inceasing, some highe casses stat to decease thei utiities because the incease of vaue does not compensate fo thei high cost θ k T λ s + p due to high vaues of θ k and p. Fig. 5 shows the agest cass θ K that can be suppotabe using Agoithm. In both settings, we can see that the highe vaue, the highe cass-k SUs that can be admitted into the monopoy netwok. We aso obseve that the UniExp setting with owe channes vaiabiity can suppot highe cass SUs than ExpE does. Fig. 6 iustates Lemma 3 in the setting ExpE when = 2. With the unique vaue µ, we can see that L = 2 whee [Φ µ] + = 0 fo = 3, 4, 5 and [Φ µ] + > 0 fo =, 2, which eads to s > 0, =, 2, in the top eft gaph of Fig. 4a with = c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.

14 This atice has been accepted fo pubication in a futue issue of this jouna, but has not been fuy edited. Content may change pio to fina pubication. Citation infomation: DOI 0.09/JSAC , IEEE Jouna on Seected Aeas in Communications 4 TABLE I MONOPOLY AND DUOPOLY REVENUE COMPARISON, 2 π mo, π2 mo π na, π2 na π co, π2 co π co, π2 co setting i setting ii 30, 28 3, 303, 3 5, 26 49, 82 3, 29 32, 35, 32 5, 32 52, 85 32, 30 34, 326, 32 5, 37 54, 88 33, 3 35, 337, 32 5, 42 56, 9 34, 32 37, 349, 33 6, 47 59, 93 35, 33 38, 349 2, 44 8, 59 60, At ine 2 of Agoithm 2, we see that θ K depends on 2, given fixed channes distibutions. Hence, θ K deceases when 2 deceases in Fig. 8a and θ K inceases when 2 inceases in Fig. 8b. Futhemoe, the noncoopeation aways maintains a highe index K than that of the coopeation case since noncoopeative opeatos ty to attact as many SUs casses as possibe eading to thei pice eduction, wheeas the coopeative ones consevativey educe the agest suppotabe cass K to incease thei pices to thei tota evenue. B. Duopoy The duopoy pefomance is pesented in Fig. 7, whee is fixed at 50 and 2 is vaied fom 4 to 47 whee the stating points ae maked by back squaes. We note that these vaues of and 2 satisfy 79 and 80. Fig. 7a shows the evenue compaison between noncoopeation and coopeation. In the case of coopeation, we conside thee diffeent weight settings w, w 2 to chaacteize the effect of shaing contact: setting i coesponds to the evenues without shaing contact, setting ii eates to the intinsic quaity pe cost and setting iii can be consideed as pice pe QoS. We can see that the coopeation aways gains moe evenues fo both opeatos than the non-coopeation; especiay when 2 is age, the gain is significant. When 2 inceases, the Nash uiibium of both opeato s evenues decease, since they competitivey educe thei ow pices to attact moe SUs in Fig. 7b. Fig. 7b aso shows that whie O 2 inceases its pice, O deceases its pice to coopeativey thei socia evenue. In this case, Fig. 7a shows that the socia evenue of setting i keeps deceasing O s evenue and inceasing O 2 s evenue, which is ceay not favoed by O ; wheeas settings ii and iii dive thei socia evenue in a simia diection that can satisfy both opeatos. We aso obseve that a casses ae suppoted by both opeatos with a vaues 2 i.e. θ K = 3. We continue to compae the evenue gain between monopoy and duopoy in Tabe I, whee we incease the pai of, 2 such that thei diffeence is a fixed vaue and satisfies Poposition 8. Since shaing the maket means osing evenue, we ceay see that the sum of monopoy evenues is age than that of the duopoy in a cases. To iustate the effect of the agest cass K admission, we change the setting to 40 casses of SUs with θ, θ 2,..., θ 40 = 0.5,, and Λ k foows a unifom distibution on [0, 20] fo k =,..., 40. With this new setting, Fig. 8 shows the agest cass vaue θ K that can be suppotabe by O and O 2 using Agoithm VI. CONCLUSION Taditionay, picing and oad baancing ae designed sepaatey fo dynamic spectum access conto. We pefom a joint optimization in two netwok makets: monopoy and duopoy. In both scenaios, we fist addess the heteogeneous muti-cass SUs uiibium behavio as a constaint in the opeatos evenue optimization pobem that can be decomposed into smae pobems eveaing thei convex stuctues. Based on that, we next povide the unique optima picing and oad baancing soutions not ony fo the monopoist s evenue but aso fo the duopoy s Nash uiibium and socia evenues. We finay popose two agoithms to find the agest suppotabe SUs cass in both scenaios. Numeica esuts ae povided to vaidate ou anaysis and show that by coopeation, both opeatos can enhance thei evenues significanty when compaed with noncoopeation. Ou mode can be extended to the cases whee both opeatos ae shaed-use o both ae excusive. Howeve, fo an oigopoy case with any finite opeatos, the mode becomes compicated and needs a diffeent appoach fo tactabe anaysis. REFERENCES [] Q. Zhao and B. Sade, A suvey of dynamic spectum access, Signa Pocessing Magazine, IEEE, vo. 24, no. 3, pp , May [2] M. Buddhikot, Undestanding dynamic spectum access: Modes, taxonomy and chaenges, in Poc. IEEE DySPAN, Dubin, Ieand, Ap. 2007, pp [3] E. Hossain, D. Niyato, and Z. Han, Dynamic Spectum Access and Management in Cognitive Radio Netwoks. Cambidge Univesity Pess, [4] Y. Xing, R. Chandamoui, and C. Codeio, Pice dynamics in competitive agie spectum access makets, IEEE J. Se. Aeas Commun., vo. 25, no. 3, pp , Ap [5] D. Niyato and E. Hossain, Competitive picing fo spectum shaing in cognitive adio netwoks: Dynamic game, inefficiency of Nash uiibium, and cousion, IEEE J. Se. Aeas Commun., vo. 26, no., pp , Jan [6] L. Yang, H. Kim, J. Zhang, M. Chiang, and C. W. Tan, Picingbased spectum access conto in cognitive adio netwoks with andom access, in Poc. IEEE INFOCOM, Shanghai, Ap. 20, pp c 203 IEEE. Pesona use is pemitted, but epubication/edistibution uies IEEE pemission. See fo moe infomation.