Energy-Efficient Resource Allocation for Heterogeneous ... - IEEE Xplore

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 11, NO. 11, NOVEMBER 2012

Energy-Efficient Resource Allocation for Heterogeneous Cognitive Radio Networks with Femtocells Renchao Xie, F. Richard Yu, Hong Ji, and Yi Li

Abstract—Both cognitive radio and femtocell have been considered as promising techniques in wireless networks. However, most of previous works are focused on spectrum sharing and interference avoidance, and the energy efficiency aspect is largely ignored. In this paper, we study the energy efficiency aspect of spectrum sharing and power allocation in heterogeneous cognitive radio networks with femtocells. To fully exploit the cognitive capability, we consider a wireless network architecture in which both the macrocell and the femtocell have the cognitive capability. We formulate the energy-efficient resource allocation problem in heterogeneous cognitive radio networks with femtocells as a Stackelberg game. A gradient based iteration algorithm is proposed to obtain the Stackelberg equilibrium solution to the energy-efficient resource allocation problem. Simulation results are presented to demonstrate the Stackelberg equilibrium is obtained by the proposed iteration algorithm and energy efficiency can be improved significantly in the proposed scheme. Index Terms—Cognitive radio, femtocell, energy efficiency, Stackelberg game, Stackelberg equilibrium.

I. I NTRODUCTION APIDLY rising energy costs and increasingly rigid environmental standards have led to an emerging trend of addressing “energy efficiency” aspect of wireless communication technologies [1], [2]. In a typical wireless cellular network, the radio access part accounts for up to more than 70 percent of the total energy consumption [3]. Therefore, increasing the energy efficiency of radio networks is very important to meet the challenges raised by the high demands of traffic and energy consumption. Cognitive radio technology, originally proposed to improve the spectrum efficiency [4], can play an important role in

R

Manuscript received August 10, 2011; revised November 26, 2011 and April 21, 2012; accepted July 31. 2012. The associate editor coordinating the review of this paper and approving it for publication was V. K. N. Lau. R. Xie is with the Key Laboratory of Universal Wireless Communication, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing, P.R. China, and also with the Dept. of Systems and Computer Eng., Carleton University, Ottawa, ON, Canada. F. R. Yu is with the Dept. of Systems and Computer Eng., Carleton University, Ottawa, ON, Canada (e-mail: Richard [email protected]). H. Ji, and Y. Li are with the Key Laboratory of Universal Wireless Communication, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing, P.R. China. This work was jointly supported by the State Key Program of National Natural Science of China (Grant No. 60832009), the Natural Science Foundation of Beijing, China (Grant No. 4102044), the National Natural Science Foundation for Distinguished Young Scholar (Grant No. 61001115), the National Natural Science Foundation of China (Grant No. 61101113), Huawei Technologies Canada CO., Ltd., and the Natural Sciences and Engineering Research Council of Canada. Digital Object Identifier 10.1109/TWC.2012.092112.111510

improving energy efficiency in radio networks [5]. In cognitive radio networks, secondary users (SUs) can monitor the surrounding radio environment, dynamically adapt transmission parameters, and opportunistically utilize the temporarily unused spectrum resource licensed to primary users (PUs) [6]. The cognitive abilities have a wide range of properties, including spectrum sensing and learning-empowered adaptive transmission, which are beneficial to improve the tradeoff among energy efficiency, spectrum efficiency, bandwidth, and deployment efficiency in wireless networks [3], [7], [8]. Some works have been done to consider energy efficiency in cognitive radio networks. In [9], the authors study the hierarchy in energy games for cognitive radio networks. The problem is to maximize the energy-efficiency for each selfish SU. The authors of [10] study the distributed power control game to maximize the transmission energy-efficiency for SUs in cognitive radio networks, where the problem of optimal power control is formulated as a repeated game. Energyefficient power control and receiver design in cognitive radio networks are studied in [11], where a noncooperative power control game for maximum energy efficiency for SUs is proposed under the constraints of fairness and interference threshold. On the other hand, femtocell has been considered as a promising technique, and has been integrated in current and future radio access networks [12]. The authors of [13] study the problem of downlink power allocation to maximize the capacity in a cellular network where a bi-level hierarchy exists. Then the Stackelberg game model is used to formulate the optimization problem and Stackelberg equilibrium is solved. The distributed power allocation strategies for a spectrumsharing femtocell network are considered in [14], and the Stackelberg game is formulated to jointly consider the utility maximization of the macrocell and the femtocells. Due to its short transmit-receive distance property, femtocell technique can also reduce energy consumption, prolong handset battery life, and increase network coverage [15], [16]. The problem of energy efficiency for femtocell base station is studied in [17], where a novel energy saving procedure is designed for femtocell base stations. In [18], resource sharing and access control in OFDMA femtocell networks are studied, where users’ selfish characteristics are considered and an incentive mechanism is designed for subscribers to share the resource of femtocell base stations. Combining cognitive radio with femtocell can further improve the system performance [19]–[22]. The smart cognitive

c 2012 IEEE 1536-1276/12$31.00

XIE et al.: ENERGY-EFFICIENT RESOURCE ALLOCATION FOR HETEROGENEOUS COGNITIVE RADIO NETWORKS WITH FEMTOCELLS

femtocell is considered in [20], where the tradeoff between the macrocell throughput and the aggregate femtocell throughput is studied. In [21], the problem of cross-tier interference in autonomous femtocell networks is studied by using cognitive radio technology to realize the cognitive radio resource management. And a strategic game is proposed to implement the interference mitigation. The interference management for long term evolution (LTE) networks with femtocells using cognitive radio technology is investigated in [22]. Then based on a distributed architecture for LTE networks, the authors recommend to use two game theoretical mechanisms to mitigate the cochannel interference. Although some works have been done for heterogeneous cognitive radio networks with femtocells, most of previous works are focused on spectrum sharing and interference avoidance. Consequently, the energy efficiency aspect in this setting is largely ignored. In addition, most of previous works assume that only the femtocell base station (FBS) has the cognitive capability, without considering the cognitive capability of the macrocell base station. In this paper, our work is different from previous works. Some distinct features of this paper are as follows. • We focus on the energy efficiency aspect of spectrum sharing and power allocation in heterogeneous cognitive radio networks with femtocells. Since both cognitive radio and femtocell are promising technologies to enable energy efficiency in wireless networks, the interplay between them merits further research. We use bits/Hz per Joule, which is commonly used in wireless networks [23], to measure the performance of energy efficiency. • To fully exploit the cognitive capability, we consider a wireless network architecture in which both the macrocell and the femtocell have the cognitive capability, which allows the macrocell and femtocells to dynamic utilize the spectrum resource licensed to primary network and sense the surrounding channel state information. This network architecture is first proposed in [19], which is of importance in practical cellular femtocell networks, where the macrocell base stations can sense the TV band. • We formulate the energy-efficient resource allocation problem in heterogeneous cognitive radio networks with femtocells as a Stackelberg game, which has been successfully used in relay selection and power allocation problems in cooperative communication networks [24], among others. Then a gradient based iteration algorithm is proposed to obtain the Stackelberg equilibrium solution to the energy-efficient resource allocation problem. The rest of this paper is organized as follows. In Section II, the system model is given, and the Stackelberg game model is formulated. In Section III, the Stackelberg equilibrium solution is presented. Simulation results are presented and discussed in Section IV. Finally, we conclude this study in Section V. II. S YSTEM D ESCRIPTION In this section, the heterogeneous cognitive radio network with femtocells is presented. Then the problem of spectrum sharing and resource allocation for energy-efficient communications is formulated as a three-stage Stackelberg game.

PU

PU

PU

PU

PU

PBSl

PBS1

w1

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PBSL

PU

wL

wl

MSU1

MSU i

MSU I

FSU1

FBS1

FBSK

FSU K

Fig. 1. System model for femtocell-based cognitive radio networks (PBS: primary base station; PU: primary user; wl : spectrum resource from primary network l; MSU: macrocell secondary user; FBS: femtocell base station; FSU: femtocell secondary user).

A. System Model Consider a communication system that consists of primary networks and a femtocell-based heterogeneous cognitive radio network, as shown in Fig. 1. The primary networks may operate on the different frequency spectrum.In this case, we will not consider the interference among the primary networks. Each primary network can offer a spectrum selling price and sell part of spectrum resource to the heterogeneous cognitive radio network with femtocells to earn additional profit. In the heterogeneous cognitive radio network, there are multiple macro secondary users (MSUs), a cognitive base station (BS), multiple femtocells, and multiple femtocell secondary users (FSUs). The cognitive BS and femtocells have the cognitive capability and can sense the channel state information [25]. Then the cognitive BS allocates the spectrum resource bought from the primary networks to femtocells or MSUs directly based on the channel quality condition to maximize its revenue. In each femtocell, there is a femtocell base station (FBS) to provide service for FSUs, where the FBS is connected to the cognitive BS over a broadband connection, such as cable modem or digital subscriber line (DSL). The whole system is operated in a time-slotted manner, and the primary networks and the femtocell-based cognitive radio network are assumed to be perfectly synchronized. To simplify the analysis of the problem, without loss generality, we assume that there is only one FSU serviced by the FBS in each time slot. Under this framework, we assume that there are L primary networks. Each primary network l is willing to offer a spectrum selling price cl and sell its part spectrum resource wl of total spectrum Wl to the heterogeneous cognitive radio network to maximize its profit. The cognitive BS can buy the spectrum resource wl from primary network l depending on the spectrum price. Then the cognitive BS allocates the spectrum wl to the femtocells or MSUs directly to gain its revenue. Here, we assume that each femtocell and MSU are accessed in the form of frequency division multiple access (including OFDMA), and in each time slot they could only be allocated one spectrum resource bought from a certain primary network. Assume there are Ktot femtocells and Itot MSUs directly served by the cognitive BS in the heteroge-

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l

w1

c1

wl

cl

xl1

Kl 1

xlk

Klk

FBS1

FBSk

L

wL

cL

xlK

KlK

FBSK

Fig. 2. Three-stage Stackelberg game modeling (FBS: femtocell base station; wl : spectrum resource from primary network l; cl : spectrum selling price from primary network l; ηlk : energy efficiency in femtocell k for spectrum resource wl ; xlk ∈ {0, 1} is the spectrum allocation index, where xlk = 1 means that the spectrum bought from primary network l is allocated to femtocell k, otherwise xlk = 0).

neous cognitive radio network. Usually the total number of femtocells and MSUs requesting to connect to the cognitive BS is larger than the total number of spectrum resource bought from the L primary networks, i.e., (Ktot + Itot ) ≥ L. Due to the limited spectrum resource bought from primary networks, we assume that there are at most L out of (Ktot + Itot ) femtocells and MSUs that can be accessed to the cognitive BS in each time slot, which can be realized by scheduling and admission control. Under this assumption, without loss generality, we assume there are K femtocells and I MSUs accessed to the cognitive BS, where K + I = L. In each time slot, when a certain spectrum wl is used by a femtocell, the aim of the femtocell is to maximize its energy-efficient communications by allocating its power. Similarly, when the spectrum is used by MSUs, the cognitive BS performs the energy-efficient power allocation. Based on the discussion above, we can formulate the problem of resource allocation for energy-efficient communications in the heterogeneous cognitive radio network as a three-stage Stackelberg game problem. B. Problem Formulation In this subsection, we formulate the energy-efficient resource allocation problem as a three-stage Stackelberg game, as shown in Fig. 2, which consists of leaders and followers. Here, we view the up-stage as the leaders that move first, then down-stage as the followers that move subsequently by observing the leaders’ strategies. Therefore, in Stage I, the primary networks are the leaders, and multiple primary networks offer the spectrum selling price cl to the cognitive BS. In Stage II, the cognitive BS, as the follower in Stage I, decides to buy the spectrum size wl from primary network l depending on the price cl offered by primary network l. Next, the cognitive BS, as the leader in Stage II, allocates the spectrum to FBSs or MSUs directly, and performs power allocation for the MSUs to gain its revenue. The FBSs in stage III performs power allocation for FSUs. Based on the above analysis, we know that each stage’s strategy will affect other stages’ strategies, and the leaders should take the followers’ actions into account when deciding their strategies. Therefore, to formulate the three-stage Stackelberg game model, we can use the backward induction method as follows.

When femtocell k obtains spectrum resource wl from the cognitive BS, the FBS aims to maximize energy efficiency in power allocation, which can be expressed as h2 pk log2 1 + lk 2 σ ηlk = , (1) pa + pk where pa denotes the additional circuit power consumption of devices during transmissions [26] (e.g., digital-to-analog converters, filters, etc), which is independent to the data transmission power. hlk and pk are the channel gain and power allocation on spectrum wl for FBS k, respectively. σ 2 is the additive Gaussian white noise with zero mean and unit variation. In (1), bits/Hz per Joule is used as the energy efficiency metric, which is commonly used in wireless networks [23], [27]. In our work, we assume that the channels are fixed over the time-period of interest. This simplifying assumption has been commonly adopted by the game-theoretic studies in the communication literature [14]. To improve energy efficiency, less energy should be used to transmit more information data. Based on (1), the aim of FBS k is to maximize its utility given spectrum wl , which can be expressed as follows. L L πk = xlk (ςk wl ηlk − cb wl ηlk ) = (ςk − cb ) xlk wl ηlk , l=1

l=1

(2) where wl is the spectrum resource bought from primary network l, ςk is the revenue for FBS k, and cb is the cost charged by the cognitive BS due to the allocated spectrum. xlk ∈ {0, 1} is the spectrum allocation index, where xlk = 1 means that the spectrum bought from primary network l is allocated to FBS k; otherwise xlk = 0. As the energy efficiency criterion is used, the first term denotes the revenue gained from the FSU for energy efficient transmission, and the second term denotes the cost charged by the cognitive BS for using the spectrum resource. Here, we have ςk > cb . Otherwise, the FBSs will not request to access. When the FBSs finish the energy-efficient power allocation, there is feedback information to the cognitive BS. For the cognitive BS, the aim is to buy the size of spectrum from primary networks and allocate them to the FBSs or MSUs directly to maximize its revenue. Here, we notice that the cognitive BS also does the energy-efficient power allocation for MSUs when the spectrum is allocated to MSUs, which is similar to (1). If we assume that spectrum demand for cognitive BS from the primary networks is satisfied the linear demand structure, we can use the following quadratic utility function [28] for the cognitive BS. K L I wl cb xlk ηlk + ξi xli ηli cb xpk ηk πb (w) = i=1 l=1 k=1 L L − 21 wl2 + 2θ wl wq − cl wl , l=1

q=l

l=1

(3) where w = {w1 , w2 , ..., wL } denotes the spectrum bought from primary networks, ηli is the energy-efficient transmission parameter for MSUs, ξi is the cost paid by MSUs to the cognitive BS, cl is the price offered by primary network l, and θ ∈ [−1, 1] is the spectrum substitutability parameter. θ = 1 means that the FBSs or MSUs can switch among the spectrum;


θ = 0 means that FBSs or MSUs cannot switch among the spectrum; θ < 0 means spectrum used by FBSs or MSUs is complementary [28]. xli ∈ {0, 1} is the spectrum allocation index, where xli = 1 means that the spectrum bought from primary network l is allocated to MSU i; otherwise xli = 0. Here, to avoid the cross-tier interference, we assume that one spectrum resource bought from a certain primary network could only be allocated to one FBS or MSU as proposed in K I [29], i.e., xlk + xli = 1. The motivations to use the k=1

i=1

utility function in (3) mainly are as follows. First, the utility function is concave about the spectrum demand, which is easy to denote the maximum revenue of cognitive BS. Second, we can get a linear spectrum demand function by differentiating the utility function (3). For the primary networks, they can sell portion of spectrum to the cognitive BS to gain additional profit. Therefore, the aim is to maximize its revenue by offering the spectrum price cl depending on the spectrum demand from the cognitive BS. We can adopt the following utility function to denote the revenue of primary network l. πl (c) = α1 (Wl − wl ) κl + cl wl ,

(4)

where α1 is a positive design parameter, which denotes the weight for the revenue from the PUs. Wl is the total spectrum licensed to primary network l, and κl denotes the spectrum transmission efficiency of primary network l. Usually, we have cl ≥ α1 κl ; otherwise, the primary networks is not willing to sell its spectrum to the heterogeneous cognitive radio network. The intuitive explanation of the utility function (4) is that the primary networks will earn the additional revenue from cognitive radio network by selling portion of its spectrum at the cost of degradation its communication performance. III. S TACKELBERG E QUILIBRIUM S OLUTION In this section, we use the backward induction method to solve the Stackelberg equilibrium for the three-stage Stackelberg game formulated above. In Subsection III-A, we first give the power allocation strategy in Stage III to realize energy-efficient communications and maximize the utility of FBSs. Then the spectrum demand for the cognitive BS is derived in Stage II to maximize its revenue, which is presented in Subsection III-B. Next, in Subsection III-C, the price determination for primary networks is solved in Stage I to maximize the utility. Finally, the gradient based iteration algorithm for energy-efficient resource allocation to get the Stackelberg equilibrium solution is given in Subsection III-D. A. Energy-Efficient Power Allocation for FBSs When spectrum wl is allocated to FBS k, FBS k aims to maximize its utility function (2) to realize energy-efficient power allocation. However, utility function (2) is a nonlinear function about the transmission power, which makes it hard to directly solve (2). Therefore, to reduce the computation complexity, we propose a gradient assisted binary search algorithm to realize the energy-efficient power allocation. Before giving the algorithm, we first prove πk (pk ) is a quasi-concave

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function and give the definition of quasi-concave function as follows. Definition 1: A mapping function f : Ω → R, defined on a convex set Ω of real n-dimensional vectors, is strictly quasiconcave, if, for all x, y ∈ Ω, x = y and 0 < λ < 1, we have f (λx + (1 − λ) y) > min {f (x) , f (y)} . L h2 pk (ςk − cb ) xlk wl log2 1 + lk . Let Rk (pk ) = σ2 l=1

Therefore, based on the above definition, we can give the following properties for πk (pk ). Lemma 1: If Rk (pk ) is strictly concave in pk , πk (pk ) is strictly quasi-concave. Furthermore, πk (pk ) is either first strictly increasing and then strictly decreasing in any pk or strictly decreasing. Inspired by [30], we can prove Lemma 1 as follows. Proof: Denote the α-sublevel sets of function πk (pk ) as Sα = {pk > 0|πk (pk ) ≥ α}

(5)

Based on the propositions, πk (pk ) is strictly quasi-concave if and only if Sα is strictly convex for all α. In this case, when α < 0, there are no points satisfying πk (pk ) = α. When α = 0, only pk = 0 satisfies πk (0) = α. Therefore, Sα is strictly convex when α ≤ 0. When α > 0, we can rewrite the Sα as Sα = {pk > 0|α (pa + pk ) − Rk (pk ) ≤ 0}. Since Rk (pk ) is strictly concave in pk , which means that −Rk (pk ) is strictly convex in pk , therefore Sα is also strictly convex. hence, πk (pk ) is strictly quasi-concave function. Next, we can obtain the derivative of πk (pk ) with pk as

∂πk (pk ) R (pk ) (pa + pk ) − Rk (pk ) φ (pk ) = k = 2 2, ∂pk (pa + pk ) (pa + pk ) (6) k (pk ) where Rk (pk ) denotes the first derivative ∂R∂p . Based on k

∂πk (pk )

∗ = 0, the above Lemma, if there is pk satisfying ∂pk

∗ pk =pk

p∗k is unique. Now we only need to solve the problem when p∗k exists. The derivative of φ (pk ) is

φ (pk ) = Rk (pk ) (pa + pk ) ,

(7)

where Rk (pk ) is the second derivative of Rk (pk ) with respect to pk . Due to Rk (pk ) < 0, φ (pk ) < 0. Therefore, φ (pk ) is strictly decreasing. In addition, according to the L’Hopital’s rule [30], we know that lim φ (pk ) = lim Rk (pk ) (pa + pk ) − Rk (pk ) pk →∞ pk →∞ Rk (pk )(pa +pk )−Rk (pk ) = lim p k pk pk →∞ Rk (pk )(pa +pk ) = lim pk , 1 pk →∞

(8) due to Rk < 0, pk > 0, and pa > 0, thus we have lim φ (pk ) < 0. Recall that pa denotes the additional

pk →∞

circuit power during transmissions, and pk denotes the power allocation for FBS k. When pk = 0, this means that there is no transmission and reception, therefore the value of pa should be zero as well. Hence, when pk → δ for arbitrarily small

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number δ approximating to 0, we have lim φ (pk ) = lim Rk (pk ) (pa + pk ) − Rk (pk ) pk →δ pk →δ (δ) (δ) (δ) = Rk pk pa + pk − Rk pk ,

efficient power allocation in FBSs, the cognitive BS can maximize its utility as presented in the following subsection. (9)

(δ)

where pk = δ. Therefore, we can getthe following two cases: (δ)

(δ)

(δ)

pa + pk − Rk pk ≥ 0, (1) When Rk pk lim φ (pk ) ≥ 0. Together with (8), we can observe that p∗k

pk →δ

exists and πk (pk ) is first strictly increasing and then strictly decreasing in pk . (δ) (δ) (δ) (2) When Rk pk pa + pk − Rk pk < 0, lim φ (pk ) < 0. Together with (7) and (8), we know that

B. The Solution of Spectrum Demand for Cognitive BS After the FBSs finish energy-efficient power allocation, the cognitive BS can decide the size of spectrum resource to buy from different primary networks. By observing (3), we know that the utility function for the cognitive BS is a concave function about spectrum demand, therefore we can differentiate πb (w) in (3) with respect to wl as follows. K I ∂πb (w) = cb xlk ηlk + ξi xli ηli − wl − θ wq − cl = 0. ∂wl i=1 k=1 q=l (10)

pk →δ

p∗k does not exist. In this case, the maximum value should be obtained at pk = 0. This is because when pk = 0, R (p ) k) = k 2 k |pk =0 < 0. we have pa = 0, and lim (pφ(p +p )2 pk →0

a

k

Hence, πk (pk ) strictly decreases in pk , the maximum value is obtained in the left endpoint. Therefore, Lemma 1 is proved. We know that if there is a local maximum solution for the quasi-concave functions, the solution is also globally optimal [31]. Hence, we can give Theorem 1 according to the proof of Lemma 1. Theorem 1: If Rk (pk ) is strictly concave, there exists a unique globally optimal power allocation p∗k , for (2), where p∗k is obtained by

(δ) Rk pk

∂Rk (pk )

, ∂πk (pk )

(1) When ∂pk

≥ = (δ) (δ) ∂pk ∗ pa +pk

pk =pk

0.

(2) When

∂Rk (pk )

(δ) ∂pk

pk =pk

0; (1) k (pk )

2) Then do pk = pk , μ1 ← dπdp

(1) and α > 1. k pk

3) If μ1 < 0, do Repeat (2)

(1)

(1)

a) pk ← pk , pk ←

(1)

pk α

, and μ1 ←

b) Until μ1 ≥ 0 (2) (1) Else do pk ← pk ∗ α and μ2 ←

dπk (pk )

dpk

dπk (pk )

dpk

(2)

pk

Repeat (1) (2) (2) (2) a) do pk ← pk ,pk ← pk ∗ α and μ2 ← b) Until μ2 ≤ 0 4) While no convergence a) do pk ← (1)

(1)

(2)

pk +pk 2

,μ ←

(1)

pk

dπk (pk )

dpk

(2)

pk

dπk (pk )

dpk p

k

(2)

b) let pk = pk when μ > 0; otherwise, pk = pk 5) Output pk The above algorithm can maximize utility function (2) and realize energy-efficient communications. After the energy-

Thus we can obtain the size of spectrum bought from primary networks l for the cognitive BS by solving (10) as follows.

wl∗ =

K

k=1

θ

cb xlk ηlk +

q=l

I i=1

ξi xli ηli −cl (θ(L−2)+1)

(1−θ)(θ(L−1)+1) K I cb xqk ηqk + ξi xqi ηqi −cq

k=1

i=1

(1−θ)(θ(L−1)+1)

−

(11)

.

Based on (11), the cognitive BS can maximize its utility function by buying the spectrum from primary networks and allocating the spectrum to FBSs or MSUs. Here, we must notice that for the different spectrum allocation xlk , xli , k xlk + i xli = 1, to FBSs or MSUs, we could get the different size of spectrum demand wl∗ in (11) and obtain the different revenue πb (w) of the cognitive BS in (3). Hence, how to allocate the spectrum to FBSs or MSUs is an important problem. We can use the following spectrum allocation algorithm. Spectrum Allocation Algorithm to FBSs or MSUs 1). Initialization: the set of FBSs and MSUs Ωk = {1, 2, ..., K}, Ωi = {1, 2, ..., I} and the set of primary networks ΩL = {1, 2, ..., L}. 2). Do repeat Find υ ∗ = arg max {cb ηlk , ξi ηli }, then let xlυ∗ = 1, ∀k,i

Ωk = Ωk − υ ∗ or Ωi = Ωi − υ ∗ . And let Ωl = Ωl − l; Until the Ωl = ∅, end repeat. 3) Output the spectrum allocation index xlk , xli . In the spectrum allocation algorithm above, when the spectrum is allocated to MSUs, the cognitive BS can do the energyefficient power allocation for MSUs as described in Subsection III-A. The intuitive explanation of the above algorithm is that the cognitive BS expects to allocate the spectrum to FBSs or MSUs with higher energy-efficient transmission, then more revenue could be obtained by the cognitive BS, which will be shown in the simulation section. To realize the above spectrum allocation algorithm, the cognitive BS only needs to know the channels of the FBSs and MSUs and the revenue of FBSs and MSUs. To get this information, the cognitive BS can get feedback information from FBSs and MSUs. C. Prices Determination for the Primary Networks After the spectrum demand is derived by the cognitive BS, for the primary networks, the revenue of each primary


network is affected not only by its price cl but also by the prices c−l offered by other primary networks, where c−l = (c1 , ..., cl−1 , cl+1 , ..., cL ) are the prices offered by all the primary networks except primary network l. Therefore, the prices determination among the primary networks is a price competitive game G = {N , {cl } , {πl (·)}}, where N = {1, 2, ..., L} is the set of players, cl is the strategy set and πl (·) is the payoff function of primary network l. Hence, the Nash equilibrium of a game is a strategy equilibrium that no primary network can increase its revenue by choosing a different strategy, given other players’ strategy [32]. In this case, we can use the best response function to find the Nash equilibrium. When the other’s strategy c−l is given, the best respond function of primary network l can be defined as follows. (12) B (c−l ) = arg max π l (cl , c−l ) . cl

∗

(c∗1 , c∗2 , ..., c∗L )

Let c = denote the Nash equilibrium of competitive price game. Thus c∗ must satisfy the following condition (13) c∗l = B c∗−l , ∀l, where c∗−l denotes the set of best responses for player q, q = l. When the spectrum demand strategy (11) of the cognitive BS is given, to solve the best response function of primary networks, we can substitute (11) into the revenue function for primary networks and rewrite (4) as at the top of next page. Therefore, we can differentiate (14) with respect to cl as at the top of next page Let

A (c−l ) = θ

K k=1

q=l

cb xlk ηlk +

I i=1

ξi xli ηli (θ(L−2)+1)

(1−θ)(θ(L−1)+1) K I cb xqk ηqk + ξi xqi ηqi −cq

k=1

i=1

(1−θ)(θ(L−1)+1)

and B= And let

∂πl (c) ∂cq

−

(16)

,

(θ (L − 2) + 1) . (1 − θ) (θ (L − 1) + 1)

(17)

= 0. Hence, we can rewrite (15) as

∂πl (c) = α1 κl B − 2cl B + A (c−l ) = 0 ∂cl

cl →∞

∂πl (c) ∂cl

= α1 κl B −2cl B +A (c−l ) = −∞ < 0,

thus utility function πl (c) is first strictly increasing, and then strictly decreasing. This means that utility function πl (c) is a concave function. We know that a strictly concave function is also strictly quasi-concave. Hence, based on [33], there exists a Nash equilibrium for the price competition game G. Theorem 3 (Uniqueness): The price competition game G for the primary networks has a unique Nash equilibrium. Proof: From Theorem 2, we know that there exists a Nash equilibrium for the price competition game G. Let c∗ be the Nash equilibrium, which must satisfy the best response (12). Therefore,

we only need to prove that function B c∗−l = α1 κl B + A c∗−l /2B is a standard function. A function is said to be a standard function when the following properties are satisfied [34]: (1). Positivity: B c∗−l > 0; ∗ ∗ ∗ (2). Monotonicity: if c > c then B c−l ≥ −l −l ∗ B c−l ;

(3). Scalability: for all λ > 1, λB c∗−l > B λc∗−l . The positivity property ∗ is obviously satisfied for the best response function B c−l . To prove the monotonicity, ∗we know that A c∗−l in (16) is an increasing function . of c −l ∗ ∗ ∗ ∗ Thus when c−l > c−l , we have A c−l > A c−l , which is equal to B c∗−l ≥ B c∗−l . Therefore, the monotonicity is proved. For scalability, for all λ > 1, we have

α1 κl B+A(c∗ −l ) λB c∗−l − B λc∗−l = λ − 2B ∗ (λ−1)α1 κl B+λA(c−l )−A(λc∗ −l ) = . 2B

α1 κl B+A(λc∗ −l ) 2B

(19) From (16), we get the equation (20) at the top of next page From (11),we have 0 ≤ wl∗ ≤ Wl . K I Let Γ = cb xlk ηlk + ξi xli ηli (θ (L − 2) + 1) − i=1 K k=1 I θ q=l cb xqk ηqk + ξi xqi ηqi , thus we can get i=1

k=1

0 ≤ Γ − cl (θ(L−2)+1) + θ

(18)

Therefore, we can obtain the Nash equilibrium solution c∗l by solving the above set of linear equations (18). After obtaining the set of prices c∗ at the Nash equilibrium, the size of spectrum bought for the secondary network from primary network l can be obtained relying on wl∗ (c∗ ). For the prices competition game G among primary networks, it’s necessary to prove the existence and uniqueness of Nash equilibrium. Thus, we give the following theorem. Theorem 2 (Existence): A Nash equilibrium exists in the price competition game G for the primary networks. Proof: Firstly, price cl is a nonempty convex and compact subset of Euclidean space. For utility function πl (c), we know that the first partial derivative of πl (c) with respect to cl is (18), and the second partial derivative is πl (c) = −2B. ∂πl (c) Obviously, B > 0, thus πl (c) < 0. Hence ∂cl is strictly = α1 κl B + A (c−l ) ≥ α1 κl B + decreasing. And lim ∂π∂cl (c) l cl →0

cl B > 0, lim

3915

q=l

cq ≤ Wl (1 − θ)(θ(L − 1)+1) . (21)

Obviously, Γ ≥ 0 must be satisfied. Otherwise, we can prove (21) is contradict if Γ < 0. To satisfy (21) when Γ < 0, we must have cq − cl (θ (L − 2) + 1) ≥ −Γ > 0. (22) θ q=l

To satisfy (22), if and only if θ = 1 and cq = cl , ∀q, l are satisfied. In this case, we can obtain Γ − cl (θ (L − 2) + 1) + θ q=l cq = Γ < 0, which is contradict with

(21). Thus

we must have Γ ≥ 0. Hence we have λA c∗−l − A λc∗−l ≥ 0. Therefore, substituting (20) into (19), we have the equation (23) at the top of next page Thus, the scalability of best response function is proved. The best response function of price competition game G is a standard function. Then based on [34], the fixed point c∗ is unique for a standard function. Therefore, the Nash equilibrium of the price competition game is unique.

3916


⎛

πl (c) = α1 ⎝Wl −

+cl

K

k=1

K

k=1

cb xlk ηlk +

I i=1

cb xlk ηlk +

I i=1

K ⎞ I ξi xli ηli −cl (θ(L−2)+1)−θ q=l cb xqk ηqk + ξi xqi ηqi −cq (1−θ)(θ(L−1)+1)

K I ξi xli ηli −cl (θ(L−2)+1)−θ q=l cb xqk ηqk + ξi xqi ηqi −cq (1−θ)(θ(L−1)+1)

i=1

k=1

⎠ κl

i=1

k=1

(14) .

(θ(L−2)+1) ∂πl (c) cl (θ(L−2)+1) ∂cl = α1 κl (1−θ)(θ(L−1)+1) −(1−θ)(θ(L−1)+1) K K I I cb xlk ηlk + ξi xli ηli −cl (θ(L−2)+1)−θ q=l cb xqk ηqk + ξi xqi ηqi −cq

+

i=1

k=1

λA c∗−l − A λc∗−l =

(15) .

K K I I λ cb xlk ηlk + ξi xli ηli (θ(L−2)+1)−λθ q=l cb xqk ηqk + ξi xqi ηqi −c∗ q

(λ−1)

i=1

k=1

=

(1−θ)(θ(L−1)+1)

i=1

k=1

−

K

k=1

cb xlk ηlk +

i=1

k=1

I i=1

(1−θ)(θ(L−1)+1) K I ξi xli ηli (θ(L−2)+1)−θ q=l cb xqk ηqk + ξi xqi ηqi −λc∗ q i=1

k=1

(1−θ)(θ(L−1)+1) K K I I cb xlk ηlk + ξi xli ηli (θ(L−2)+1)−θ q=l cb xqk ηqk + ξi xqi ηqi i=1

k=1

(1−θ)(θ(L−1)+1)

i=1

k=1

.

λB c∗−l − B λc∗−l =

K K I I (λ−1) α1 κl (θ(L−2)+1)+ cb xlk ηlk + ξi xli ηli (θ(L−2)+1)−θ q=l cb xqk ηqk + ξi xqi ηqi k=1

i=1

2B(1−θ)(θ(L−1)+1)

D. Gradient Based Iteration Algorithm to Obtain the Stackelberg Equilibrium When the backward induction is given to solve the threestage Stackelberg game in the above subsection, in this Subsection, we propose a gradient based iteration algorithm to get the Stackelberg equilibrium. We first prove the existence and uniqueness of the equilibrium for the three-stage Stackelberg game. We give Proposition 1 as follows. Proposition 1: The optimal energy-efficient power allocation scheme in Subsection III-A, the spectrum demand wl∗ in (11), and the price determination strategy c∗l in Subsection III-C are the subgame perfect equilibrium, respectively, in each stage. Proof: We could view the three-stage Stackelberg game as an extensive game, and each stage is a subgame. Thus, in Stage III, the energy-efficient power allocation for FBSs is a subgame. Based on the discussion in Subsection III-A, we know that the solution of energy-efficient power allocation can maximize the utility function in (2) when the spectrum is allocated by the cognitive BS. That means there is no other power allocation strategy that each FBS is willing to take. Therefore, the power allocation for FBS in Subsection III-A is the subgame equilibrium. Similarly, for Stage II and Stage I, the spectrum demand in (11) and the price strategy proposed in Subsection III-C are also the subgame equilibrium. Therefore, Proposition 1 is proved. Based on Proposition 1, we know that every subgame perfect equilibrium is a Nash equilibrium [32]. Therefore, there exists a Stackelberg equilibrium for the three-stage Stackelberg game. Because each subgame perfect equilibrium solved in the above is unique, the three-stage Stackelberg equilibrium

k=1

(20)

i=1

(23) > 0.

is also unique. Hence, the existence and uniqueness of the Stackelberg equilibrium are proved. Inspired by [35], we can use the following gradient based iteration algorithm to obtain the three-stage Stackelberg game equilibrium. Algorithm 1: Gradient Iteration Algorithm for Stackelberg Equilibrium (1). Initialization: given a price cl by primary network l. (2). Repeat the iteration (a). Do the energy-efficient power allocation for FBSs or MSUs, and the cognitive BS decides the spectrum wl and allocates it to FBSs or MSUs to maximize its revenue (3). (b). The primary networks update the prices as c (t + 1) = c (t) + μ∇π (c (t)) .

(24)

(c). Until c (t) − c (t − 1) / c (t − 1) ≤ ε end iteration. Here, μ is the iterative step size of the price, π = (c(t)) {π1 , ..., πL }, ∇π (c (t)) is the gradient with ∂πl∂c . l In the proposed algorithm, for each iteration, the cognitive BS can decide the spectrum demand and allocate the spectrum to FBSs or MSUs when the prices are updated. Then until the prices are converged, the algorithm is stopped. The algorithm proposed above can get the Stackelberg game equilibrium when the prices are converged, which will be shown in the simulation part of this paper. In practical networks, the proposed gradient-based iteration algorithm to obtain the three-stage Stackelberg game equilibrium can be implemented as follows: (1). The primary networks first randomly offer spectrum


selling prices and broadcast the information. (2). Then the cognitive BS receives the price information, and notices the number of primary networks to FBSs and MSUs, then FBSs and MSUs feedback the channel state information to the cognitive BS via backhaul link. (3). The cognitive BS decides to buy the size of spectrum and allocates the spectrum to FBSs or MSUs. (4). Then the primary networks update the prices and repeat steps (2), (3) until the prices are converged. In the above iteration process, the following feedback information is required to reach the Stackelberg game equilibrium. The primary networks first broadcast the information of spectrum selling prices to cognitive BS. Then the cognitive BS collects the channel state information of MSUs and femtocells, and decides the spectrum demand from the primary networks based on the spectrum prices information. Finally, the cognitive BS feedback the spectrum demand information to primary networks.

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7 Offered by Primary Network 2 Offered by Primary Network 1 6.5

Price

6

5.5

5

4.5

0

5

10 Iteration Step

15

20

(a) Price offered by primary networks versus iteration step.

14

12

From Primary Network 1 From Primary Network 2

Spectrum Demand

10

IV. S IMULATION R ESULTS AND D ISCUSSIONS

8

6

4

2

0

0

5

10 Iteration Step

15

20

(b) Spectrum demand from the cognitive BS versus iteration step. Fig. 3.

Convergence of the proposed iteration algorithm.

90 80

Revenue of Primary Network 1

In this section, we use computer simulations to evaluate the performance of the energy-efficient resource allocation scheme. For ease of illustration, we first consider a simple network scenario with two primary networks and two femtocells, and there is one user accessed in each femtocell. Then, we consider multiple primary networks, multiple femtocells, and multiple users later. The simulation parameters are set as follows. The total spectrum licensed by each primary network is 25M Hz, the spectrum efficiency of primary networks is 2 and the design parameter α1 = 1. The coverage radius of the cognitive BS is 500m, and the coverage radius of the femtocell is 20m. Users are randomly located in the cell. The revenue for the FBS is ςk = 3, and the cost of using spectrum is cb = 1, the additional circuit power consumption is pa = 0.1W . Fig. 3 shows the convergence of the proposed gradient based iteration algorithm for Stackelberg equilibrium. We study the performance of prices determination and spectrum demand over iteration step. From Fig. 3(a), we can observe that given the initial prices offered by primary networks, the price competition game between primary networks can get a Nash equilibrium after several iteration steps. As the price offered by primary network 1 decreases, the spectrum demand from primary network 1 increases. Similarly, the spectrum demand from primary network 2 decreases when the price offered by primary network 2 increases. When the prices offered by primary networks get a Nash equilibrium, the spectrum demand for the cognitive BS also gets a Nash equilibrium in Fig. 3(b). For the primary networks, there is a price competition game G, which means that the primary network should take the prices offered by others into account when determining its price to maximize the revenue. In Fig. 4, we evaluate the performance of the revenue of a primary network when the prices of other primary networks are given. From the figure, we can see that the revenue of primary network 1 first increases with its price increasing. Then when the price reaches a certain value, the revenue of the primary network begins to decrease. As the prices offered by other primary

70 60 50 Best Response 40 30 Price c2=6

20

Price c2=4 Price c2=2

10 0

Fig. 4.

0

2

4

6 Price c1

8

10

12

Revenue of primary network 1 versus the price.

networks increase, the revenue of primary network 1 increases. This is because that the cognitive BS will buy more spectrum from the primary network when the other primary networks have higher prices. Also we can see that as the prices of other primary networks increase, the primary network can offer a higher price to get the Nash equilibrium, which makes the best response of the primary network has a higher price, and can bring a higher revenue. Here, we notice that the player should know actions of other players when getting its best response in Fig. 4. To obtain the prices information of other players, we can assume that there is a central controller as proposed

3918


12

30

10

25 Price c2=9

Nash equilibrium Spectrum Demand

Price c2

8

6

4

B1 (1=2=4dB)

Price c2=5

20

Price c2=2 15

10

B2 (1=2=4dB) 2

5

B1 (1=2=10dB) B2 (1=2=10dB)

0

0

2

4

6 Price c1

8

10

0

12

Fig. 5. Best response of the primary networks (γ1 and γ2 denote the channel state information from FBS1 and FBS2 to users, respectively).

Fig. 7.

0

2

4

6 Price c1

8

10

12

Spectrum demand from the cognitive BS.

4.5 140 Energy Efficiency of FBS (bits/Hz/Joule)

4

Revenue of Cognitive Base Station

120

100

80 The Number of Primary Networks=4 The Number of Primary Networks=3 The Number of Primary Networks=2

60

40

3.5 3 2.5 2 =7 dB =5 dB =3 dB

1.5 1 0.5

20 0 0

2

4

6

8

10 12 14 Spectrum Demand

16

18

20

0

0.2

0.4

0.6

0.8

1

Power (W)

Revenue of the cognitive BS versus the spectrum demand.

Fig. 8. Energy efficiency of the femtocell base station (γ denotes the channel state information from FBS to user).

in [36] to collect the price charged by primary networks. In this case, when the primary network takes its strategy, it can get the actions of other players from the central controller. In Fig. 5, we give the best response function of primary networks for the different channel condition of the FBS. In h2 the simulations, γk = σkl2 , k ∈ 1, 2, denotes the channel state information from the FBS to the user in the femtocell. The intersection of the price curves denotes the Nash equilibrium point. From the figure, we can observe that the best response of primary networks has a higher price when the channel is better. This is because that the cognitive BS expects to get more spectrum to maximize its revenue by allocating the bought spectrum to the FBS when the channel condition of the FBS is good. Thus the primary networks can have a chance to offer a higher price to sell its spectrum. From (3), we know that the revenue of the cognitive BS is a concave function of the spectrum demand. We study the revenue performance of the cognitive BS over the spectrum demand in Fig. 6. From the figure, we can see that the revenue increases for the cognitive BS when the number of primary networks increases. This is not hard to understand. Because when there are more primary networks, the cognitive BS can buy more spectrum from primary networks with lower price due to the price competition among primary networks. Thus, the revenue of the cognitive BS increases, and there is more spectrum demand to get the maximum value of the revenue when the number of primary networks increases.

In Fig. 7, we analyze the relationship between the spectrum demand from the cognitive BS and the prices offered by primary networks. We can observe that the spectrum demand is a linear function of the prices, which is also shown from our theory analysis of the spectrum demand in (11). We can see that when the price offered by primary network 1 increases, the spectrum from the primary network 1 bought by the cognitive BS decreases. When the price offered by primary network 2 increases, there will be more spectrum from primary network 1 bought by the cognitive BS. The performance of FBS in terms of its energy efficiency over the power budget is evaluated in Fig. 8. From the figure, we can observe that the energy efficiency first increases with its power increasing, then when the power reaches a certain point, the energy efficiency begins to decrease. This is because that there is a tradeoff between transmission capacity and power consumption for the energy-efficient power allocation. The better the channel condition, the more the energy efficiency obtained in the FBS. Also, we can see from the figure that less power is needed for the FBS with good channel condition to obtain the same energy efficiency for the FBS. In Figs. 9 and 10, we study the performance of average energy efficiency of the heterogeneous cognitive radio network. In the simulations, we assume that there are 8 primary networks. We also compare the proposed scheme with an existing scheme [37]. A random spectrum allocation scheme is used to allocate spectrum to FBSs or MSUs in the existing

Fig. 6.


A gradient based iteration algorithm has been proposed to obtain the Stackelberg equilibrium solution. Simulation results have been presented to demonstrate the performance of the proposed scheme. In practice, it is difficult to have the perfect knowledge of a dynamic channel. In our future work, we will consider imperfect channel state information in energy-efficient resource allocation for heterogeneous cognitive radio networks with femtocells.

Average Energy Efficiency of Network (bits/Hz/Joule)

3.8 3.6 Proposed Shceme Existing Scheme

3.4 3.2 3 2.8 2.6 2.4 2.2

3919

ACKNOWLEDGMENT 0

0.2 0.4 0.6 0.8 Number of Femtocells/Total Accessed Users and Femtocells

1

Fig. 9. Average energy efficiency of the heterogeneous cognitive radio network.

We thank the reviewers for their detailed reviews and constructive comments, which have helped to improve the quality of this paper. R EFERENCES

Average Energy Efficiency of Network (bits/Hz/Joule)

3.5 Proposed Shceme Existing Scheme 3

2.5

2

1.5

1

0.5

Fig. 10. network.

4

6

8 10 12 14 16 Distance between Femtocell to User (m)

18

20

Average energy efficiency of the heterogeneous cognitive radio

scheme. From the figure, we can observe that as the ratio of the femtocells increases, the average energy efficiency of the heterogeneous cognitive radio network increases. This is because that the femtocells have short transmission distance, which has more chance to get high transmission rate with low power due to better channel condition. Moreover, we know that as the distance increases, the channel condition becomes worse, and more power is needed. Thus the performance of energy efficiency of network decreases as distance increases. In addition, the proposed scheme always has better energy efficiency than the existing scheme due to the optimal resource allocation considered in the proposed scheme. V. C ONCLUSIONS AND F UTURE W ORK In this paper, we have studied the issues of spectrum sharing and resource allocation for heterogeneous cognitive radio networks with femtocells to improve energy efficiency. We have formulated the resource allocation problem as a three-stage Stackelberg game. In Stage I, the primary network offers the spectrum selling price to the cognitive BS. In Stage II, the cognitive BS decides to buy the spectrum size from a primary network and allocates the spectrum to femtocells or macro secondary users. In Stage III, the femtocell base station perform power allocation for the femtocell secondary users. Then we have used the backward induction method to solve the resource allocation in each stage and proved the existence and uniqueness of the Stackelberg game equilibrium.

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F. Richard Yu (S00-M04-SM08) received the PhD degree in electrical engineering from the University of British Columbia (UBC) in 2003. From 2002 to 2004, he was with Ericsson (in Lund, Sweden), where he worked on the research and development of 3G cellular networks. From 2005 to 2006, he was with a start-up in California, USA, where he worked on the research and development in the areas of advanced wireless communication technologies and new standards. He joined Carleton School of Information Technology and the Department of Systems and Computer Engineering at Carleton University in 2007, where he is currently an Associate Professor. He received the Carleton Research Achievement Award in 2012, the Ontario Early Researcher Award (formerly Premiers Research Excellence Award) in 2011, the Excellent Contribution Award at IEEE/IFIP TrustCom 2010, the Leadership Opportunity Fund Award from Canada Foundation of Innovation in 2009 and the Best Paper Awards at IEEE/IFIP TrustCom 2009 and Intl Conference on Networking 2005. His research interests include cross-layer design, security and QoS provisioning in wireless networks. Dr. Yu is a senior member of the IEEE. He serves on the editorial boards of several journals, including IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY, IEEE C OMMUNICATIONS S URVEYS & T U TORIALS , ACM/Springer Wireless Networks, EURASIP Journal on Wireless Communications Networking, Ad Hoc & Sensor Wireless Networks, Wiley Journal on Security and Communication Networks, and International Journal of Wireless Communications and Networking, and a Guest Editor for IEEE Systems Journal for the special issue on Smart Grid Communications Systems. He has served on the Technical Program Committee (TPC) of numerous conferences, as the TPC Co-Chair of IEE Globecom13, CCNC’13, INFOCOM-CCSES2012, ICC-GCN2012, VTC2012S, Globecom11, INFOCOM-GCN2011, INFOCOM-CWCN’2010, IEEE IWCMC’2009, VTC’2008F and WiN-ITS’2007, as the Publication Chair of ICST QShine 2010, and the Co-Chair of ICUMT-CWCN’2009. Hong Ji received the B.S. degree in communications engineering and the M.S. and Ph.D. degrees in information and communications engineering from the Beijing university of Posts and Telecommunications (BUPT), Beijing, China, in 1989, 1992, and 2002, respectively. From June to December 2006, she was a Visiting Scholar with the University of British Columbia, Vancouver, BC, Canada. She is currently a Professor with BUPT. She also works on national science research projects, including the HiTech Research and Development Program of China (863 program), The National Natural Science Foundation of China, etc. Her research interests include heterogeneous networks, peer-to-peer protocols, and cognitive radio. Yi Li is a lecturer in the School of Information and Communication Engineering of Beijing University of Posts and Telecommunications. He received his BE degree from Xi’an Technological University, his ME degree from China Academy of Telecommunication Technology, and his Ph.D. degree from Beijing University of Posts and Telecommunications. His current research interests include wireless communications, ubiquitous networks, cognitive networks, and Internet of things.