Dusit Niyato 1 and Ekram Hossain 2

1. IEEE Transactions on Wireless Communications, Vol. 7, No. 11, Nov. 2008 Market-Equilibrium, Competitive, and Cooperative Pricing for Spectrum Sharing in Cognitive Radio Networks: Analysis and Comparison Dusit Niyato1 and Ekram Hossain2 1An assistant professor in the School of Computer Engineering, at the Nanyang Technological University, Singapore. 2An Associate Professor in the Department of Electrical and Computer Engineering at University of Manitoba, Winnipeg, Canada. Presented by: Ming-Lung

2. Outline • Introduction • Three Pricing Models • Solutions • Distributed Implementation & Stability Analysis • Numerical Performance Analysis • Conclusion • Comment

3. Introduction • For spectrum trading, which involves spectrum selling and buying processes, one of the major issues is pricing. • Pricing impacts the incentive of the primary service provider (in selling the spectrum) and the satisfaction of the secondary users (in buying the spectrum). • Spectrum price must be properly chosen (through a pricing model) based on the preferences and objectives of both primary and secondary users.

4. Contribution • Three different pricing models and solutions • Distributed algorithms • Stability analysis • Performance analysis which • reveals their interesting performance behaviors and • gives insight into the spectrum trading problem

5. Cognitive Wireless Network Model • We consider a cognitive radio system with • N primary services • Primary service i owns the frequency spectrum denoted by Fi • Primary service iserves Mi primary users • The price (per unit spectrum) of Fi is denoted by pi

6. Wireless Transmission • With adaptive modulation, the transmission rate can be dynamically adjusted based on the channel quality. • The spectral efficiency can be obtained from [22] • γis the signal-to-noise ratio (SNR) • BERtar is target bit-error-rate (BER) • Spectral efficiency (in bit/s/Hz, or (bit/s)/Hz): • refers to the information rate that can be transmitted over a given bandwidth in a specific communication system. • is a measure of how efficiently a limited frequency spectrum is utilized. [22] A. J. Goldsmith and S.-G. Chua, “Variable rate variable power MQAM for fading channels,” IEEE Trans. Commun., vol. 45, no. 10, pp. 1218-1230, Oct. 1997

7. Pricing Models • We consider three different pricing models which encompass different degrees of competition and cooperation among the primary service providers. • Market-equilibrium pricing model • Competitive pricing model • Cooperative pricing model

8. Pricing models: market-equilibrium • Market-equilibrium pricing model • It is assumed that the primary service is not aware of others. • In an actual environment, this could be due to the lack of any centralized controller or information change among primary services. • In this spectrum trading, market-equilibrium price denotes the price for which spectrum supplied by the primary service is equal to the spectrum demand from the secondary service. • This market-equilibrium price ensures that there is no excess supply in the market and spectrum supply meets all spectrum demand.

9. Pricing models: competitive • Competitive pricing model • It is assumed that a primary service is aware of the existence of other primary services and all of the primary services compete with each other to achieve the highest individual profit. • Given the spectrum prices offered by other primary services, one primary service chooses the price for its own spectrum so that its individual profit is maximized.

10. Pricing models: cooperative • Cooperative pricing model • It is assumed that all of the primary services know each other and they fully cooperate to obtain the highest total profit by selling spectrum to the secondary service. • In an actual environment, to achieve this full cooperation, extensive communication would be required among all primary services.

11. Utility of Secondary Service • We consider the utility gained by the secondary users • b is a vector of shared spectrum sizes from all the primary services, i.e., b = [b1 … bi … bN]. • pi is the price offered by primary service i. • ki(s) denotes the spectral efficiency of wireless transmission by the secondary users using frequency spectrum Fi owned by primary service i. • The utility function takes spectrum substitutability into account through the parameter ν (0.0 ≦ ν ≦ 1.0). (?) (!!) Unit problem

12. Spectrum substitutability • If a secondary user uses a multi-interface network adaptor, it is able to switch among the frequency spectra freely depending on the offered price. • This spectrum substitutability parameter νis defined as follows: • When ν = 0.0, a secondary user cannot switch among the frequency spectra, • while for ν = 1.0, the secondary user can switch among the operating frequency spectra freely.

13. Demand function • The demand function for spectrum Fi at the secondary service can be obtained using as follows: • p denotes a vector of prices offered by all primary services in the market. • The demand function in (2) can be written as follows: • p-i denotes the vector of prices of all primary services except service i.

14. Demand function (cont’d) • D1(p-i) and D2 are constants for given pj for i ≠ j which are given as follows:

15. Revenue and Cost Functions for a Primary Service • We assume that the primary users are charged at a flat rate for a guaranteed amount of bandwidth. • If the required bandwidth cannot be provided, a primary service offers “discount” to the primary users, and this is considered as the cost of sharing spectrum with the secondary service. • Let Ril denote the revenue gained from primary users served by primary service i, • Ris denote the revenue gained from sharing spectrum with secondary users, • Ci denote the cost due to QoS degradation of primary users.

16. Revenue and Cost Functions for a Primary Service (cont’d) • The revenue and cost functions can be defined as follows: • c1 and c2 denote constant weight for the revenue and cost functions at the primary service • Bireq denotes bandwidth requirement per user • Wi denotes spectrum size • Mi denotes the number of ongoing primary users • ki(p) denotes spectral efficiency of wireless transmission for primary service i.

17. Profit function • Based on the revenue and cost, profit Pi of a particular primary service i owning spectrum Fi can be expressed as follows:

18. Supply function • The spectrum supply function can be derived based on a profit maximization problem. • To obtained the optimal spectrum size to be shared, we differentiate the profit function with respect to bi (where pi is given) as follows: • Then the supply function can be expressed as follows:

19. Solution of Market-Equilibrium Pricing Model • The market-equilibrium (i.e., solution) is defined as the price pi* at which supply equals spectrum demand, i.e., • The vector p* = [… pi* …]T denotes the market equilibrium prices for all primary services.

20. Solution of Competitive Pricing • We consider the Nash equilibrium as a solution • The Nash equilibrium is obtained by using the best response function which is the best strategy of one player given others’ strategies. • The best response function of primary service i, given a vector of prices offered by other primary services p-i, is defined as follows: • The vector p* = [ … pi* …]T denotes a Nash equilibrium of this game on competitive pricing for

21. Solution of Competitive Pricing (cont’d) • Mathematically, to obtain the Nash equilibrium, we have to solve the following set of equations: • The size of the shared bandwidth bi in the individual profit function is replaced with spectrum demand Di(p), and the profit function can be expressed as follows:

22. Solution of Competitive Pricing (cont’d) • Then, using , we obtain • The solution pi*, which is a Nash equilibrium, can be obtained by solving the above set of linear equations by using a numerical methodwhen all the parameters in (10) are available. • Then, given a vector of price p* at the Nash equilibrium, the size of the shared spectrum can be obtained from the spectrum demand function Di(p*).

23. Solution of Cooperative Pricing • An optimization problem is formulated to obtain the optimal price which provides the highest total profit for all primary services. • This optimization problem can be expressed as follows:

24. Solution of Cooperative Pricing (cont’d) • The Lagrangian can be expressed as follows: • λj, μk, and σl are Lagrange multipliers for the constraints in (12) and (13), respectively. • Using Kuhn-Tucker conditions, we can obtained the vector of optimal prices p* such that the total profit of all the primary services is maximized.

25. Kuhn-Tucker conditions • In mathematics, the Karush–Kuhn–Tucker conditions (also known as the Kuhn–Tucker or KKT conditions) are necessary for a solution in nonlinear programming to be optimal. • Consider the following nonlinear optimization problem • Minimize f(x) • Subject to: gi(x) ≦ 0, hj(x) = 0 • If x* is a local minimum, then there exist constants μi and λj such that

26. Distributed Implementation • In a practical cognitive radio environment, a primary service may not have the complete network information. • Therefore, a primary service must learn the behavior of other entities from the history, and a distributed price adjustment algorithm is required which would gradually reach the solution.

27. Distributed Implementation: Market-Equilibrium Pricing • The price offered by each primary service is gradually adjusted in a direction that minimizes the difference between spectrum demand and supply. • This process works as follows: • The spectrum price is initialized to pi[0] and this price is sent to the secondary service. • The secondary service replies with the size of spectrum demand which is computed from the demand function Di(p[t]) for spectrum Fi • Then, the primary service computes the size of the supplied spectrum Si(pi[t]). • The price adjustment in each iteration can be expressed as • This process repeats until the difference of prices in current iteration t and next iteration t+1 becomes less than the threshold ε (e.g. ε = 10-5).

28. Distributed Implementation: Competitive Pricing • We assume that a primary service cannot observe the prices of other primary services. • The process works as follows: • The spectrum price is initialized to pi[0] and is sent to the secondary service • The secondary service replies with the size of spectrum demand • The price in next iteration is given by

29. Distributed Implementation: Competitive Pricing (cont’d) • The marginal spectrum demandcan be approximated by a small variation in price ξ (e.g. ξ = 10-4) • P.S. the following equation is needed to get the profit

30. Distributed Implementation: Cooperative Pricing • We assume that primary services can exchange information on current profit among each other. • The process works as follows: • The spectrum price is initialized to pi[0] and is sent to the secondary service • The secondary service replies with the size of spectrum demand • The price in next iteration is given by

31. Distributed Implementation: Cooperative Pricing (cont’d) • The marginal total profit can be estimated by

32. Information Exchange Protocol • Market-equilibrium: • In each iteration, each primary service sends the price to the secondary service. • The secondary service responds with spectrum demand for each primary service. • 2 messages are exchanged. • Competitive: • Each primary service sends pi[t] – ξ and pi[t] + ξ to the secondary service. • The secondary service respond 2 messages respectively for each primary service. • 4 messages are exchanged. • Cooperative: • Each primary service sends pi[t] – ξ and pi[t] + ξ to the secondary service. • The secondary service responds to all primary services. • Each primary service exchanges its profit with each other. • 4N messages are exchanged, where N is the number of primary services.

33. Stability Analysis: Discrete-Time Linear Control System Representation • The distributed pricing algorithms can be represented by discrete-time linear control systems, and subsequently, we can use the classical control theory techniques to analyze the behavior of the algorithms. • The block diagrams of these pricing models are shown in Fig. 2 for the case of two primary services.

34. Stability Analysis: Discrete-Time Linear Control System Representation (cont’d) • The state space representation of a general control system is as follows: • u represents a unit input. • For market-equilibrium pricing

35. Stability Analysis: Discrete-Time Linear Control System Representation (cont’d) • For competitive pricing

36. Stability Analysis: Discrete-Time Linear Control System Representation (cont’d) • For cooperative pricing

37. Stability Analysis: Local Stability Analysis • The simplest way to analyze stability is to consider the eigenvalues of the Jacobian matrix of the self-mapping function • By definition, the self-mapping function is stable if and only if the eigenvaluesei are all inside the unit circle of the complex plane (i.e., |ei| < 1) • Jacobian matrix • Eigen values

38. Stability Analysis: Local Stability Analysis (cont’d) • For market-equilibrium pricing

39. Stability Analysis: Local Stability Analysis (cont’d) • For competitive pricing

40. Stability Analysis: Local Stability Analysis (cont’d) • For cooperative pricing

41. Stability Analysis: Local Stability Analysis (cont’d) • The following observations can be made: • Stability depends mainly on the parameters of the primary services, and these parameters can be adjusted locally. • For market-equilibrium pricing, the stability condition also depends on channel quality of the primary users. • Distributed cooperative pricing is less stable than competitive pricing. • When the number of ongoing primary users increases, the distributed algorithms for competitive and cooperative pricing become more stable while the distributed algorithms for market-equilibrium pricing becomes less stable.

42. Numerical Performance Analysis: Parameter Setting • We consider a cognitive radio environment with two primary services and a group of secondary users. • The total frequency spectrum available to each primary service is 20 MHz (i.e., Wi = 20) • The number of primary users is set to M1 = M2 = 10. • The target BER for the secondary users is BERtar = 10-4 • The bandwidth requirement of each of the primary users is 2 Mbps (i.e., Bireq = 2) • The channel quality (i.e., SNR at receiver) for the secondary users varies between 9 to 22dB • c1 = 5, c2 = 10, ν = 0.7 • For distributed pricing algorithms, p1[0] = p2[0] = 1

43. Numerical Results: Efficiency of Pricing Solutions • Sharing spectrum can gain higher profit • Cooperative > Competitive > Market-equilibrium

44. Numerical Results: Existence of Pricing Solutions • Under variation in offered price from primary service one • Spectrum supply depends largely on the number of primary users and their bandwidth requirements • The solution exists only for certain range of values of bandwidth requirement

45. Numerical Results: Existence of Pricing Solutions (cont’d) • Best responses of the primary services in case of competitive pricing • The existence of Nash equilibrium depends on the number of primary users and their bandwidth requirement

46. Numerical Results: Existence of Pricing Solutions (cont’d) • For cooperative pricing (no figure) • The solution does not exist when the number of primary users and/or their bandwidth requirements become large • This is because in this case the allocated spectrum size may become zero.

47. Numerical Results: Variations in Price and Profit under Different Channel Qualities • Variations in price under different channel qualities for the spectrum offered by primary service one (γ1) • When γ1increases, the corresponding demand becomes larger • Therefore, primary service one can increase its price to gain higher profit. • On the other hand, demand for primary service two decreases, and corresponding price decreases

48. Numerical Results: Impact of Spectrum Substitutability • Set γ1= γ2 = 13 dB • Assume p1 = p2 • As ν increases, for all pricing models, spectrum price decreases. • For ν = 0, there is no competition, the price of competitive and cooperative pricing equals • For ν = 1, the price of competitive pricing decreases to attract more demand, and converges to the price of market-equilibrium.

49. Numerical Results: Impact of Number of Primary Services • Fix the total spectrum size for all primary services to 60MHz • The number of primary users served by each primary service to 10. • For market-equilibrium and cooperative pricing, the spectrum price increases. • For competitive pricing, the spectrum price decreases due to higher degree of competition.

50. Numerical Results: Stability Region and Convergence of the Distributed Algorithms • Stability region under different learning rate • Market-equilibrium > competitive > cooperative