Joint Spectrum Access and Pricing in Cognitive Radio Networks with Elastic Traffic

Transcription

1 Joint Spectrum Access and Pricing in Cognitive Radio Networks with Eastic Traffic Joceyne Eias University of Bergamo E-mai: Fabio Martignon University of Bergamo E-mai: Abstract This paper studies the economic interactions between Secondary Users and Primary Operators in a Cognitive Radio Network scenario. Secondary Users transmit their traffic, eventuay spitting it over mutipe avaiabe frequency spectra, each owned by an independent primary network operator. Users are charged a fixed price per unit of bandwidth used, and face spectrum access costs. The transmission rate of each secondary user is assumed to be function of network congestion (ike for TCP traffic) and the price per bandwidth unit. Primary operators se spare bandwidth to secondary users, and set spectrum access prices to maximize their revenue. We provide sufficient conditions for the existence and uniqueness of the Nash equiibrium considering a pecuiar cass of spectrum pricing functions, viz. poynomia functions, which ead to efficient spectrum aocation, and we derive optima price and spectrum aocation settings. Finay, we discuss numerica cognitive radio network exampes that provide insights into the mode s soution. Index Terms: - Cognitive Radio Networks, Spectrum Access, Pricing, Stackeberg Game, Eastic Traffic. I. INTRODUCTION Cognitive Radio Networks (CRNs) can provide high bandwidth to mobie users via heterogeneous wireess architectures and dynamic spectrum access techniques. The inefficient usage of the avaiabe spectrum [1] can be improved through opportunistic access to the icensed bands without interfering with the existing users []. In a CRN, a primary (or icensed) user has a icense to operate in a certain spectrum band; his access is generay controed by the Primary Operator (PO) and shoud not be affected by the operations of any other unicensed user. On the other hand, unicensed (secondary) users have no spectrum icense, and they impement additiona functionaities to share the icensed spectrum band without interfering with primary users. This work focuses on the joint dynamic spectrum access and pricing probems in cognitive radio networks using Game Theory. We consider mutipe primary operators, which are owners of spectrum bands and set prices for their icensed spectrum, and a set of secondary users, which compete in a non-cooperative way for the spectrum bands eft avaiabe by primary users. As a consequence, Game Theory is the natura framework to study the interactions among such users. Non-cooperative games for competitive spectrum access in cognitive radio networks have been recenty considered in [], [3], [4], [5], [6], [7], [8]. The works in [7], [8], which propose static and dynamic spectrum sharing schemes as we as spectrum pricing techniques, are reated to our work, but they present the foowing imitations: (1) ony one primary operator is considered, whie it woud be more reaistic to take into account different primary operators that provide spectrum access opportunities to the secondary users, and () infinitesimay sma secondary users are assumed; secondary users form a secondary service, which is represented by a demand function. However, it is more interesting to study a more genera scenario, where mutipe primary operators coexist and provide spectrum access to a finite set of secondary users characterized by eastic traffic demands. This paper overcomes these imitations by proposing a nove game theoretic mode that soves the joint probem of spectrum aocation and price setting, considering both mutipe Primary Operators (POs), which set prices for their icensed spectrum, and a finite set of Secondary Users (SUs), which are characterized by eastic traffic demands that can be transmitted over one or mutipe frequency spectra. We mode the interaction between POs and SUs as a Stackeberg game [9]: POs set their prices and SUs respond by presenting a certain amount of fow to the network. Secondary users do not cooperate among themseves, thus eading to a Nash game. We study anayticay the existence and uniqueness of the Nash Equiibrium Point (NEP) for our Stackeberg game. Next, we obtain expicit expressions for the secondary users fows and the optima prices, aong with the conditions under which the NEP exits and is unique. Finay, we anayze and discuss severa numerica exampes that iustrate the foowing resuts: (1) the optima price that primary operators shoud impose is independent of the number of secondary users, and it is proportiona to the average utiity per traffic unit experienced by such users; () as the number of secondary users grows, the revenue achieved by primary operators increases unti it reaches a saturation point. The paper is structured as foows: Section II introduces the cognitive radio network mode, incuding POs and SUs objective functions, as we as the considered cost functions. Section III demonstrates the existence and uniqueness of the Nash Equiibrium Point, whie Section IV computes the NEP aong with the conditions under which such equiibrium exists and is unique. Section V determines the NEP for the specia case of symmetrica secondary users. Section VI discusses numerica exampes that iustrate how our mode captures in-

2 teresting aspects of the interaction between primary operators and secondary users. Finay, Section VII concudes this paper. II. NETWORK MODEL The adopted mode is a cognitive radio wireess system withasetn {1,...N} of Primary Operators (POs), each operating on a separate frequency spectrum, F n, and a set U {1,...I} of Secondary Users (SUs), wiing to utiize the frequency spectra {F 1,...F N }. Each Secondary User can transmit simutaneousy over mutipe spectrum bands, spitting his traffic over the set of avaiabe channes, thus choosing which primary operators wi transport his traffic. Users demands are eastic (ike for TCP traffic), in the sense that they are function of the prices set by POs and the costs due to ink congestion, as we as of the utiity perceived in transmitting the traffic over the avaiabe channes. Let f i denote the expected fow that SU i sends on wireess channe. The secondary user fow configuration f i {f1,...,f i N i } is caed a spectrum access strategy of SU i, and the set of strategies H i {f i R N : f i, N}is caed the spectrum access strategy space of SU i. The system fow configuration f {f 1,...,f I } is caed a spectrum access strategy profie, and takes vaues in the product strategy space H. Furthermore, et f i represent the fow configuration of a users except SU i. Each secondary user i U maximizes his degree of satisfaction (his objective function F u,i, defined in the foowing), which we assume has a component reated to the throughput (utiity) and another one reated to costs (disutiity). On the other hand, each Primary Operator behaves sefishy, and pays to maximize its own profit. Each PO defines its own price strategy p R, where p represents the price per bandwidth unit charged by PO for a the traffic that passes through wireess channe F. The coection of the PO strategies buids up the price strategy profie p (p 1,...p N ) R N.Letp represent the price strategy of a POs except PO. The vector (f,p) R IN+N grouping both the spectrum access and price strategy profies is referred to as network strategy profie, and represents the soution of the game where both SUs and POs operate simutaneousy. A. Secondary User Objective Function We associate to SU i Uthe objective function F u,i, which is a function of the fow transmitted on each wireess channe as we as of the prices set by Primary Operators: F u,i (f i,f i,p)q i (f i ) [ f i J i (f i,f i )+ N + f i p (f i,f i )]. (1) N The first term, Q i (f i ), represents the utiity for transmitting a tota amount of fow N f i f i. The second term encompasses two components: the first one, N f i J i(f i,f i ),is the disutiity due to ink deay and the second one, N f i p (f i,f i ) corresponds to the price paid by secondary user i to the Primary Operators. Reca that p (f i,f i ) is the price per bandwidth unit set by the th PO. In the foowing we assume that Q i (f i ) is equa to N αi f i and p (f i,f i ) is constant and equa to p ; α i therefore represents the utiity of SU i per unit of transmitted fow, on wireess channe. Hence, the objective function of SU i becomes as foows: F u,i (f i,f i,p) α i f i f i [J i (f i,f i )+p ]. () N N Each SU i maximizes his objective function F u,i over a his fow configurations: { max α f i i f i } f i [J i (f i,f i )+p ], (3) N N where p is the optima price set by Primary Operator and f i are the optima fows of a secondary users j U, with j i. B. Primary Operator Utiity Function The utiity function of the th Primary Operator, U PO,is given by U PO (f,p,p )p f, where f f i is the tota amount of fow on wireess channe. Each SP maximizes its utiity U PO over a its price strategies: max p {p f }. C. Poynomia Link Costs Foowing the guideines of [1], this work considers a cass of poynomia ink cost functions originay adopted in the context of road traffic modeing [11]. Such costs have appeaing properties that ead to predictabe and efficient network fows, ensuring the uniqueness of the Nash equiibrium point. More specificay, we assume that each wireess channe is characterized by the foowing cost function: cost() a (f ) β() + b, N, (4) where a, b and β() are channe-specific positive parameters, and f, as defined above, is the tota amount of fow that is transmitted over wireess channe. This is the cost adopted by the US Bureau of Pubic Roads [11]. The additive term b here can be interpreted as an additiona fixed to per traffic unit for the use of wireess channe. III. EXISTENCE AND UNIQUENESS OF THE NASH EQUILIBRIUM POINT We now demonstrate the existence and uniqueness of the Nash Equiibrium Point (NEP), considering the poynomia cost function (4) defined above. Let us denote by F u,i (f i,f i,p) the secondary user objective function on wireess channe, which is equa to:

3 F u,i (f i,f i,p)α i f i (f i [a (f ) β() + b + p ]) (5) We have therefore the foowing objective function for each SU i U: F u,i (f i,f i,p) α i f i f i [a (f ) β() + b + p ]. N N (6) The i-th secondary user objective function (6) is continuous in f {f 1,...,f I } and concave in f i : the second partia derivatives of F u,i (f) with respect to f i are equa to a β() (f ) [β() ] [f +(β() 1) f i ] and are therefore negative f i, f and β() >. These properties ensure the existence of the Nash equiibrium [1]. Having setted the question of existence of a NEP, it can be shown that the NEP is indeed unique under appropriate conditions [1]. The demonstration is not reported due to space constraints. IV. COMPUTING THE SOLUTION This section is dedicated to computing the Nash equiibrium point. For a given price vector p (p 1,p,..., p N ) R N, we have a non-cooperative spectrum access game between secondary users. Each SU i maximizes his objective function F u,i, which has the foowing expression, considering the cost function iustrated in Section II-C: F u,i (f i,f i,p) α i f i f i [a (f ) β() + b + p ]. N N The optima fows of the users can be obtained by soving the set of first-order conditions: for i U, { ( / f i )F u,i,iff i > <,iff i which yieds the foowing expression of the i th secondary user fow on wireess channe : f i (p ) αi [a (f ) β() + b + p ]. (7) a β()(f ) [β() 1] If we sum over a i U we obtain: [ f (p ) αi I (b ] 1/β() + p ), (8) (I + β())a and the condition f > impies that p < ˆp α i I b. At this stage, Primary Operator shoud sove the probem of maximizing the objective function p f (p ) p [ ] 1/β() αi I (b +p ) (I+β())a with respect to p. The optima vaue of p is: p β() 1+β() [ α/i i b ] (9) and the vaue of the optima fow f is: [ f αi I b ] 1/β() (1) a (1 + β()) (I + β()) provided that αi /I > b, which ensures that f > and p >. It is aso easy to check that p < ˆp. The optima vaues of f i are: A D where A and D have the foowing expressions: ( α i AαI(1 i + β())(i + β()) ) [I + Iβ()+β() ] Iβ()b DIβ()[a (1 + β())(i + β())] 1/β() [ ] (1 1/β()) α i Ib. (11) Now we must derive the conditions under which the individua fows are positive. It is easy to check that the denominator is positive provided that αi /I > b.sof i > [I + Iβ()+β() ](α i αi /I)+β()αi >β()b. For β() 1, 1 a [ α i ( ) (1 + I) I(1 + I) α i b (1 + I) ], (1) and f i Note that, for β() 1, the condition αi /I > b impies that > α i > (1+I) ( ) (1+I) ( αi /I αi /I )+b (I+1). +b (I+1) > b, and from the previous > (1+I) ( expression (α i (1+I) ( (I+1) ), we can write the foowing: α i > αi )+b /I (I+1) >b. V. SOLUTION FOR A SPECIAL CASE: SYMMETRICAL USERS αi /I )+b We consider here a specia case of the probem formuated above, where a secondary users have the same utiity per unit of fow on wireess channe, i.e., α i α, i U, N, and as a consequence they share the same objective function. This specia case permits to derive simper equiibrium expressions, providing further insights into the game equiibria. Proposition 1: In the considered cognitive radio wireess system with symmetrica users, the price and fow vaues at the Nash Equiibrium Point are given by the foowing expressions: p β() 1+β() (α b ) (13) f I, (14)

4 [ where f I(α b ) 1/β(). a (1+β())(I+β())] Furthermore, for β() 1, p and f become: and p α b (15) f I(α b ) a (I +1). (16) Proof: The above expressions of the equiibrium prices and fows can be easiy derived foowing the same procedure used in Section IV for computing the genera soution. VI. NUMERICAL RESULTS This section anayzes and discusses the numerica resuts obtained by soving our proposed game, testing the sensitivity of the achieved network equiibria to different parameters, namey the SUs per-bandwidth utiity (α i ) and the number of Secondary Users (I). A. Effect of the Utiity (α i) We first consider a Cognitive Radio Network scenario with identica SUs that share a singe frequency spectrum F (i.e., N 1), owned by a PO. The parameter vaues are set as foows: α i 1, i U, N, a 1, b.5 and β() 1, N. The Nash Equiibrium soution is in this case p 1/4 and f 1 f 1/1. Figure 1 iustrates the secondary user objective function (which is identica for both users) F u,i α i f i f i [a (f ) β() + b + p ] as a function of the users fow vaues (f i). In this Figure, the price p was set to 1/4, so that the Primary Operator maximizes its revenue. scenario with N wireess channes, owned by two POs, and I 1secondary users, 5 of which having α i 4+ɛ, the other 5 having α i 4 ɛ, with ɛ in the to.1 range. In this way, since αi 4constant, the tota fow on each wireess channe F is aways the same for each ɛ vaue (see expression (1)). The other parameters, a, b and β(), are set as in the previous scenario. Figure (a) reports the equiibrium fow (f i ) transmitted over frequency spectrum F by two users beonging to each one of the two groups, as a function of ɛ. Furthermore, Figure (b) iustrates the ratio between the fow transmitted by users having α i 4+ɛ and those having αi 4 ɛ. User fow (f i ) α i 4+ α i 4 (a) 3.5 F u,i.. Fow Ratio f Fig. 1. Secondary user objective function in the two-user network scenario as a function of f 1 and f, with p p 1/4. We then measured the amount of fow transmitted into the network by SUs having different α i vaues, i.e., different per bandwidth unit utiities. To this aim, we considered a CRN f (b) Fig.. CRN scenario with I 1secondary users, 5 of which having α i 4+ɛ, the other 5 having αi 4 ɛ: (a) equiibrium fow sent by each user on wireess channe F, and (b) ratio between the fow routed by secondary users having α i 4+ɛ and those having αi 4 ɛ. It can be observed that the amount of fow f i transmitted by each secondary user increases consistenty with increasing α i vaues, so that even sma increases in ɛ ead to quite arge

5 differences in. A variation of this scenario is then considered, where I 1 secondary users have different per-fow utiity vaues on the wireess channes: 5 SUs have α i 1 3+ɛ on channe 1 and α i 4+ɛ on channe, the other 5 have α i 1 3 ɛ on channe 1 and α i 4 ɛ on channe. Figure 3 iustrates the equiibrium fows sent by each SU beonging to these two groups on each wireess channe, thus permitting to evauate the fow variation due to the different utiity eves perceived by such users on the two avaiabe wireess channes. Primary Operator Revenue (α b ) 4a..5 α i Number of Secondary Users (I) User fow (f i ) α i 1 3+ α i 1 3 α i 4 Fig. 4. Primary Operator revenue as a function of the number of users, I (Symmetrica secondary users case). We proposed the utiization of poynomia pricing functions, which ead the system to unique and efficient Nash equiibrium points. Finay, we iustrated severa numerica exampes that provide insights into our proposed game Fig. 3. Equiibrium fow sent by each user on N wireess channes with different per-fow utiity vaues: 5 SUs have α i 1 3+ɛ on channe 1 and α i 4+ɛ on channe, the other 5 have αi 1 3 ɛ and αi 4 ɛ on channes 1 and, respectivey. B. Effect of the number of Secondary Users (I) We further measured the revenue achieved by a PO as a function of the number of SUs, I, considering a CRN scenario with a singe frequency spectrum, α i 4, a 1, b.5 and β() 1. Figure 4 iustrates the PO s revenue at the Nash Equiibrium point, p f, as a function of I. For simpicity, we consider the symmetrica users case studied in Section V. The equiibrium price, p, is independent of the secondary users number, since it is equa to p α b, whie the equiibrium fow on wireess channe, f, increases with increasing I vaues, f I(α b ) a (I+1). Hence, the revenue achieved by the -th Primary Operator increases with increasing I vaues, and is upper bounded by (α b ) 4a (which is approximatey equa to 3.6 in this scenario). VII. CONCLUSION This paper studied the economic interaction between secondary users and primary network operators in Cognitive Radio Networks. Secondary users are characterized by eastic traffic, and face spectrum access costs. Primary Operators aocate the avaiabe spectrum setting appropriate prices to maximize their revenue. This probem was modeed with a two-stage (Stackeberg) game, where Primary Operators set prices per bandwidth unit and users respond by presenting a certain amount of fow to the network. ACKNOWLEDGMENT This work was partiay supported by MIUR and FP7 in the framework of the PRIN SESAME and PrimeLife projects. REFERENCES [1] F.C.C.S.P.T. Force. Report of the spectrum efficiency working group. Federa Communications Commission, Tech. Report, pages 155,. [] I.F. Akyidiz, W.Y. Lee, M.C. Vuran, and S. Mohanty. NeXt generation/dynamic spectrum access/cognitive radio wireess networks: a survey. Computer Networks, pages , vo. 5, no. 13, 6. [3] Z. Ji and K.J.R. Liu. Dynamic spectrum sharing: a game theoretica overview. IEEE Comm. Magazine, pages 88 94, vo. 45, no. 5, 7. [4] F. Khozeimeh and S. Haykin. Dynamic spectrum management for cognitive radio: an overview. Wireess Communications and Mobie Computing, artice in press, 9. [5] Y. Wu, B. Wang, and K.J.R. Liu. Repeated Spectrum Sharing Game with Sef-Enforcing Truth-Teing Mechanism. In Proc. of IEEE Internationa Conference on Communications, ICC 8, pages , Beijing, China, May 8. [6] F. Wang, M. Krunz, and S. Cui. Price-based spectrum management in cognitive radio networks. In Proc. of the nd Internationa Conference on Cognitive Radio Oriented Wireess Networks and Communications (CrownCom), pages 7 78, Orando, Forida, USA, Juy-August 7. [7] D. Niyato and E. Hossain. Competitive spectrum sharing in cognitive radio networks: a dynamic game approach. IEEE Transactions on Wireess Communications, pages , vo. 7, no. 7, 8. [8] D. Niyato and E. Hossain. Competitive pricing for spectrum sharing in cognitive radio networks: dynamic game, inefficiency of Nash equiibrium, and cousion. IEEE Journa on Seected Areas in Communications, pages 19, vo. 6, no. 1, January 8. [9] T. Basar and G.J. Osder. Dynamic noncooperative game theory. SIAM Series in Cassics in Appied Mathematics, Phiadephia, PA, [1] E. Atman, T. Basar, T. Jimenez, and N. Shimkin. Competitive routing in networks with poynomia costs. IEEE Transactions on Automatic Contro, pages 9 96, vo. 47(1),. [11] O.F.P.R. BUREAU. Traffic Assignment Manua. US Department of Commerce, Washington, DC, [1] J.B. Rosen. Existence and uniqueness of equiibrium points for concave n-person games. Econometrica, pages 5 534, vo. 33(3), 1965.