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A Non-Cooperative Power Control Game for Secondary Spectrum Sharing Juncheng Jia and Qian Zhang Department of Computer Science and Engineering Hong Kong University of Science and Technology, Hong Kong, China {jiajc, qianzh}@cse.ust.hk

Abstract—Limited spectrum resources, inefficient spectrum usage and increasing wireless communication necessitates a paradigm shift from the current fixed spectrum management policy to a more flexible one. With the Federal Communications Commission’s (FCC) spectrum policy reform, secondary spectrum sharing has become a viable and promising option. In this paper, we study power control for spectrum sharing among secondary users with interference temperature limit (ITL) constraints at measurement points. Each secondary user will adjust its own transmission power to make sure the overall interference at the measurement points does not exceed the required ITL. Under the assumption that each secondary user is selfish and rational while there is only limited coordination between primary users and secondary users, we study the distributed power control scheme for secondary users. In our proposed system model, secondary users generating the highest interference at a measurement point will back off their transmissions if the ITL is exceeded. A non-cooperative power control game among secondary users is proposed to maximize each user’s utility. We identify the Nash equilibrium of the proposed game and analyze its property. Simulations are conducted to demonstrate that the proposed solution can achieve a satisfactory performance in terms of the total transmitting rate of all the secondary users.

I. I NTRODUCTION Much of the wireless technological innovation is happening in the unlicensed bands, and as a result these small frequency ranges are becoming crowded, with many personal electronic devices interfering with each other. In contrast, white space has been identified in most of the licensed bands. As the investigation of the current broadcast television frequency bands in USA shows, on average only 8 channels (out of 68 total channels) are used in any given TV market, or roughly 12%. If unlicensed devices were allowed to coexist with broadcast television, an additional 350 MHz of prime spectrum real estate would be available. Experiments conducted by the Shared Spectrum Company indicate that there is as much as 62% white space below 3GHz band even in the most crowed area near downtown Washington DC, where both government and commercial spectrum usage are intensive [1]. To help alleviate such an imbalance and improve utilization efficiency, the FCC has begun considering other ways of managing spectrum. Rather than static allocations based on detailed site surveys, a more real-time, dynamic approach must be adopted. To that end, they have new policy [1] that

allows cognitive radios [3] to operate in licensed frequency ranges, provided they are smart enough to sense their RF environment and steer clear of frequencies where they detect licensed carriers. This offers many new frequency bands for communications. However, in the original FCC-proposed model a channel is either occupied or it is not. If it is occupied by a licensed signal, an unlicensed device may not use it. A logical extension to this binary model is one of true coexistence, where unlicensed transceivers can operate on the same frequencies as licensed signals, provided they can quantify and bound the additional interference. To that end, the FCC proposed the interference temperature model [3], which provides a metric to measure interference experienced by licensed receivers. In this paper, we focus our study based on such an interference temperature model. We consider such a case where the secondary users who observe the channel availability dynamically, coexist with primary users to conduct data transmission. Here, the term secondary users refer to spectrum users who are not owners of the spectrum and operate based on agreements/etiquettes imposed by the primary users/ owners of the spectrum. A certain interference temperature limit is set to protect the primary users. Transmission power control is one of the most important issues in such a model. First, the secondary users (transmitter and receiver pairs) sharing the spectrum have to use appropriate transmission power control so that the interference temperature limit at a certain measurement point is not violated. Second, the satisfaction of elastic-data users is directly related to SINR at the receivers, which is a result of the transmission power of secondary users. Therefore, in this paper, we study the power control issue. We assume the interaction between the primary users and secondary users is quite limited. The only interaction that occurs is when the measurement point notifies the secondary user who is generating the highest received power that the interference temperature limit has been violated. Here we assume this secondary user will back off its transmission upon receiving such a notification. If the interference temperature limit is still violated, the next highest user will be notified to back off. Considering the selfishness of secondary users, we model this as a non-cooperative power control game where each user/player wants to maximize its own utility. Under the assumption that each user is rational, a unique Nash equilibrium can be identified in this game which is for each secondary

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user to pick the transmit power generating the same received power at the measurement point. The performance of this Nash equilibrium is presented by some simulation. As the result shows, such an equal allocation is good compared with the social optimal solution. The rest of this paper is organized as follows. In the next section, we present some basic concepts for spectrum sharing and related work in both spectrum sharing and power control. In section III, we propose our spectrum sharing framework. Section IV includes the game theoretical modeling of the behaviors of secondary users in our framework, identification of Nash equilibrium and performance analysis. We give the numerical results in Section V.

II. P RELIMINARY F OR S PECTRUM S HARING A ND P OWER C ONTROL A. Spectrum Sharing Some exploration has been done in the field of spectrum sharing. The idea of real-time secondary markets for spectrum is proposed in [7], where secondary users ask the cellular license-holder for temporary access to spectrum as needed. Recently, two spectrum sharing models have been proposed to make more efficient use of spectrum: the exclusive use model and the commons model [2]: (1) In the exclusive use model, a licensee (primary user) has exclusive and transferable rights to the user of a specified spectrum within a defined geographic area, with flexible spectrum use rights that are governed primarily by technical rules to protect spectrum against interference. (2) In the commons (open access) model, unlimited numbers of unlicensed users are allowed to share frequencies, with usage rights that are governed by technical standards or etiquettes but with no right to protection from interference. In general, spectrum sharing models differ from each other in how protection priority is assigned. For exclusive use model, a new metric called interference temperature is used to quantify and manage the interference in the environment and provide protection to primary users. Based on such a metric, a licensee sets up an interference temperature limit under which secondary users can coexist with primary users (i.e., the interference temperature limit is not violated at primary users’ receivers). The interference temperature is a measurement of the power and bandwidth occupied by interference. Interference temperature TI is specified in Kelvin and is define as TI (fc , B) =

PI (fc , B) , kB

(1)

where PI (fc , B) is the average interference power in Watts centered at fc , covering bandwidth B measured in Hertz Boltzmann’s constant k is 1.38 × 10−23 Joules per Kelvin degree [10]. The FCC would establish an interference temperature limit (ITL) for a given geographic area. This value would be a maximum amount of tolerable interference for a given frequency band in a particular location. Any unlicensed

transmitter utilizing this band must guarantee that their transmissions added to the existing interference must not exceed the interference temperature limit at a licensed receiver. Several works is done under the above IT model. Authors in [5] study the coexistence of primary and secondary users under interference temperature constraints, where auction mechanisms are used to achieve weighted max-min fairness or system optimization. In [6] QoS issue is considered based on the system model of [5]. Our paper shares the similar system model as [5][6], but we consider the situation where the explicit allocation of primary user does not exist and secondary users wants to maximize its own benefits instead of system utility. In this paper, we share the similar system model with [5][6]. B. Power Control General transmission power control in wireless networks is well studied in literatures. In [8] an overview is given of the challenges, solutions and open issues related to power control in wireless ad hoc networks. The main objectives are to reduce the total energy consumed in packet delivery and/or increase network throughput by increasing the channel’s spatial reuse. The selection of transmission power is coupled with routing and media access control layers, which makes the problem much more complicated. In [9], a variety of power control problems about single hop transmissions are formulated as nonlinear optimization with system-wide objectives and individual users’ constraints are studied. Our work is different in that we assume the users are selfishly maximizing their own utilities other than optimizing the system utility cooperatively.

III. F RAMEWORK F OR S PECTRUM S HARING In this paper, we focus on the exclusive user model. Suppose a licensee has deployed a primary system using a spectrum band within a certain geographic area. However, the deployment is not utilizing the spectrum fully: either only a few primary devices are deployed which leaves the spectrum with too much space idle, or the radio technology of primary devices can tolerate much more noise than the noise currently appearing in the radio environment. Such an underutilization of spectrum gives other spectrum users the opportunity to invite new services or improve the quality of current services, either with a spatial reuse approach or with an underlay approach The main challenges for such spectrum sharing can be identified. For primary users, protection is the main concern. For secondary users, how to share the spectrum among them fairly and efficiently is the main challenge. Furthermore management cost is a big issue. Taking these factors into consideration, we propose a spectrum sharing framework. In our framework, there exist some nodes called measurement points which monitor the real-time interference temperature of the concerned band of spectrum at their locations. These measurement points are deployed in a careful plan, such that


Fig. 1: System model for spectrum sharing.

as long as the interference temperature limit set on each point is not violated, protection of primary users can be guaranteed. The topic of how to place the measurement points and what the appropriate interference temperature limit should be is out of the scope of this paper. The protection of primary users is achieved by limited interaction between the measurement point and the secondary users. We assume a measurement point has the ability to trace secondary users who impose the maximum received power at its location besides monitoring the interference temperature. As long as the interference temperature limit is not violated, the measurement point will not interfere in the secondary users’ operation. If the limit is exceeded, the measurement point will send a message to the secondary users generating the highest interference. These secondary users upon received such a message will back off their data transmissions. When there are several secondary users generating the same highest interference, random pick-up is used to break the tie. If the interference temperature limit is still violated, the next highest user will be notified to back off. Such a backing off will continue until the interference temperature is no longer violated.

IV. N ON -C OOPERATIVE P OWER C ONTROL G AME In this section, the above system model is investigated in a non-cooperative game frame. We give the formulation of the power control game model. Then we identify a unique Nash Equilibrium and analyze the properties of this Nash Equilibrium. A. Utility Function We make the following assumptions with respect to the communication network. We assume at a geographic area a spectrum licensee offers a portion of its spectrum with bandwidth W to be shared among a set of N = {1, . . . , N } secondary users. Each secondary user is a pair of transmitter and a dedicated receiver. Here we simplify the problem such that all secondary users use a spread spectrum signaling format over the whole shared band.

Such a spectrum sharing is base on the guarantee that the interference temperature limit set by the licensee is not violated. The interference temperature limit here is interpreted as a threshold of the total receiver power at a specified measurement point, denoted by T . The system model is illustrated in Fig. 1. We denote the transmit power for user , pmax ]. In this paper, we let i by pi and pi ∈ Pi = [pmin i i Pi = (0, +∞), which is interpreted as the achievable power of secondary users can generate interference much higher than the power threshold at the measurement point. The link gain from user i’s transmitter to user j’s receiver is hij and the link gain from i’s transmitter to the measurement point is hi0 . Then the constraint that the power threshold is not violated is expressed as N pi hi0 ≤ T. (2) i=1

Secondary users have elastic data applications in which throughput of each user are determined by the SINR at its receiver. When user i is not backed off by the measurement point, the quality of service enjoyed by user i is characterized by a function ui (γ(i)), where γ(i) is the receiver SINR at user i’s receiver, pi hii (3) γi (p) = 1 σi + L j=i pi hji where p = (p1 , . . . , pN ) is the transmit power profile, σi is the background noise at secondary receiver i and L is the normalized spreading sequence length. In this paper, we define the function ui as ui (γi ) = ln(γi ),

(4)

which presents the throughput of communication systems in the high SINR regime. If user i is backed off, then its utility is 0. Denote the set of secondary users that back off transmissions as B. B includes the users with the largest interference at the measurement point. We use pb to present the power vector after the backing off process. In general, Ui is used to denoted user i’s utility  N    u (p), if pi hi0 ≤ T  i    i=1    N  Ui (p) = ui (pb ), if pi hi0 > T and i ∈ / B (5)   i=1     N     if pi hi0 > T and i ∈ B  0, i=1

With such a general utility function, the social optimal solution can be found with a centralized algorithm. However, in our setting the only possible central point, the measurement point, can only pass quite limited information about radio environment. It is not capable to conduct complicated power allocation computation. Further more, such an approach requires the cooperation with secondary users which may not be reasonable in reality. Therefore we discuss the non-cooperative behaviors


of secondary users within the above system model in the next section. B. Game Model We assume secondary users behave selfishly in that each secondary user wants to maximize its own utility: it wants to use large transmission power to obtain high SINR at the receiver side; however, too high power may mean it is the one generating the largest interference at the measurement point when the ITL is violated. We consider a game in strategic form. Let G = [N , {Pi }, {Ui }] denote the non-cooperative power control game. Here the players in the game correspond to the secondary users in N . Each player picks a transmit power from the strategy space Pi which is continuous. The joint strategy space P = P1 × P2 × . . . × PN , is the Cartesian product of the individual strategy sets for N players. Each player receives a payoff Ui (p) which is define in the previous subsection. Here the power vector p becomes a power profile (strategy profile), and the power profile of user i’s opponents is defined as p−i = (p1 , . . . , pi−1 , pi+1 , . . . , pN ). Therefore, the power profile p = (pi , p−i ). C. Nash Equilibrium of Our Game We use the concept of Nash equilibrium to analyze the stability of this game. The best response of a player is an action that maximizes its utility function for a given action tuple of the other players. Definition 1: Bi is a best response by player i to p−i if Bi (p−i ) = arg max Ui (pi , p−i ). pi ∈Pi

N

(6)

pi hi0 = T.

(8)

i=1

With the definition of the best response function, Nash equilibrium can be defined as follows: Definition 2: A power profile p∗ is a Nash equilibrium (NE) of G if it is a fixed point of the best responses, i.e., Ui (p∗i , p∗−i ) ≥ Ui (pi , p∗−i ),

which results payoff 0 for it. Therefore, at that point, none of the users have the incentive to deviate from the power profile p∗ unilaterally. Further this Nash equilibrium is unique. Suppose there exists another Nash equilibrium (p1 , . . . , pi , . . . , pN ). (1) If N i=1 pi hi0 < T , any user can increase its payoff by increasing transmit power N with the power of the other ones being fixed. (2) If i=1 pi hi0 > T , there must be some user backed off and it can do better by transmit at lower power such that the constraint is not violated. (3) the only Therefore N possible case for (p1 , . . . , pi , . . . , pN ) is i=1 pi hi0 = T and ∃i = j, pi hi0 < pj hj0 . For this, user i can definitely increasing its transmission power by a small ∆p, which causes the T threshold exceeded. Then some other users will back off and user i will enjoy a better payoff. So another Nash equilibrium does not exist. Therefore this Nash equilibrium is the only possible outcome of the game. The concept of Pareto-efficient is usually used to characterize the goodness of a Nash equilibrium. Definition 3: The outcome of a game is Pareto-efficient if no other outcome makes every player at least equally as well off and at least one player strictly better off. For our game, we assess the Nash equilibrium with respect to Pareto-optimality. Lemma 1: A power profile is Pareto-efficient if and only if the total received power threshold is reached [5], i.e.,

(7)

for any p∗−i and any user i. According to the definition, in a Nash equilibrium, none of the players can improve its payoff by unilaterally changing its strategy. Theorem 1: There exists a unique Nash equilibrium for the T for non-cooperative power control game, which is p∗i hi0 = N all i ∈ N . Proof : We can identify a Nash equilibrium which is for each secondary user to choose a transmit power such that the received power at the measurement point is an equal T . distribution of the received power threshold, i.e., p∗i hi0 = N The reason this point is a Nash equilibrium is explained as follows. Suppose user i changes its transmission power while the other users adhere to the original ones. Since the utility function Ui (pi , p−i ) is a strictly increasing function, user i can only improve its utility by increasing its transmission power user i increases its power to pi > p∗i , then p i . However, if N N ∗ i=1 pi hi0 > i=1 pi hi0 = T , which means the interference temperature limit is violated at the measurement point. According to the rule, user i will back off its transmission

Proof : It is immediate since if the power constraint is not tight, then each secondary user can increase its transmit power T , which increases the SINR for by a factor of α = N i=1 pi hi0 each user. Therefore if a power profile is Pareto-efficient, then the power threshold is reached. Theorem 2: This Nash equilibrium is Pareto-efficient. Proof : Since the power threshold is reached at the Nash equilibrium, according to Lemma 1, it must be Pareto-efficient. D. Performance of Nash Equilibrium Although it is Pareto-efficient, it may not be social optimal. The reason is that the link gains hij ’s are independent and the power distribution for social optimal depends on these hij ’s. Here we consider a special case where all the secondary receivers are co-located with the measurement point. This models the secondary network with a single access point. Here hij = hi0 for all i, j ∈ N . Theorem 3: The Nash equilibrium in the case of co-located receivers is social optimal. Proof : Social optimal implies Pareto-efficient in our system which in turn results in a tight power constraint. Therefore, Ui (pi ) = ln

pi hi0 . σ0 + T − pi hi0

(9)


Denote pri = pi hi0 , the system optimization problem is N

maximize

Ui (pri )

i=1 N

subject to

pri = T

i=1

Using the Lagrangian multiplying: L(pr , λ)

=

N

N Ui (pri ) − λ( pri − T )

i=1

=

N i=1

i=1

ln

pri

σ0 + T − pri

N − λ( pri − T ). i=1

(10)

Fig. 2: System utilities for NE and social optimal with different system sizes

We examine the first-order condition of the optimal point ∂L(pr , λ) = 0, ∂pri

i ∈ N.

(11)

Since symmetric property exists here, it is easy to get pri = For the second-order condition, it satisfies ∂ 2 L(pr , λ) < 0, ∂(pri )2

i ∈ N.

T N.

(12)

T Therefore, pri = N , which is the Nash equilibrium in this special case, is actually the social optimal point. For the general cases, we use simulations to show the results.

V. N UMERICAL R ESULTS In this section, we present the numerical results for the performance of general spectrum sharing cases, i.e. the secondary receivers are not co-located with the measurement point. The range of simulated area is 300m × 300m. The transmitters of secondary users are randomly placed within this area. The distance between the transmitter and its intended receiver is within [10m, 50m]. The bandwidth of shared spectrum is 1 MHz with the AWGN noise σi is the same, 1.0 × 10−10 and the spreading gain L is 128. Path gains are obtained by simple path loss model hij = K( ddij0 )α where K = 1.0 × 10−6 , d0 = 10m, α = 3 and dij is the distance between transmitter i and receiver j. A. System Utility The results of game solutions are compared with social optimal solutions. Here optimal solutions are obtained by transferring the original maximization problem to a convex optimization problem with p = log p, which is solved by existing efficient optimization tools [4]. The SINR for each simulated case is greater than 1, so that the utility is nonnegative. Fig. 2 shows the system utility with different system size. When the system size is small (below 5), the system utility at the Nash equilibrium is approaching the social optimal. With

Fig. 3: System utilities for NE and social optimal with different power thresholds the system size increasing, the system utility is also increased. The result of Nash equilibrium is 10% worse than the social optimal. We also investigate the influence caused by an interference temperature limit setup. In the simulation, the system utility is observed with different received power thresholds at the measurement point. In Fig. 3, a single simulation is conducted where the locations of 10 secondary user pairs are fixed. The power threshold is increased from 1.0 × 10−8 to 5.0 × 10−6 . As Fig. 3 shown, the system utility for both Nash equilibrium and social optimal is encountering a relatively faster increase with low power thresholds. With high power thresholds, the utility turns to be stable. It is because of the background noise counting much in low cases and negligible in high cases B. Individual Utilities We use a single simulation to examine individuals’ performance. A sample secondary user placement is plotted in Fig. 4. Each secondary user is numbered from 1 to 10. The distance from transmitters to the measurement point at (0, 0) is increased with indexes. We set the distance between each transmitter and its receiver to be a constant (30m). The utility for each secondary user is shown in Fig. 5. Roughly speaking,


temperature limit constraints at measurement points. This is a case of coexistence of primary users and secondary users, which requires limited coordination between primary users and secondary users. We study the distributed power control among secondary users in such a system, where each secondary user adjusts its own transmission power to make sure the overall interference at the measurement point does not exceed the required interference temperature limit (ITL). Under the assumption that each secondary user is selfish and rational, a non-cooperative power control game among secondary users is proposed to maximize each user’s utility. We prove that a unique Nash equilibrium exists in this game. Moreover, the performance of such a solution is good and close to the social optimal. Fig. 4: A sample secondary user placement R EFERENCES

Fig. 5: Utility for each secondary user in Fig. 4.

at Nash equilibrium the utility for users farther away from the measurement point enjoy higher utility. However, the utility also depends on the crowding factor. User 9 and 10 are the farthest users, but they are the users with the highest utility, since they are nearby and cause each other high interference. If the transmitter-receiver distances are not fixed, individuals’ utility will have greater variance. Such unfairness is caused by the system model, topology and the selfishness of the secondary users. If fairness objectives are desired, (e.g., max-min fairness, weighted fairness, etc), primary users have to employ more complicated management mechanisms; more explicit information exchange is needed also. Such issues will be further studied in our future work.

VI. C ONCLUSION Spectrum sharing is a promising solution for future wireless communication applications. In this paper, we study the case of spectrum sharing among secondary users with interference

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