Stackelberg Game Based Power Control with Outage Probability Constraints for Cognitive Radio Networks

Abstract

This paper firstly investigates the problem of uplink power control in cognitive radio networks (CRNs) with multiple primary users (PUs) and multiple second users (SUs) considering channel outage constraints and interference power constraints, where PUs and SUs compete with each other to maximize their utilities. We formulate a Stackelberg game to model this hierarchical competition, where PUs and SUs are considered to be leaders and followers, respectively. We theoretically prove the existence and uniqueness of robust Stackelberg equilibrium for the noncooperative approach. Then, we apply the Lagrange dual decomposition method to solve this problem, and an efficient iterative algorithm is proposed to search the Stackelberg equilibrium. Simulation results show that the proposed algorithm improves the performance compared with those proportionate game schemes.

1. Introduction

The high energy consumption and exponential growth in wireless communication networks face serious challenges to the design of more energy efficiency and spectrum efficiency green communications that should deal with the scarcity of radio resources. A promising approaches technique called cognitive radio networks (CRNs) is proposed as a key design to improve spectral efficiency of the LTE-advanced standard. Game theory is an effective tool for resource allocation in multiuser communication systems, which has been applied to CRNs, and [1] summarized the advance of game theory for CRNs. Recently, Stackelberg game has been formulated for resource allocation which addresses the relationship between PUs and SUs in [2–9]. In these two-tier femtocell networks, the PUs were served by the mobile operators, whereas the SUs are deployed by indoor users for their own interests. Their different service requirements and design objectives motivate us to adopt the framework of hierarchical game to the power allocation problem with the leader-follower structure, which is actually a multiple-leader multiple-follower Stackelberg game. Specifically, the PUs compete with each other in a noncooperative manner to maximize their utilities, all the time anticipating the response of the followers. This subgame is referred to as the upper subgame, while after the PUs apply their strategies, the SUs update their power allocation strategies in response to the PUs' strategies. The SUs also compete with each other in a noncooperative manner to maximize their own utility function. Moreover, this subgame is referred to as the lower subgame.

Reference [2] presented a joint pricing and power allocation scheme for CRNs with Stackelberg game; PUs and SUs can benefit from the channel sharing model by achieving the Stackelberg equilibrium (SE). In [3, 4], a Stackelberg game is used to study the joint utility maximization of the PUs and the SUs with a maximum tolerable interference power constraint (IPC) at the primary base station (PBS). However, the algorithm is suboptimal because it does not consider how to control the IPC among PBS. Therefore, [5] proposed an optimal price-based power algorithm for the PBS and SUs maximize their revenues by Stackelberg game with IPC. To maximize users' energy efficiency of CRNs, a new green power control scheme was studied in [6], where PUs and SUs aim at maximizing their energy efficiency with Stackelberg game. In [7], the authors formulated a Stackelberg game model to maximize the payoff of both SUs and PUs by jointly optimizing transmission powers of SUs and subband allocations of SUs. Moreover, in [8], the authors analyzed the cooperation interaction between the PUs and SUs transmitters from a Stackelberg game theoretic perspective with secrecy constraints. Moreover, [9] proposed a game theoretic framework to model the interactions among multiple PUs and multiple SUs under different system parameter settings and under system perturbation.

However, the above papers aim to maximize users' revenue without considering the outage probability of multi-PUs and multi-SUs. It is clear that both PUs and SUs need to update their transmission power frequently to maintain their Signal to Interference plus Noise Ratio (SINR) level (usually referred to as QoS) due to channel fading and interferences from other users when they transmit on the same channel. Therefore, in CRNs, the fundamental performance traits of multiple SUs power control should be investigated with multiple PUs; also the issues of multiple PUs power control should be studied in the presence of multiple SUs in fading channels and interferences. In the game problem formulation, the outage probabilities of both PUs and SUs represent their own utility below the required target SINR used to guarantee their desired QoS. Hence, the channel outage probabilities with respect to both PUs and SUs transmissions should be required in the multiple users' game framework, if users' SINR falls below a certain threshold.

In this paper, we proposed a new Stackelberg game to model the hierarchical competition between PUs and SUs in CRNs with global interference and outage constraints. We theoretically prove the existence and uniqueness of robust Stackelberg equilibrium for the noncooperative approach. Then, we employ the Lagrange dual decomposition method to solve the game problem by decomposing the game problem into independent suboptimal solution. Furthermore, we develop an efficient iterative algorithm to converge the SE.

2. System Model

We consider a CRN composed of a single PBS with a set of PUs $\mathsf{K}≔\{\mathrm{1},\dots ,K\}$ having priority to use a set of channels $\mathsf{N}≔\{\mathrm{1},\dots ,N\}$ and a secondary base station (SBS) with a set of SUs $\mathsf{L}≔\{\mathrm{1},\dots ,L\}$ allowed to share N channels from the PBS, shown in Figure 1. We define the channel gains of the PU k and the SU l on the channel n as $h_{kk}^n$ and $h_{ll}^n$ , respectively. The channel gain between the kth PU transmitter and the SU l receiver is denoted by $h_{kl}^n$ ; the channel gain from the lth SU transmitter to the kth PU receiver is $h_{lk}^n$ . The transmit power of the kth PU and the lth SU on the channel n is denoted by $p_k^n$ and $p_l^n$ . We assume each user can transmit simultaneously over multiple channels, and one channel can be served only by one PU but can be used by multiple SUs.

Figure 1

System model.

The SINR at the kth PU receiver on the nth channel can be defined as

$$\begin{array}{cc}& {\gamma }_k^n=\frac{p_k^n{h}_{kk}^n}{{\sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n{p}_l^n{h}_{lk}^n+{\delta }^{\mathrm{2}}}=\frac{p_k^n{h}_{kk}^n}{I\left({\mathbf{p}}_{-k}^n\right)},\end{array}$$

(1)

where if the PU k and the SU l transmit on the same channel n, ${\rho }_{l,k}^n=\mathrm{1}$ ; otherwise, ${\rho }_{l,k}^n=\mathrm{0}$ . ${\delta }_{}^{\mathrm{2}}$ is variance of AWGN and the interference is given by $I({\mathbf{p}}_{-k}^n)={\sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n{p}_l^n{h}_{lk}^n+{\delta }_{}^{\mathrm{2}}$ .

The SINR at the lth SU receiver on the channel n can be expressed by

$$\begin{array}{cc}& {\gamma }_l^n=\frac{p_l^n{h}_{ll}^n}{{\sum }_{j=\mathrm{1},j\ne l}^L{\rho }_{l,j}^n{p}_j^n{h}_{lj}^n+p_k^n{h}_{kl}^n+{\delta }_{}^{\mathrm{2}}}=\frac{p_l^n{h}_{ll}^n}{I\left({\mathbf{p}}_{-l}^n\right)},\end{array}$$

(2)

where $I({\mathbf{p}}_{-l}^n)={\sum }_{j=\mathrm{1},j\ne l}^L{\rho }_{l,j}^n{p}_j^n{h}_{lj}^n+p_k^n{h}_{kl}^n+{\delta }_{}^{\mathrm{2}}$ .

The required target SINR threshold ${\stackrel{-}{\gamma }}_u^n$ of the user u (PUs or SUs) which is not satisfied is defined as the outage probability on the channel n. The outage probability $\mathrm{P}\mathrm{r}({\gamma }_u^n<{\stackrel{-}{\gamma }}_u^n)$ of the user u on the channel n can be expressed as [10]

$$\begin{array}{cc}& \mathrm{P}\mathrm{r}\left({\gamma }_u^n<{\stackrel{-}{\gamma }}_u^n\right)=\mathrm{1}-{\displaystyle \prod }_{i\ne u}^{}{\left(\mathrm{1}+{\rho }_{i,u}^n\frac{{\stackrel{-}{\gamma }}_u^n{p}_i^n{h}_{iu}^n}{p_u^n{h}_{uu}^n}\right)}^{-\mathrm{1}}.\end{array}$$

(3)

Therefore, the above equation shows the outage probability of one user in the presence of multiple SUs and/or PUs. Accordingly, the goal of each PU k is to transmit with the minimum target SINR ${\stackrel{-}{\gamma }}_k^n$ on the channel n with the prescribed limited channel outage ${\xi }_k^n$ constraint

$$\begin{array}{cc}& \mathrm{P}\mathrm{r}\left({\gamma }_k^n<{\stackrel{-}{\gamma }}_k^n\right)<{\xi }_k^n,\hspace{1em}\forall k,\hspace{0.17em}\forall n,\end{array}$$

(4)

where (4) means the outage constraint of the PU k on the channel n that guarantees its required QoS. Similarly, we define every SU's minimum target SINR ${\stackrel{-}{\gamma }}_l^n$ , and the channel outage ${\xi }_l^n$ constraint of each SU l on the channel n is

$$\begin{array}{cc}& \mathrm{P}\mathrm{r}\left({\gamma }_l^n<{\stackrel{-}{\gamma }}_l^n\right)<{\xi }_l^n,\hspace{1em}\forall l,\hspace{0.17em}\forall n.\end{array}$$

(5)

For each PU k, taking (3) into (4), we can get

$$\begin{array}{cc}& {\displaystyle \prod }_l^L\left(\mathrm{1}+{\rho }_{l,k}^n\frac{{\stackrel{-}{\gamma }}_k^n{p}_l^n{h}_{lk}^n}{p_k^n{h}_{kk}^n}\right)<\frac{\mathrm{1}}{\mathrm{1}-{\xi }_k^n},\hspace{1em}\forall k,\hspace{0.17em}\forall n.\end{array}$$

(6)

After rewriting (6), we can obtain

$$\begin{array}{cc}& {\displaystyle \sum }_l^L\log \left(\mathrm{1}+{\rho }_{l,k}^n\frac{{\stackrel{-}{\gamma }}_k^n{p}_l^n{h}_{lk}^n}{p_k^n{h}_{kk}^n}\right)\le \log \left(\frac{\mathrm{1}}{\mathrm{1}-{\xi }_k^n}\right),\hspace{1em}\forall k,\hspace{0.17em}\forall n.\end{array}$$

(7)

Similarly, taking (3) into (5), rewrite (5) as follows:

$$\begin{array}{cc}& {\displaystyle \sum }_{j\ne l}^{L+\mathrm{1}}\log \left(\mathrm{1}+{\rho }_{j,l}^n\frac{{\stackrel{-}{\gamma }}_l^n{p}_j^n{h}_{jl}^n}{p_l^n{h}_{ll}^n}\right)\le \log \left(\frac{\mathrm{1}}{\mathrm{1}-{\xi }_l^n}\right),\hspace{1em}\forall l,\hspace{0.17em}\forall n.\end{array}$$

(8)

In addition, the global aggregate interference from all SUs to each channel should not be larger than the maximum interference threshold $T_k^n$ to ensure the SUs' transmission would not cause unendurable interference on every channel of each PU. Mathematically, this can be written as

$$\begin{array}{cc}& {\displaystyle \sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n{p}_l^n{h}_{lk}^n<T_k^n,\hspace{1em}\forall k,\hspace{0.17em}\forall n.\end{array}$$

(9)

3. Stackelberg Game Theoretic Approach

The PUs (leaders) price the SUs (followers) to control the interference power made by the SUs under the IPC. Each PU will offer a suitable price to maximize its revenue by selling resource to SUs. Based on the interference price provided by PUs, each SU will adjust its transmission power to maximize its revenue. PUs have higher priority than SUs, we use Stackelberg game to model the strategy between the PUs and SUs.

Then, as the leaders, PUs will maximize their utility (SINR performance plus the payment from the SUs occupying the channels). The utility function of the PU k is as follows:

$$\begin{array}{cc}& U_k\left({\mathbf{p}}_k,{\mathbf{p}}_{-k},w_k\right)={\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_k^n{\gamma }_k^n+w_k{\displaystyle \sum }_{n=\mathrm{1}}^N\hspace{0.17em}{\displaystyle \sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n{p}_l^n{h}_{lk}^n,\end{array}$$

(10)

where if the PU k transmit on the channel n, ${\rho }_k^n=\mathrm{1}$ ; otherwise, ${\rho }_k^n=\mathrm{0}$ . ${\mathbf{p}}_k=(p_k^{\mathrm{1}},p_k^{\mathrm{2}},\dots ,p_k^N)$ is the transmission power vector over the transmitted channels of the PU k; ${\mathbf{p}}_{-k}$ represent the transmit power vector of all users except the PU k. $w_k$ denotes that the PU k charges the price over all SUs if the SUs transmit on the channel of the PU.

Hence, the revenue utility optimization problem for the PU k is as follows:

$$\begin{array}{cc}& \mathbf{P}\mathbf{1}\hspace{0.17em}\text{:\hspace{0.17em}\hspace{0.17em}}\mathrm{M}\mathrm{a}\mathrm{x}\hspace{1em}{U}_k\left({\mathbf{p}}_k,{\mathbf{p}}_{-k},w_k\right),\\ \text{s.t.}\hspace{1em}\left(\mathrm{8}\right)\left(\mathrm{10}\right),\hspace{0.17em}{p}_k^n\ge \mathrm{0},\hspace{0.17em}{\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_k^n{p}_k^n\hspace{0.17em}{\le P}_k^{\mathrm{m}\mathrm{a}\mathrm{x}},\end{array}$$

(11)

where $P_k^{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum transmission power of the PU k.

Then, for SUs (followers), we set the revenue utility of the lth SU with two parts: the first one is the income from the SINR achieved from the PUs. The second one is the payment for the PUs. Then, the revenue utility function of the lth SU is as follows:

$$\begin{array}{cc}& U_l\left({\mathbf{p}}_l,{\mathbf{p}}_{-l},{\mathbf{w}}_l\right)={\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_l^n{\gamma }_l^n-{\displaystyle \sum }_{n=\mathrm{1}}^N\hspace{0.17em}{\displaystyle \sum }_{k=\mathrm{1}}^K{\rho }_{l,k}^n{w}_k{p}_l^n{h}_{lk}^n,\end{array}$$

(12)

where if the SU l transmit on the channel n, ${\rho }_l^n=\mathrm{1}$ ; otherwise, ${\rho }_l^n=\mathrm{0}$ . ${\mathbf{p}}_l=(p_l^{\mathrm{1}},p_l^{\mathrm{2}},\dots ,p_l^N)$ is the transmission power vector over the transmitted channels of the SU l; ${\mathbf{p}}_{-l}$ represent the transmission power vector of all users except the SU l. ${\mathbf{w}}_l=(w_{\mathrm{1}},w_{\mathrm{2}},\dots ,w_K)$ denotes that the SU l pays the price vector for all PUs; if the SU l does not transmit on the channel of the PU k, $w_k=\mathrm{0}$ .

Hence, the optimization problem for the SU l is as follows:

$$\begin{array}{cc}& \mathbf{P}\mathbf{2}\text{:\hspace{0.17em}\hspace{0.17em}}\mathrm{M}\mathrm{a}\mathrm{x}\hspace{1em}{U}_l\left({\mathbf{p}}_l,{\mathbf{p}}_{-l},{\mathbf{w}}_l\right)\\ \text{s.t.}\hspace{1em}\left(\mathrm{9}\right)\left(\mathrm{10}\right),\hspace{0.17em}{p}_l^n\ge \mathrm{0},\hspace{0.17em}{\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_l^n{p}_l^n\le P_l^{\mathrm{m}\mathrm{a}\mathrm{x}},\end{array}$$

(13)

where $P_l^{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum transmission power of the lth SU.

4. Solution of the Proposed Stackelberg Game

The optimization problems $\mathbf{P}\mathbf{1}$ and $\mathbf{P}\mathbf{2}$ taken together are the proposed Stackelberg game problem with several constraints. The goal of the proposed Stackelberg game is to achieve the SE, in which point both PUs and SUs have no incentive to deviate [3], so, the distributive algorithms convergence to the SE is difficult. Thus, we depart the game problem into suboptimal independent solution and employ an iterative algorithm to achieve the SE.

4.1. Global Efficiency of the RSE

In noncooperative games, the existence and uniqueness of equilibrium are not always achieved [11] due to multiple players' competition. Hence, for the multifollower subgame, we should study the existence and uniqueness of the global SG response to the leaders' price. In particular, a variational equilibrium (VE) [12] is applied to analyze the global SG for our case. This is because a VE is more stable than any other generalized Nash equilibrium under parameter uncertainty [13]. Particularly, a number of SUs aim to achieve their QoS requirement through applying the resource from PUs in cognitive radio networks; VE is regarded as an appropriate solution.

For a market fixed price at the PUs, all the SUs aim to maximize their own utility by buying the resource through PUs. Thus, we formulate a new objective function for all SUs; the new utility is the sum utilities of all SUs and can be expressed as follows:

$$\begin{array}{cc}& {\tilde{U}}_{\text{Sum}}^F={\displaystyle \sum }_{l=\mathrm{1}}^L\hspace{0.17em}{\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_l^n{\gamma }_l^n-{\displaystyle \sum }_{l=\mathrm{1}}^L\hspace{0.17em}{\displaystyle \sum }_{n=\mathrm{1}}^N\hspace{0.17em}{\displaystyle \sum }_{k=\mathrm{1}}^K{\rho }_{l,k}^n{w}_k{p}_l^n{h}_{lk}^n.\end{array}$$

(14)

Therefore, in order to achieve the actable outcome of the proposed SG, our goal is guarantying the existence and uniqueness of the SG when maximizing (14). The corresponding Lagrangian function and the KTT conditions for the SU l expressed in and the KTT conditions for the SU l are given by

$$\begin{array}{cc}& {\nabla }_{{\mathbf{p}}_l}{U}_l\left({\mathbf{p}}_l^{},{\mathbf{p}}_{-l}^{},{\mathbf{w}}_l\right)-{\nabla }_{{\mathbf{p}}_l}\left({\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_l^n{p}_l^n-P_l^{\mathrm{m}\mathrm{a}\mathrm{x}}\right){\upsilon }_l-{\nabla }_{{\mathbf{p}}_l}\left({\displaystyle \sum }_{n=\mathrm{1}}^N{\omega }_l^n\left({\displaystyle \sum }_{j\ne l}^{L+\mathrm{1}}\log \left(\mathrm{1}+{\rho }_{j,l}^n\frac{{\stackrel{-}{\gamma }}_l^n{p}_j^n{h}_{jl}^n}{p_l^n{h}_{ll}^n}\right)-\log \left(\frac{\mathrm{1}}{\mathrm{1}-{\xi }_l^n}\right)\right)\right)-{\nabla }_{{\mathbf{p}}_l}{\displaystyle \sum }_{n=\mathrm{1}}^N{\phi }_l^n\left({\displaystyle \sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n{p}_l^n{h}_{lk}^n-T_k^n\right){\upsilon }_l\left({\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_l^n{p}_l^n-P_l^{\mathrm{m}\mathrm{a}\mathrm{x}}\right)=\mathrm{0},\\ {\displaystyle \sum }_{n=\mathrm{1}}^N{\omega }_l^n\left({\displaystyle \sum }_{j\ne l}^{L+\mathrm{1}}\log \left(\mathrm{1}+{\rho }_{j,l}^n\frac{{\stackrel{-}{\gamma }}_l^n{p}_j^n{h}_{jl}^n}{p_l^n{h}_{ll}^n}\right)-\log \left(\frac{\mathrm{1}}{\mathrm{1}-{\xi }_l^n}\right)\right)=\mathrm{0},\\ {\displaystyle \sum }_{n=\mathrm{1}}^N{\phi }_l^n\left({\displaystyle \sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n{p}_l^n{h}_{lk}^n-T_k^n\right)=\mathrm{0}.\end{array}$$

(15)

We note that the robust followers' game shows a jointly convex generalized SE problem; therefore, the solution of the SE problem with constraints in (13) is a variational inequality $\mathrm{V}\mathrm{I}\left(\mathbf{P},\mathbf{F}\right)$ , where P is the set of joint convexity. It is important to determine a vector $z^{\ast }\in \mathbf{P}\subset R^n$ , such that $〈\mathbf{F}({\mathbf{z}}^{\ast }),\mathbf{z}-{\mathbf{z}}^{\ast }〉\ge \mathrm{0}$ , for all $z\in \mathbf{P}$ and $\mathbf{F}(\mathbf{p})=-({\nabla }_p{\tilde{U}}_l({\mathbf{p}}_l^{}){)}_{l=\mathrm{1}}^L$ [13]. Then, the solution of $\mathrm{V}\mathrm{I}\left(\mathbf{P},\mathbf{F}\right)$ is a variational SE.

In this paper, we only focus on the power control in cognitive radio networks by assuming the channel assignment has already been done. Then, we can divide the variational inequality $\mathrm{V}\mathrm{I}\left(\mathbf{P},\mathbf{F}\right)$ into N subproblems; each subproblem denotes $\mathrm{V}\mathrm{I}\left({\mathbf{P}}_n,{\mathbf{F}}_n\right)$ on the subchannel n and they are independent. Therefore, on the subchannel n, the KKT conditions can be expressed as [12]

$$\begin{array}{cc}& {\mathbf{F}}_n\left(\mathbf{p}\right)+{\upsilon }_n{\nabla }_p\left({\displaystyle \sum }_{l=\mathrm{1}}^L{\rho }_l^n{p}_l^n-{\displaystyle \sum }_{l=\mathrm{1}}^L{\rho }_l^n{P}_l^{\mathrm{m}\mathrm{a}\mathrm{x}}\right)+{\omega }_n{\nabla }_p\left({\displaystyle \sum }_{l=\mathrm{1}}^L\left({\displaystyle \sum }_{j\ne l}^{L+\mathrm{1}}\log \left(\mathrm{1}+{\rho }_{j,l}^n\frac{{\stackrel{-}{\gamma }}_l^n{p}_j^n{h}_{jl}^n}{p_l^n{h}_{ll}^n}\right)\right)-{\displaystyle \sum }_{l=\mathrm{1}}^L\log \left(\frac{\mathrm{1}}{\mathrm{1}-{\xi }_l^n}\right)\right)+{\phi }_n{\nabla }_p\left({\displaystyle \sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n{p}_l^n{h}_{lk}^n-T_k^n\right).\end{array}$$

(16)

Now from the definition of [12], we have

$$\begin{array}{cc}& {\mathbf{F}}_{\mathrm{1}}=\left[\begin{array}{c}{\rho }_{\mathrm{1},k}^{\mathrm{1}}\left(w_k{h}_{\mathrm{1}k}^{\mathrm{1}}-\frac{\mathrm{1}}{I\left(p_{-\mathrm{1}}^{\mathrm{1}}\right)}\right)\\ {\rho }_{\mathrm{2},k}^{\mathrm{1}}\left(w_k{h}_{\mathrm{2}k}^{\mathrm{1}}-\frac{\mathrm{1}}{I\left(p_{-\mathrm{2}}^{\mathrm{1}}\right)}\right)\\ ⋮\\ {\rho }_{L,k}^{\mathrm{1}}\left(c_k{h}_{Lk}^{\mathrm{1}}-\frac{\mathrm{1}}{I\left(p_{-L}^{\mathrm{1}}\right)}\right)\end{array}\right],\\ ⋮\\ {\mathbf{F}}_n=\left[\begin{array}{c}{\rho }_{\mathrm{1},k}^n\left(c_k{h}_{\mathrm{1}k}^n-\frac{\mathrm{1}}{I\left(p_{-\mathrm{1}}^n\right)}\right)\\ {\rho }_{\mathrm{2},k}^n\left(c_k{h}_{\mathrm{2}k}^n-\frac{\mathrm{1}}{I\left(p_{-\mathrm{2}}^n\right)}\right)\\ ⋮\\ {\rho }_{L,k}^n\left(c_k{h}_{Lk}^n-\frac{\mathrm{1}}{I\left(p_{-L}^n\right)}\right)\end{array}\right].\end{array}$$

(17)

Therefore, the Jacobian of ${\mathbf{F}}_n$ is

$$\begin{array}{cc}& {\mathbf{J}}_1=\left[\begin{array}{cccc}{\rho }_{\mathrm{1},k}^{\mathrm{1}}{h}_{\mathrm{1}k}^{\mathrm{1}}& \mathrm{0}& \cdots & \mathrm{0}\\ \mathrm{0}& {\rho }_{\mathrm{2},k}^{\mathrm{1}}{h}_{\mathrm{2}k}^{\mathrm{1}}& \cdots & \mathrm{0}\\ \mathrm{0}& & ⋮& \\ \mathrm{0}& \mathrm{0}& \cdots & {\rho }_{L,k}^{\mathrm{1}}{h}_{Lk}^{\mathrm{1}}\end{array}\right],\\ ⋮\\ {\mathbf{J}}_n=\left[\begin{array}{cccc}{\rho }_{\mathrm{1},k}^n{h}_{\mathrm{1}k}^n& \mathrm{0}& \cdots & \mathrm{0}\\ \mathrm{0}& {\rho }_{\mathrm{2},k}^n{h}_{\mathrm{2}k}^n& \cdots & \mathrm{0}\\ \mathrm{0}& & ⋮& \\ \mathrm{0}& \mathrm{0}& \cdots & {\rho }_{L,k}^n{h}_{Lk}^n\end{array}\right].\end{array}$$

(18)

Each ${\mathbf{F}}_n{\mathbf{J}}_n$ is a diagonal matrix and all the diagonal elements are positive. Therefore, ${\mathbf{F}}_n{\mathbf{J}}_n$ is positive definition on ${\mathbf{P}}_n$ , and so ${\mathbf{F}}_n$ is strictly monotone. Hence, the global SG problem admits a unique global variational equilibrium solution [12]. Due to the jointly convex nature of the global SE problem, the variational equilibrium is the unique global maximizer of (14) [12], which completes the proof in the literature [12].

4.2. Solution of the Optimization for SUs (Followers)

For the $\mathit{P}\mathit{2}$ , the utility function of each SU is a concave function of $p_l^n$ and the constraints are all linear, so a partial Lagrange dual decomposition method (LDDM) [4] for the problems is used.

For the problem in (13), the corresponding Lagrangian function for the SU l on the subchannel n can be expressed as

$$\begin{array}{cc}& {\mathcal{L}}_l^{}\left({\mathbf{p}}_l,{\mathbf{w}}_k,{\omega }_l^{},{\upsilon }_l,{\mathit{\phi }}_l\right)={\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_l^n{\gamma }_l^n-{\displaystyle \sum }_{n=\mathrm{1}}^N\hspace{0.17em}{\displaystyle \sum }_{k=\mathrm{1}}^K{\rho }_{l,k}^n{w}_k{p}_l^n{h}_{lk}^n-{\displaystyle \sum }_{n=\mathrm{1}}^N{\omega }_l^n\left({\displaystyle \sum }_{j\ne l}^{L+\mathrm{1}}\log \left(\mathrm{1}+{\rho }_{j,l}^n\frac{{\stackrel{-}{\gamma }}_l^n{p}_j^n{h}_{jl}^n}{p_l^n{h}_{ll}^n}\right)-\log \left(\frac{\mathrm{1}}{\mathrm{1}-{\xi }_l^n}\right)\right)-{\upsilon }_l\left({\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_l^n{p}_l^n-P_l^m\right)-{\displaystyle \sum }_{n=\mathrm{1}}^N{\phi }_l^n\left({\displaystyle \sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n{p}_l^n{h}_{lk}^n-T_k^n\right),\end{array}$$

(19)

where ${\omega }_l^n$ , ${\upsilon }_l$ , and ${\phi }_l^n$ are the nonnegative dual variables of the constraints in (13).

We decompose the optimization problem into N independent subproblems. Then, on the subchannel n, taking the Karush-Kuhn-Tucker (KKT) condition [4],

$$\begin{array}{cc}& \frac{\partial {\mathcal{L}}_l^{}}{\partial p_l^n}=\frac{h_{ll}^n}{I\left({\mathbf{p}}_{-l}^n\right)}-w_k{h}_{lk}^n+{\omega }_l^n{{\displaystyle \sum }}_{j\ne l}^{L+\mathrm{1}}\left({\rho }_{j,l}^n\frac{{\stackrel{-}{\gamma }}_l^n{p}_j^n{h}_{jl}^n}{p_l^n\left(p_l^n{h}_{ll}^n+{\stackrel{-}{\gamma }}_l^n{p}_j^n{h}_{jl}^n\right)\ln \mathrm{2}}\right)-{\upsilon }_l-{\phi }_k{h}_{lk}^n.\end{array}$$

(20)

Simply, in (20), we assume that the jth user causing the interference power $p_j^n{h}_{jl}^n$ to the SU l on the subchannel n can be denoted as the average interference power except the SU l: $G_l^n=E[{\sum }_{j=\mathrm{1},j\ne l}^{L+\mathrm{1}}{\rho }_{j,l}^n{p}_j^n{h}_{jl}^n]$ . Hence, we rewrite (20) as follows:

$$\begin{array}{cc}& \frac{\partial {\mathcal{L}}_l^{}}{\partial p_l^n}=\frac{h_{ll}^n}{I\left({\mathbf{p}}_{-l}^n\right)}-w_k{h}_{lk}^n+{\omega }_l^n\frac{{\stackrel{-}{\gamma }}_l^n{G}_l^n\left({\sum }_{j\ne l}^{L+\mathrm{1}}{\rho }_{j,l}^n\right)}{p_l^n\left(p_l^n{h}_{ll}^n+{\stackrel{-}{\gamma }}_l^n{G}_l^n\right)\ln \mathrm{2}}-{\upsilon }_l-{\phi }_k{h}_{lk}^n.\end{array}$$

(21)

Set (21) to zero and get the optimal transmission power of the SU l if it transmits on the channel n:

$$\begin{array}{cc}& {p_l^n}^{\ast }=\frac{-{\stackrel{-}{\gamma }}_l^n{G}_l^n+\sqrt{\mathrm{4}{\omega }_l^n{h}_{ll}^n{\stackrel{-}{\gamma }}_l^n{G}_l^n\left({\sum }_{j\ne l}^{L+\mathrm{1}}{\rho }_{j,l}^n\right)/X\ln 2+{\stackrel{-}{\gamma }}_l^n{G}_l^n}}{\mathrm{2}{h}_{ll}^n},\end{array}$$

(22)

where $X_l^n=(w_k+{\phi }_l^n)h_{lk}^n+{\upsilon }_l-h_{ll}^n/I\left({\mathbf{p}}_{-l}^n\right)$ .

The transmission power of the SU l is zero if the interference price for it is larger than payoff threshold $Q_l^n$ on the channel n. Then, setting ${p_l^n}^{\ast }=\mathrm{0}$ , we can get the payoff threshold of the SU l if it transmits on the channel n:

$$\begin{array}{cc}& Q_l^n=\frac{\mathrm{1}}{h_{lk}^n}\left(\frac{\mathrm{4}{\omega }_l^n{h}_{ll}^n{\stackrel{-}{\gamma }}_l^n{G}_l^n\left({\sum }_{j\ne l}^{L+\mathrm{1}}{\rho }_{j,l}^n\right)}{\left({\left({\stackrel{-}{\gamma }}_l^n{G}_l^n\right)}^{\mathrm{2}}-{\stackrel{-}{\gamma }}_l^n{G}_l^n\right)\ln \mathrm{2}}-{\upsilon }_l-{\phi }_l^n{h}_{lk}^n+\frac{h_{ll}^n}{I\left({\mathbf{p}}_{-l}^n\right)}\right).\end{array}$$

(23)

From (23), if the price $w_k^{}>Q_l^n$ , the price is above the payoff threshold of the SU l, and it will stop transmitting on the channel n without buying the interference power.

4.3. Solution of the Optimization for PUs (Leaders)

In order to maximize its own utility, each PU needs to adaptively offer an interference price to SUs based on transmit power response of the SUs. $\mathbf{P}\mathbf{1}$ can be decomposed into two subproblems: fix $w_k$ to get the optimal transmission power of each PU k, and then search the optimal $w_k$ . The optimal transmission power of the PU k can be applied by the previous LDDM.

Thus, for $\mathbf{P}\mathbf{1}$ in (11), the corresponding Lagrangian function can be given as

$$\begin{array}{cc}& {\mathcal{L}}_k^{}\left({\mathbf{p}}_k,w_k,{\mathbf{\lambda }}_k^{},{\mu }_k,{\mathbf{\nu }}_k\right)={\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_k^n{\gamma }_k^n+w_k{\displaystyle \sum }_{n=\mathrm{1}}^N\hspace{0.17em}{\displaystyle \sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n{p}_l^n{h}_{lk}^n-{\displaystyle \sum }_{n=\mathrm{1}}^N{\lambda }_k^n\left({\displaystyle \sum }_l^L\log \left(\mathrm{1}+{\rho }_{l,k}^n\frac{{\stackrel{-}{\gamma }}_k^n{p}_l^n{h}_{lk}^n}{p_k^n{h}_{kk}^n}\right)-\log \left(\frac{\mathrm{1}}{\mathrm{1}-{\xi }_k^n}\right)\right)-{\mu }_k\left({\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_k^n{p}_k^n-P_k^m\right)-{\displaystyle \sum }_{n=\mathrm{1}}^N{\nu }_k^n\left({\displaystyle \sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n{p}_l^n{h}_{lk}^n-T_k^n\right),\end{array}$$

(24)

where ${\lambda }_k^n,{\mu }_k$ , and ${\nu }_k^n$ are the dual variables of the constraints in (11).

Similarly, in (24), we assume the lth SU causing the interference $h_{lk}^n{p}_l^n$ to the PU k on the channel n can be denoted as the average interference power: $G_k^n=E[{\sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n{h}_{lk}^n{p}_l^n].$ According to the KKT conditions, we obtain the optimal transmission power of the PU k if it transmits on the channel n:

$$\begin{array}{cc}& {p_k^n}^{\ast }=\frac{\mathrm{1}}{\mathrm{2}{h}_{kk}^n}\left(-{\stackrel{-}{\gamma }}_k^n{G}_k^n+\sqrt{\frac{\mathrm{4}{\lambda }_k^n{h}_{kk}^n{\stackrel{-}{\gamma }}_k^n{G}_k^n\left({\sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n\right)}{\left({\mu }_k-h_{kk}^n/I\left({\mathbf{p}}_{-k}^n\right)\right)\ln \mathrm{2}}+{\stackrel{-}{\gamma }}_k^n{G}_k^n}\right).\end{array}$$

(25)

Since ${\mathcal{L}}_k^{}\left({\mathbf{p}}_k,w_k,{\mathbf{\lambda }}_k^{},{\mu }_k,{\mathbf{\nu }}_k\right)$ is a stepwise function with breakpoints at $Q_l^n$ for the SU l, we should discuss the existence of the optimal price $w_k$ first. So we divide (24) with respect to $w_k$ with two parts on each channel n; we have ${\mathcal{L}}_{p,k}^{}(p_l^n)={\gamma }_k^n(p_l^n)$ and ${\mathcal{L}}_{p,k}^{}(w_k)=(w_k-{\nu }_k)p_l^n{h}_{lk}$ . From (22), it can be easily observed that ${\gamma }_k^n(p_l^n)$ is a concave function of $w_k$ . Therefore, we only need to discuss the situation of ${\mathcal{L}}_{p,k}^{}(w_k)$ . For the SU l, we first sort $Q_l^n$ ( $n=\mathrm{1},\dots ,N$ ) in ascending order and have N intervals $(\mathrm{0},Q_l^{\mathrm{1}})(Q_l^{\mathrm{1}},Q_l^{\mathrm{2}}),\dots ,(Q_l^{N-\mathrm{1}},Q_l^N)$ , where $Q_l^{\mathrm{1}}<Q_l^{\mathrm{2}}<\cdots <Q_l^N$ . Note, if the SU l is not allocated on the channel n, $(Q_l^{n-\mathrm{1}},Q_l^n)$ must be taken out of the order. We take $(\mathrm{0},Q_l^{\mathrm{1}})$ for an example. When $w_k\to \mathrm{0}$ , we can derive that

$$\begin{array}{cc}& {\left.\frac{\partial {\mathcal{L}}_{p,k}^{}\left(w_k\right)}{\partial w_k}\right|}_{w_k\to \mathrm{0}}>p_l^n{h}_{lk}>\mathrm{0}.\end{array}$$

(26)

Taking the second derivative of ${\mathcal{L}}_{p,k}^{}(w_k)$ with respect to $w_k$ is

$$\begin{array}{cc}& \frac{{\partial }^{\mathrm{2}}{\mathcal{L}}_{p,k}^{}}{\partial w_k^{\mathrm{2}}}=-\frac{h_{ll}^n}{\mathrm{2}}\sqrt{\frac{{\omega }_l^n{\stackrel{-}{\gamma }}_l^n{G}_l^n\left({\sum }_{j\ne l}^{L+\mathrm{1}}{\rho }_{j,l}^n\right)}{X_l^n\ln \mathrm{2}}+{\stackrel{-}{\gamma }}_l^n{G}_l^n}\left(\frac{{\omega }_l^n{h}_{lk}^n{\stackrel{-}{\gamma }}_l^n{G}_l^n\left({\sum }_{j\ne l}^{L+\mathrm{1}}{\rho }_{j,l}^n\right)}{{\left(X_l^n\right)}^{\mathrm{2}}\ln \mathrm{2}}\right)<\mathrm{0}.\end{array}$$

(27)

${\mathcal{L}}_{p,k}^{}(w_k)$ is a concave function whether $\left(\partial {\mathcal{L}}_{p,k}^{}/\partial w_k\right){|}_{w_k\to Q_l^{\mathrm{1}}}>\mathrm{0}$ or $\left(\partial {\mathcal{L}}_{p,k}^{}/\partial w_k\right){|}_{w_k\to Q_l^{\mathrm{1}}}<\mathrm{0}$ except at the nondifferentiable point $Q_l^{\mathrm{1}}$ .

Through the above analysis, ${\mathcal{L}}_k^{}\left({\mathbf{p}}_k,w_k,{\mathbf{\lambda }}_k^{},{\mu }_k,{\mathbf{\nu }}_k\right)$ is a concave function with respect to $w_k$ except at $Q_l^n$ . The ellipsoid method [4] can be employed to solve the convex optimization in each interval.

4.4. Iterative Algorithm to Find the SE

For the above discussion, we propose an iterative algorithm to search the SE. Due to the fact that computation of the dual variables is a complicated task, the subgradient method (SM) [14] is applied to obtain the global optimum SE of this problem. Then, dual variables are updated as follows:

$$\begin{array}{cc}& {\omega }_l^n\left(t+\mathrm{1}\right)={\left({\omega }_l^n\left(t\right)+\alpha \left({\displaystyle \sum }_{j\ne l}^{L+\mathrm{1}}\log \left(\mathrm{1}+{\rho }_{j,l}^n\frac{{\stackrel{-}{\gamma }}_l^n{p}_j^n{h}_{jl}^n}{p_l^n{h}_{ll}^n}\right)-\log \left(\frac{\mathrm{1}}{\mathrm{1}-{\xi }_l^n}\right)\right)\right)}^{+},\\ {\upsilon }_l\left(t+\mathrm{1}\right)={\left({\upsilon }_l\left(t\right)+\beta \left({\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_l^n{p}_l^n-p_l^{\mathrm{m}\mathrm{a}\mathrm{x}}\right)\right)}^{+},\\ {\phi }_l^n\left(t+\mathrm{1}\right)={\left({\phi }_l^n\left(t\right)+\theta \left({\displaystyle \sum }_{l=\mathrm{1}}^L{\rho }_{l,k}^n{p}_l^n{h}_{lk}^n-T_k^n\right)\right)}^{+},\end{array}$$

(28)

$$\begin{array}{cc}& {\lambda }_k^n\left(t+\mathrm{1}\right)={\left({\lambda }_k^n\left(t\right)+\vartheta \left({\displaystyle \sum }_l^L\log \left(\mathrm{1}+{\rho }_{l,k}^n\frac{{\stackrel{-}{\gamma }}_k^n{p}_l^n{h}_{lk}^n}{p_k^n{h}_{kk}^n}\right)-\log \left(\frac{\mathrm{1}}{\mathrm{1}-{\xi }_k^n}\right)\right)\right)}^{+},\\ {\mu }_k\left(t+\mathrm{1}\right)={\left({\mu }_k\left(t\right)+\chi \left({\displaystyle \sum }_{n=\mathrm{1}}^N{\rho }_k^n{p}_k^n-P_k^{\mathrm{m}\mathrm{a}\mathrm{x}}\right)\right)}^{+},\end{array}$$

(29)

where t is the iteration index and $\alpha >\mathrm{0}$ , $\beta >\mathrm{0}$ , $\theta >\mathrm{0}$ , $\vartheta >\mathrm{0}$ , and $\chi >\mathrm{0}$ are sufficiently small. The SM guarantees the convergence of the above optimal dual variables if the step sizes are chosen by following the step size policy [14].

We then design the iterative algorithm to achieve the SE shown in Algorithm 1.

Algorithm 1: Iterative algorithm for reaching the SE.

$(\mathrm{01}$ ) Initialization: set $q=\mathrm{0}$ , set initial $c_k(q)$ and ${\mathbf{p}}_k(q)$ for $k\in K$ . Set initial ${\mathbf{p}}_l(q)$ for $l\in L$ .

      Set τ, where τ is positive and sufficiently small.

(02) For each SUl

(03) Use SM to find the optimal step sizes ${\alpha }^{\ast }$ , ${\beta }^{\ast }$ and ${\theta }^{\ast }$ , and update ${\omega }_l^n$ , ${\upsilon }_l$ and ${\phi }_k$ according to (28), respectively.

(04) For the given $c_k(q)$ and ${\mathbf{p}}_k(q)$ of all PUs, each SU l responds with its transmit power vector ${\mathbf{p}}_l^{\ast }(q+\mathrm{1})$ according to (22).

(05) If $Q_l^n<c_k(q)$ , the SU l stops transmitting on the channel n of the PU k.

(06) End

(07) For each PUk

(08) Use SM to find the optimal ${\chi }^{\ast }$ and ${\vartheta }^{\ast }$ , and update ${\lambda }_k^n$ and ${\mu }_k$ by (29).

(09) For the responded ${\mathbf{p}}_l^{\ast }(q)$ of all SUs, each PU k updates its transmit power vector as ${\mathbf{p}}_k^{\ast }(q+\mathrm{1})$ , according to (25).

(10) Each PU k updates its price by the solution of

       $$c_k\left(q+\mathrm{1}\right)=\arg \underset{c_k}{\max }\left\{U_k\left({\mathbf{p}}_k\left(q+\mathrm{1}\right),{\mathbf{p}}_{-k}\left(q+\mathrm{1}\right)\right)\right\}$$

(11) End

(12) For each PU k, if $‖{\mathbf{p}}_k^{\ast }(q+\mathrm{1})-{\mathbf{p}}_k(q)‖\le \tau$ or $q>\mathrm{1}{\mathrm{0}}^{\mathrm{2}}$ , stop the algorithm. Otherwise, $q=q+\mathrm{1}$ ,

repeat steps (02) and (11) until the condition is satisfied.

5. Simulation Results and Their Analysis

In this section, several numerical examples are presented to evaluate the performances of the proposed SG by comparing the optimal price-based SG considering global interference in [5] and the nominal SG without considering global interference and outage constraints. The cell radius of 500 m with the PBS centered at the original CRN. The simulation parameters are as follows: ${\delta }^{\mathrm{2}}=\mathrm{1}{\mathrm{0}}^{-\mathrm{12}}$  W, $K=\mathrm{3}$ , $L=\mathrm{5}$ , and $N=\mathrm{10}$ . The SINR threshold of PUs and SUs is set as 7 dB and 4 dB, respectively. All PUs and SUs deploy the maximum power: $P_k^{\mathrm{m}\mathrm{a}\mathrm{x}}=\mathrm{100}$  mW and $P_l^{\mathrm{m}\mathrm{a}\mathrm{x}}=\mathrm{50}$  mW. The channel gain in this system is $h_{ij}=d_{ij}^{-\mathrm{4}}$ , with $d_{ij}$ being their corresponding distance. The outage probability thresholds of both PUs and SUs are 0.001. We set the interference-to-noise ratio (INR) as $T/{\delta }^{\mathrm{2}}$ .

Firstly, we illustrate the convergence of the proposed algorithm for achieving an SE of the proposed game. From Figure 2, the three PUs and five SUs iteratively update their utilities and obviously converge to the SE. The proposed algorithm converges quickly in terms of PUs, only about ten times. In addition, due to the larger number of SUs, the convergence of the SUs is slower than that of the PUs.

Figure 2

The convergence of utility of PUs (leaders), setting price of PUs, and utility of SUs (followers).

5.1. Impact of INR

In this subsection, we set the number of PUs and SUs as 3 and 5, and INR changes from −20 dB to 20 dB which means that IPC changes from $\mathrm{1}{\mathrm{0}}^{-\mathrm{14}}$  W to $\mathrm{1}{\mathrm{0}}^{-\mathrm{10}}$  W.

We then consider the sum rate of PUs and SUs for the three solutions with different tolerant interference constraints shown in Figures 3 and 4. For the performance of sum rate of SUs, Figure 3 shows the nominal SG scheme outperforms other two schemes when the interference temperature level is stringent but is inferior to the two schemes when it is loose. The proposed SG performs the worst, because of the demand of satisfying the outage probability constraint. Once the interference constraints are loose enough to be not active, accordingly, our proposed solution works better than the others. For the leaders, the sum rate of the nominal SG performs the worst because of not including the global interference constraints. This is because the performance of PUs may be degraded with the increases of the interference.

Figure 3

Sum rate of PUs versus INR.

Figure 4

Sum rate of SUs versus INR.

Figure 5 presents the outage probability of the system with different INR. It is observed that the proposed scheme achieves much lower outage probability than other schemes; in particular, the performance gap becomes larger with the increase of INR. This is because our proposed algorithm works best by considering the outage probability constraints of users, which prevents the outage events well.

Figure 5

Outage probability versus INR.

5.2. Impact of Different Number of SUs

In this subsection, the INR is set to be 10 dB. The number of SUs changes from 2 to 20. All the other simulation parameters are the same as the beginning part of this section.

For the PUs, from Figure 6, the sum rate of the SG solution performs the worst because of not including the global interference constraints. So the PUs may refuse to sell more spectrum resource because of protecting their own communication QoS. The proposed scheme outperforms most in terms of sum rate of SUs because the algorithm allows more SUs to share their radio resource so that it increases PUs' utilities with considering the channel uncertainty and global interference constraints.

Figure 6

Sum rate of PUs versus number of SUs.

Figure 7 shows the sum rate of SUs versus the number of SUs. The sum rate of SUs performance of our proposed algorithm works better than other two schemes; this is because the proposed scheme is able to support more SUs at the BS shown in Figure 7, so that more SUs have opportunities to transmit, which increase the sum rate. In addition, the performance gap between the proposed scheme and the other algorithms increases when the network grows larger, by which it can be concluded that the proposed scheme is more suitable for application in larger networks. Because the scheme sets the adaptive punishment parameter among all served SUs to control their behavior, more SUs can be served at the BS, thus achieving a higher sum rate.

Figure 7

Sum rate of SUs versus number of SUs.

Figure 8 shows the number of outage probability comparison versus the number of SUs for different algorithms. The proposed algorithm is able to support more SUs than other schemes. This is because we develop the channel assignment scheduling scheme to decrease the probability of the unserved SUs, so more SUs are admitted to serve at the BS without causing unendurable interference to PUs. In addition, the interference among SUs is taken into the revenue utility function to void serious interference for some SUs who have bad channel condition.

Figure 8

Outage probability versus number of SUs.

6. Conclusion

In this paper, we propose a Stackelberg game for power control problem in CRNs with channel outage constraints and global interference constraints. We employ LDDM to solve the problem by decomposing it into independent subproblems and develop an iterative algorithm to achieve SE. Simulation results show that the proposed algorithm improves the performance compared with other game algorithms.