Price-Based Power Allocation in Two-Tier Spectrum Sharing Heterogeneous Cellular Networks

doi:10.1007/s41650-018-0021-6

Price-Based Power Allocation in Two-Tier Spectrum Sharing Heterogeneous Cellular Networks

Junhui Zhao^,, Qiuping Li^,, Yi Gong^,, Yongqiang Ning^,, Feifei Gao^,

School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China;School of Information Engineering, East China Jiaotong University, Nanchang 330013, China

作者简介 About authors

Junhui Zhao[corresponding author](S‘00-M‘04SM‘09) received his M S and Ph D degrees from Southeast University, Nanjing, China, in 1998 and 2004, respectively From 1998 to 1999, he worked with Nanjing Institute of Engineers at ZTE Corporation Then, he worked as an Assistant Professor at the Faculty of Information Technology in 2004, Macao University of Science and Technology, and continued to work there till 2007 as an Associate Professor In 2008, he joined Beijing Jiaotong University as an Associate Professor, where he is currently a Professor at the School of Electronics and Information Engineering Since 2016, he has also been with the School of Information Engineering in East China Jiaotong University Meanwhile, he was also a short term Visiting Scholar at Yonsei University, South Korea in 2004 and a Visiting Scholar at Nanyang Technological University, Singapore from 2013 to 2014 His current research interests include wireless and mobile communications and the related applications, which contain 5G mobile communication technology, high-speed railway communications, vehicle communication network, wireless localization, and cognitive radio He received the IEEE WCSP 2017 Best Paper Award , E-mail：junhuizhao@hotmail.com.

Qiuping Li received her B E degree in electronic information science and technology from China West Normal University, Sichuan, China, in 2015 She is currently pursuing her Ph D degree in communication and information systems from Beijing Jiaotong University, Beijing, China Her research interests include heterogeneous cellular network, vehicular networks, mobile edge computing, resource allocation, and optimization techniques , E-mail：16111029@bjtu.edu.cn.

Yi Gong (S‘99-M‘03-SM‘07) received his Ph D degree in electrical engineering from the Hong Kong University of Science and Technology, Hong Kong, China in 2002 He joined as a Member of Professional Staff with the Hong Kong Applied Science and Technology Research Institute, Hong Kong, China He was employed to work with Nanyang Technological University, Singapore, which he continues to actively collaborate with He is currently a Professor with Southern University of Science and Technology of China, Shenzhen, China His current research interests include cognitive radio, cooperative communications, MIMO, OFDM, and cross-layer design for wireless systems Since 2006, he has been serving on the editorial board of the IEEE Transactions on Wireless Communications and the IEEE Transactions on Vehicular Technology , E-mail：gongy@sustc.edu.cn.

Yongqiang Ning received his B E degree in communication engineering from Beijing Jiaotong University, Beijing, China, in 2015 He is currently pursuing his M S degree in communication and information systems from Beijing Jiaotong University, Beijing, China His research interests include signal processing, heterogeneous cellular network, resource allocation , E-mail：15120121@bjtu.edu.cn.

DrGao has served as the symposium cochair of the 2015 IEEE Conference on Communications,the 2014 IEEE Global Communications Conference,and the 2014 IEEE Vehicular Technology Conference Fall He has served as a technical committee member for many other IEEE conferences He has served as an Editor of the IEEE Transactions on Wireless Communications,the IEEE Communications Letters,the IEEE Signal Processing Letters,the IEEE Wireless Communications Letters,the International Journal on Antennas and Propagations,and China Communications , E-mail：feifeigao@ieee.org.

Abstract

Attributable to the using of the same spectrum resources, heterogeneous cellular networks have serious interference problems, which greatly restricts the performance of the network. In this paper, the price-based power allocation for femtocells underlaying a macrocell heterogeneous cellular network is investigated. By exploiting interference pricing mechanism, we formulate the interference management problem as a Stackelberg game and make a joint utility optimization of macrocells and femtocells. Specially, the energy consumption of macrocell users and the transmission rate utility of femtocell users are considered in this utility optimization problem. In the game model, the macrocell base station is regarded as a leader, which coordinates the interference from femtocell users to the macrocell users by pricing the interference. On the other hand, the femtocell base stations are modelled as followers. The femtocell users obtain their power allocation by pricing. After proving the existence of the Stackelberg equilibrium, the non-uniform and

Keywords： heterogeneous cellular networks ; interference management ; power allocation ; interference price,game theory

PDF (1381KB) Metadata Metrics Related articles Export EndNote| Ris| Bibtex Favorite

Cite this article

Junhui Zhao. Price-Based Power Allocation in Two-Tier Spectrum Sharing Heterogeneous Cellular Networks. [J], 2018, 3(2): 28-34 doi:10.1007/s41650-018-0021-6

Ⅰ. INTRODUCTION

Heterogeneous cellular networks(HCNs)have emergedas a promising solution for improving the quality of indoor wireless communication as well as enhancing spectral efficiency, and it has been widely viewed as a key enabling technology for the fifth generation (5G)mobile networks^[1]. HCNs are composed of traditional macrocell base station (MBS) and femtocell base station (FBS), such as long term evolution(LTE)-Advanced heterogeneous network scenarios with macrocells and femtocells. The FBSs are deployed with short-range, low-cost, low-power base stations. They are connected to the macrocell networks by digital subscriber line and cable modem^[2]. Femtocell technology improves cellular coverage within a building by providing a high-quality short-distance link to the user equipment. Consequently, it is a promising solution to reduce energy consumption for wireless networks.

On the other hand, with the explosive growth of wireless data traffic, energy consumption of wireless networks has increased greatly^[3,4]. Therefore, energy-efficient design in wireless networks has become increasingly important. However, macrocells and femtocells share the same spectrum. Moreover, dense deployments of femocells in an existing macrocell will inevitably arise cross-tier interference and cotier interference^[5,6]. Consequently, interference management is one of the most important challenges for the deployment of HCNs^[7,8].

Over the past few years, several works have appeared to deal with the power allocation for two-tier spectrum sharing HCNs. In Refs. [9,10], a price-based resource allocation strat egy was studied in the two-tier femtocell networks, but it did not consider the energy consumption of macrocell user equipment (MUE), and it was not conducive to the improvement of energy efficiency. In Refs. [11,12], an algorithm based on energy efficiency was presented for power allocation of femtocell. The proposed algorithm took energy consumption of MUE and femtocell user equipments (FUEs) into account, but it ignored the transmission rate utility for FUEs. Similarly, in Ref. [13], the authors utilized the Stackelberg game to propose a distributed resource allocation strategy in deviceto-device(D2D)communications underlaying a cellular network. However, it mainly maximized the sum data rates of D2D users and it didn’t consider the energy consumption of cellular users. In Ref. [14], a Stackelberg equilibrium (SE) was studied and a distributed non-uniform pricing power optimization algorithm was proposed, the major drawback of this scheme was the slow convergence and it took a great number of iterations to achieve optimization transmit power.

In this paper, price-based power allocation in HCNs is investigated. We model the power allocation problem as a Stackelberg game. In the game model, the MBS works as a leader, which decides the price of the interference on femtocell users to maximize its revenue. Subsequently, FBSs act as followers, and the FUEs calculate their power allocation to maximize the revenue based on the interference prices. By exploiting interference pricing mechanism, we control the aggressive interference of MBS below the interference threshold. Moreover, we not only consider the energy consumption of MUE but also take the transmission rate utility of FUEs into account in the game. Then, we decompose the interference management and power allocation problem into two sub-problems, and the SE is derived by solving those sub-problems. The non-uniform pricing scheme and uniform pricing scheme are proposed. We design the distributed interference pricing algorithm for the uniform interference pricing scheme and prove its convergence.

The remainder of the paper is organized as follows. In section Ⅱ, we introduce the system mode of a two-tier HCN. In section Ⅲ, we formulate the optimization problem and then propose the utility function of Stackelberg game. The interference pricing and power control scheme are introduced in section Ⅳ. In section V, simulation results are presented. Finally, the conclusion is presented in section Ⅵ.

Ⅱ. SYSTEM MODEL

Consider the system model for the uplink of a two-tier heterogeneous cellular network in Fig.1. The network consists of one central MBS which is underlaid with N randomly distributed FBSs. For analytical tractability, we assume that each base station has only one active user. Besides, during each signaling slot, the MUE and FUEs all operate over the same frequency band.

In the HCNs, let p_m denote the transmit power of the MUE and p_i denote the transmit power of the ith FUE. Denote h_mm as the channel gain from the MUE to the MBS. h_im denotes the channel gain from the ith FUE to the MBS. $σ_{m}^{2}$ is the complex white Gaussian noise of MUE with zero mean and unit variance. The signal-to-interference-plus-noise-ratio(SINR) of the MUE is

γ_{m} = \frac{p_{m} h_{m m}}{\sum_{i} p_{i} h_{i m} + σ_{m}^{2}}, (1)

where ∑_i p_ih_im denotes the aggregate cross-tier interference from all femtocell users.

Figure 1

New window| Download| PPT slide

Figure 1 System model for the uplink of a two-tier heterogeneous network

Let p_i denote the transmit power of the ith FUE. While h_ii denotes the channel gain from ith FUE to the ith FBS, and h_mi denotes the channel gain from MUE to the ith FBS. The $σ_{i}^{2}$ is the complex white Gaussian noise with zero mean and unit variance. The SINR of the ith FUE can be expressed as

γ_{i} = \frac{p_{i} h_{i i}}{\sum_{j \neq i} p_{i} h_{j i} + p_{m} h_{m i} + σ_{i}^{2}}, \forall i, (2)

where p_mh_mi denotes the interference suffered by MUE, and∑_j=i p_jh_ji represents the co-tier interference from all the other FUEs.

To avoid excessive interference to the MUE, we assume that the aggregate interference from all the femtocell users should not be larger than the interference tolerance threshold, which is expressed as

\sum_{i} p_{i} h_{i m} \leq T, (3)

where T denotes the maximum interference tolerance threshold of the MUE.

Ⅲ. PROBLEM FORMULATION

In this section, we exploit interference pricing mechanism and formulate the interference management problem as a Stackelberg game. In the Stackelberg game, players are differentiated as leaders and followers according to their action priorities^[15]. In this paper, we model the MBS as the leader and FUEs act as followers. The MBS imposes prices on femtocell users to obtain the interference revenue. Then, the FUEs calculate their power allocation based on the interference prices.

As for the MBS, we take the transmission rate and energy cost into account. It is easy to observe that the objective of MBS is to maximize its revenue by charging interference price from femtocell users. We set interference price as λ_i, the parameter u_m denotes additional power cost factor and w denotes the utility gain per unit capacity. Then, the revenue of the MBS can be calculated as follows

R_{n} = w \log (1 + γ_{m}) + \sum_{i = 1}^{N} λ_{i} p_{i} h_{i m} - u_{m} p_{m} . (4)

As for the FBSs, each FUE maximizes its utility by the offered price from the MBS. So, the revenue of each FUE can be defined as its achievable transmission rate minus the payment for interference quota. Thus, the revenue of ith FBS can be expressed as

R_{i} = w \log (1 + γ_{i}) - λ_{i} p_{i} h_{i m} . (5)

Based on the game model, we decompose the game problem into two sub-optimization problems: interference pricing problem and power control problem. Then, the revenue optimization problem can be summarized as below.

$P r o b l e m 1$ The interference pricing problem of MBS can be written as

\max_{λ \geq 0} R_{m} (λ_{i}, p_{m}, p_{i})

s . t . \sum_{i} p_{i} h_{i m} \leq T (6)

In the Stackelberg game, FUEs decide their power allocation to maximize the revenue. Moreover, transmit power of FUEs should be higher than 0.

$P r o b l e m 2$ The power control problem of FBSs can be modelled as

\max_{λ \geq 0} R_{i} (p_{i}, λ_{i})

s . t . p_{i} \geq 0 ， \forall_{i} . (7)

Ⅳ. POWER ALLOCATION AND INTERFERENCE PRICING

In the Stackelberg game, MBS and FBS implement the best strategy to achieve the maximum revenue. As for the MBS, by increasing the transmit power of MUE, MBS can increase its interference tolerance and more FUEs could access to the shared spectrum, which will subsequently increase its revenue^[16]. Besides, MBS can obtain its revenue from FUEs by interference pricing. Thus, interference prices can be optimized with power allocation and the tradeoff between revenue and energy consumption can be adjusted by the cost of energy. As for the FBS, its revenue depends upon power allocation of FUE. The FUEs have to optimize their transmit power on sharing spectrum according to interference prices.

In general, SE can be obtained by solving the subgame problems^[17]. Let λ^∗be a solution for Problem 1 and p^∗be a solution for Problem 2. We suppose that the price vector λ^∗=[λ₁, λ₂, …, λ_N]and power vector p^∗=[p₁, p₂, …, p_N]are the SE. Then, the strategy(λ^∗, p^∗)is an SE for the proposed Stackelberg game, and it is defined as the stable point after that none of the players can gain individually by unilaterally altering its own strategy. The following conditions are satisfied

R_{m} (λ^{*}, p^{*}) \geq R_{m} (λ, p^{*}), (8)

R_{i} (p_{i}^{*}, p_{- i}^{*}, λ^{*}) \geq R_{i} (p_{i}^{*}, p_{- i}^{*}, λ^{*}), \forall i, (9)

where $p_{- i}^{*} = [p_{1}, p_{2}, \dots, p_{i - 1}, p_{i + 1}, \dots, p_{N}]$ denotes the transmit power vector of the other FUEs.

We consider the scenario that FBSs are sparsely deployed within MBS coverage. Due to path loss and penetration loss, co-tier interference between femtocells can be ignored, i. e. , h _ji=0, ∀j=i^[18]. Eq. (2)can be rewritten as

γ_{i} = \frac{p_{i} h_{i i}}{p_{m} h_{m i} + σ_{i}^{2}}, \forall i . (10)

The above formula is related to transmit power of FUEs, the channel power gain as well as the cross-tier interference from MUE. Then we consider two pricing schemes: non-uniform interference pricing and uniform interference pricing, and determine the solutions of power allocation formulas for the two pricing schemes, respectively.

For the non-uniform pricing scheme, the MBS sets different interference prices for all FUEs. For Problem 2, FUEs should optimize their transmit power in order to deduce the optimal strategy. Considering the sparse deployment of femtocells, substituting Eq. (10)into Eq. (5), we can obtain the revenue of the ith FUE as

R_{i} = w \log (1 + \frac{p_{i} h_{i i}}{p_{m} h_{m i} + σ_{i}^{2}}) - λ_{i} p_{i} h_{i m}, \forall i . (11)

It is obvious that Problem 2 is a convex problem. The optimal solution must satisfy the Karush-Kuhn-Tucker (KKT) condition

\frac{\partial R_{i} (p_{i}, λ_{i})}{\partial p_{i}} = 0, \forall i . (12)

From Eqs. (11)and(12), we have the optimal power allocation strategy of the femtocell users as follow

p_{i} * = {(\frac{w}{λ_{i} h_{i m}} - \frac{p_{m} h_{m i} + σ_{i}^{2}}{h_{i i}})}^{+}, \forall i . (13)

where ${(\cdot)}^{+} ≜ \max (., 0)$ , i. e. , if the interference price $λ_{i} \geq w h i i / ((p_{m} h_{m i} + σ_{i}^{2}) h_{i m})$ , FUEs will stop transmitting on sharing spectrum. Therefore, the MBS can control the transmit power of the FUEs to the shared spectrum by controlling the interference pricing.

To solve the interference pricing problem, substituting Eq. (13)into Eqs. (3)and(4), the optimization problem can be formulated as

\underset{λ \geq 0}{\max R_{m}} =w \log (1 + \frac{p_{m} h_{m m}}{\sum_{i} p_{i} h_{i m} + σ_{m}^{2}}) +

{\sum_{i = 1}^{N} λ_{i} (\frac{w}{λ_{i} h_{i m}} - \frac{p_{m} h_{m i} + σ_{i}^{2}}{h_{i i}})}^{+} h_{i m} - u_{m} p_{m}, (14)

s . t . {\sum_{i} (\frac{w}{λ_{i} h_{i m}} - \frac{p_{m} h_{m i} + σ_{i}^{2}}{h_{i i}})}^{+} h_{i m} \leq T .

It can be derived that the objective function in Eq. (14) is concave with respect to R_m and the inequality constraints are convex, thus the Lagrangian method or other convex optimization methods can be utilized to solve the problem efficiently^[19]. It is assumed that the interference threshold is large enough so that all the FUEs can transmit on the shared spectrum. We denote µ _j as the Lagrange multiplier. The Lagrangian function can be formed to solve the problem of the interference pricing.

L_{i} (p_{i}, λ_{i}, u_{j})

= w \log (1 + \frac{p_{m} h_{m m}}{\sum_{i} p_{i} h_{i m} + σ_{m}^{2}}) +

{\sum_{i = 1}^{N} λ_{i} (\frac{w}{λ_{i} h_{i m}} - \frac{p_{m} h_{m i} + σ_{i}^{2}}{h_{i i}})}^{+} h_{i m} - u_{m} p_{m} +

u_{j} ({\sum_{j = 1}^{N} (\frac{w}{λ_{i} h_{i m}} - \frac{p_{m} h_{m i} + σ_{i}^{2}}{h_{i i}})}^{+} h_{i m} - T) . (15)

According to the KKT conditions, for the ith FBS, the following conditions should be satisfied

\frac{\partial L_{i} (p_{i}, λ_{i}, μ_{j})}{\partial λ_{i}} = 0, (16)

{\sum_{i = 1}^{N} λ_{i} (\frac{w}{λ_{i} h_{i m}} - \frac{p_{m} h_{m i} + σ_{i}^{2}}{h_{i i}})}^{+} h_{i m} - T = 0, (17)

μ_{j} \geq 0. (18)

According to Eq. (16), the optimal interference price for the ith FBS can be expressed as

λ_{i}^{*} = \frac{\sum_{i = 1}^{N} \sqrt{\frac{w h_{i m} (p_{m} h_{m i} + σ_{i}^{2})}{h_{i i}}}}{\sum_{i = 1}^{N} \sqrt{\frac{h_{i m} (p_{m} h_{m i} + σ_{i}^{2})}{h_{i i}} + T}} \sqrt{\frac{w h_{i i}}{h_{i m} (p_{m} h_{m i} + σ_{i}^{2})}} . (19)

From the formula of interference price in Eq. (19), the MBS sets different interference prices for different femtocell users. Interference prices are related to the transmit power of MUE, femtocell user channel power gain and so on. The MBS imposes higher interference prices for the FUEs that are deployed near by the MBS, and imposes lower interference prices for the FUEs that are far away from the MBS.

For the uniform interference pricing scheme, since we assume that the resource is distributed identically for all FUEs, the MBS sets the same interference price for all FUEs, i. e. , λ₁=λ₂=…=λ_N =λ. By solving Problem 2, we can obtain the optimal transmit power of the ith FUE

p i * = {(\frac{w}{λ h_{i m}} - \frac{p_{m} h_{m i} + σ_{i}^{2}}{h_{i i}})}^{+}, \forall i . (20)

The power allocation of the uniform interference pricing scheme can be achieved by the distributed interference pricing algorithm. First, the MBS broadcasts initial interference price λ to the FUEs. Then, the FUEs determine their transmit power p_i according to Eq. (20). Thirdly, the MBS measures the aggregate interference from all FUEs. When the aggregate interference ∑_i p_ih_im exceeds the interference constraint T, it will increase the price by d and if it is smaller than T, the MBS will decrease the price by d. After several iterations, the transmit power and interference price will reach the equilibrium points. The distributed interference pricing algorithm has the steps in uniform pricing. Algorithm complexity of the distributed interference pricing algorithm is O(N log(1/∆)).

Uniform pricing: distributed interference pricing algorithm

1)Initialization: λ⁰, d, ∆, and termination precision ε.

2)Determine the transmit power p_m.

3)Solve the optimization problem(20), and the MBS calculates the aggregate interference∑_i p_ih_im.

4)If∑i p_ih_im> T+ε, then

λⁱ=λⁱ⁻¹+d

if∑i p_ih_im 6 T+ε, then

λⁱ=λⁱ⁻¹−d

end

5)Update λⁱ and broadcast to FUEs.

6)If $‖ λ^{n} - λ^{n - 1} ‖$ ?< ∆, stop, otherwise go to 3).

Ⅴ. RESULTS AND DISCUSSIONS

In this section, numerical results are presented to demonstrate the performance of the proposed schemes. Assuming that there are one center MBS and three randomly deployed FBSs with their users in the sparsely scenario. Simulation parameters are assumed as follows: transmission rate income value is set to 1 for all users, and all payoff factors are equal to 0.1. Without loss of generality, we assume the average channel power gains between femtocell users and macrocell are h_1m=0.1, h_2m=0.5, h_3m=1, and h_im=h_mi, ∀i. Moreover, h_ii and h_mm are supposed as 1.

Fig.2 shows the MBS revenue versus the interference constraint under two pricing schemes. It can be seen that the revenue of the MBS increases with the adding of the interference constraint in a certain range, and the revenue for non-uniform scheme is higher than uniform scheme. In addition, it is worth noting that when interference constraint is sufficiently small or large, the revenue of the MBS becomes equal for the two pricing schemes. This is because when the interference constraint is very small or large, the number of FUEs that access in sharing spectrum reach the minimum or maximum, and the revenue of the MBS tends to be stable. The simulation result shows that non-uniform scheme is conducive to improve the MBS revenue.

Figure 2

New window| Download| PPT slide

Figure 2 Revenue of the MBS vs. interference constraint

Fig.3 illustrates the relationship between the MBS revenue and interference constraint. From the graph, it can be seen that when the MUE transmits power is 1 and 2, the revenue of the MBS firstly decreases with the increasing of the interference constraint, and then gradually increases. The reason is when the interference constraint T is relatively small, there are few FUEs accessing the sharing spectrum, and the revenues of the MBS mainly depend on the MUE’s transmission on sharing spectrum as well as energy consumption. As the interference constraint increases, the revenues of the MBS mainly depend on the profit from FUEs of interference price. So the MBS revenue gradually increases and finally becomes gentle. It can also be observed that when the interference constraint is higher than 0 dB, a lower interference constraint results in a larger revenue of MBS. While the interference constraint is smaller than 0 dB, a higher interference constraint results in a larger revenue of MBS.

Figure 3

New window| Download| PPT slide

Figure 3 Revenue of the MBS vs. interference constraint

Fig.4 shows the revenue of the FBS. It can be seen that the number of FUEs accessing the MBS as well as their revenue increases with the interference constraint T increasing. Besides, the FUEs with a small channel gain h_im have the priority to access to the sharing spectrum, and they will achieve a higher revenue than the large channel gain. For the reason that the FUEs with a small channel gain will cause smaller crosstier interference, and they have less payment for the cross-tier interference.

Figure 4

New window| Download| PPT slide

Figure 4 Revenue of the FBS vs. interference constraint

Figs. 5 and 6 depict the convergence performance of the distributed interference pricing algorithm. Fig.5 shows the uniform interference price versus the number of iterations. Through the distributed interference pricing algorithm, the MBS uses the interference constraint T to adjust the interference price, and finally achieves the optimal uniform interference price. In addition, the interference price is related to the number of FUEs accessing the shared spectrum. Similarly, Fig.6 indicates the transmit power of the FUEs versus the number of iterations. Moreover, the different FUEs converge to the different transmit powers. It is observed that the proposed algorithm converges within a small number of iterations, and it is promising for practical applications.

Figure 5

New window| Download| PPT slide

Figure 5 The uniform interference price vs. the number of iterations

Figure 6

New window| Download| PPT slide

Figure 6 The transmit power of the FUEs vs. the number of iterations

Ⅵ. CONCLUSION

In this paper, we have studied the power allocation problem for a HCN. By exploiting interference pricing mechanism, we formulate the interference management problem as a Stackelberg game, and make a joint utility optimization of macrocells and femtocells. Then, a convex optimization problem is formulated to investigate the femtocell interference problem. Non-uniform pricing scheme and uniform pricing scheme are proposed. Simulation results have been conducted to demonstrate the effectiveness of the proposed schemes in power allocation.

The authors have declared that no competing interests exist.

作者已声明无竞争性利益关系。

Reference

By original order

By published year

By cited within times

By Impact factor

[1]

Andrews

, S.

Buzzi

, W.

Choi

, et al.

What will 5G be

[J]. IEEE Journal on Selected Areas in Communications, 2014, 32(6): 1065-1082.

Journal of Communications and Information Networks