﻿ 蜂窝网络下基于max-min公平性的D2D功率分配
 计算机应用   2017, Vol. 37 Issue (4): 945-947, 953  DOI: 10.11772/j.issn.1001-9081.2017.04.0945 0

### 引用本文

NI Junhong, SHEN Zhentao, YANG Huifeng. D2D power allocation based on max-min fairness underlying cellular systems[J]. Journal of Computer Applications, 2017, 37(4): 945-947, 953. DOI: 10.11772/j.issn.1001-9081.2017.04.0945.

### 文章历史

1. 华北电力大学 电子与通信工程系, 河北 保定 071003;
2. 国网河北省电力公司 信息通信分公司, 石家庄 050021

D2D power allocation based on max-min fairness underlying cellular systems
NI Junhong1, SHEN Zhentao1, YANG Huifeng2
1. Department of Electronics and Communication Engineering, North China Electric Power University, Baoding Hebei 071003, China;
2. Information and Communication Branch, State Grid Hebei Electric Power Company, Shijiazhuang Hebei 050021, China
Abstract: Concerning the fairness problem of multiple Device-to-Device (D2D) users reusing the spectrum resources allocated to cellular subscribers, a power allocation algorithm based on max-min fairness was proposed under the premise of guaranteeing the rate of cellular users. First, the nonconvex optimization problem was transformed into a Difference between Convex functions (DC) programming problem, then the global optimization algorithm of convex approximation and the bisection algorithm were used to achieve power optimization of D2D. Simulation results show that compared with the global optimization algorithm which only uses convex approximation, the proposed algorithm has better convergence and maximizes the bottleneck rate of D2D users.
Key words: Device-to-Device (D2D)    max-min fairness    difference between convex functions programming    power optimization
0 引言

1 系统模型

 $\gamma _c^n = \frac{{p_c^nH_c^n}}{{\sum\limits_{m \in M} {p_m^n\tilde H_{m, c}^n + \sigma _0^2} }}$ (1)

 $\gamma _m^n = \frac{{p_m^nH_m^n}}{{\sum\limits_{m' \in M\backslash \{ m\} } {p_{m'}^n\tilde H_{m', m}^n + p_c^n\tilde H_{c, m}^n + \sigma _0^2} }}$ (2)

 ${R_m} = \sum\limits_{i = 1}^N {{\rm{lb}}(1 + \gamma _m^n)}$ (3)

 $\begin{array}{l} \;\;\;\;\;\mathop {\max }\limits_\mathit{\boldsymbol{ P}} \mathop {\min }\limits_{m \in M} \;{R_m}(\mathit{\boldsymbol{ P}})\\ {\rm{s}}.{\rm{t}}.\;\;\;\;{C_1}:\;{R_c}(\mathit{\boldsymbol{ P}}) \ge {R_{c,\min }}{\kern 1pt} ;\quad \forall c \in N\\ \,\quad \quad {C_2}\;:\;\sum {p_m^n} \le {p_{d,\max }};\quad \forall m \in M\\ \quad \quad \;{C_3}\;:\;p_c^n \le {p_{c,\max }};\quad \forall c \in N \end{array}$ (4)

2 功率优化

 ${R_m}(\mathit{\boldsymbol{P}}) = {f_m}(\mathit{\boldsymbol{P}}) - {g_m}(\mathit{\boldsymbol{P}})$ (5)

 ${f_m}(\mathit{\boldsymbol{P}}) = \sum\limits_{n = 1}^N {{\rm{lb}}(p_m^nH_m^n + \sum\limits_{m' \in {U_n}\backslash \{ m\} } {p_{m'}^n\tilde H_{m',m}^n + \sigma _0^2} )}$ (6)
 ${g_m}(\mathit{\boldsymbol{P}}) = \sum\limits_{i = 1}^N {{\rm{lb}}(\sum\limits_{m' \in {U_n}\backslash \{ m\} } {p_{m'}^n\tilde H_{m',m}^n + \sigma _0^2} )}$ (7)

 ${R_m}(\mathit{\boldsymbol{P}}) = {f_m}(\mathit{\boldsymbol{P}}) + \sum\limits_{j \in {U_n}\backslash \{ m\} } {{g_j}(\mathit{\boldsymbol{P}})} - \sum\limits_{j \in {U_n}} {{g_j}(\mathit{\boldsymbol{P}})}$ (8)

 ${F_m}(\mathit{\boldsymbol{P}}) = {f_m}(\mathit{\boldsymbol{P}}) + \sum\limits_{j \in {U_n}\backslash \{ m\} } {{g_j}(\mathit{\boldsymbol{P}})}$ (9)
 $G(\mathit{\boldsymbol{P}}) = \sum\limits_{j \in {U_n}} {{g_j}(\mathit{\boldsymbol{P}})}$ (10)

 ${R_m}(\mathit{\boldsymbol{P}}) = {F_m}(\mathit{\boldsymbol{P}}) - G(\mathit{\boldsymbol{P}})$ (11)

 $\begin{array}{l} \;\;\;\;\;\mathop {\max }\limits_\mathit{\boldsymbol{P}} \;F(\mathit{\boldsymbol{P}}) - G(\mathit{\boldsymbol{P}})\quad \\ {\rm{s}}.{\rm{t}}.\;\;\;\;{C_1} \sim {C_3}\rm{in}(4) \end{array}$ (12)

 $F(\mathit{\boldsymbol{P}}) = \mathop {\min }\limits_{m \in {U_n}} {F_m}(\mathit{\boldsymbol{P}})$ (13)

 $G(\mathit{\boldsymbol{P}}) \approx G(\mathit{\boldsymbol{P}}') + \left\langle {\nabla G(\mathit{\boldsymbol{P}}'),\mathit{\boldsymbol{P}} - \mathit{\boldsymbol{P}}'} \right\rangle$ (14)

 $F(\mathit{\boldsymbol{P}}) - G(\mathit{\boldsymbol{P}}) \approx F(\mathit{\boldsymbol{P}}) - G(\mathit{\boldsymbol{P}}') - \left\langle {\nabla G(\mathit{\boldsymbol{P}}'),\mathit{\boldsymbol{P}} - \mathit{\boldsymbol{P}}'} \right\rangle$ (15)

 $\begin{array}{l} \;\;\;\;\;\;\;\;\mathop {\max }\limits_{\mathit{\boldsymbol{P}},\eta } \quad \eta \\ \,\,\,{\rm{s}}.{\rm{t}}.\;\;\;\;{C_1} \sim {C_3}\;{\rm{in}}\;(4)\\ \quad \;\,\,\,{F_m}(\mathit{\boldsymbol{P}}) - G({\mathit{\boldsymbol{P}}^{(\lambda )}}) - \langle \nabla G({\mathit{\boldsymbol{P}}^{(\lambda )}}),\mathit{\boldsymbol{P}} - {\mathit{\boldsymbol{P}}^{(\lambda )}}\rangle \, \ge \eta ;\\ \;\;\;\;\;\;\;\;\;\;\forall m \in M \end{array}$ (16)

 $\begin{array}{l} \;\;\;\;\;\;\;\mathop {\max }\limits_\mathit{\boldsymbol{P}} \;\{ F(\mathit{\boldsymbol{P}}) - G({\mathit{\boldsymbol{P}}^{(\lambda )}}) - \langle \nabla G({\mathit{\boldsymbol{P}}^{(\lambda )}}),\mathit{\boldsymbol{P}} - {\mathit{\boldsymbol{P}}^{(\lambda )}}\rangle \} \\ \;\;{\rm{s}}.{\rm{t}}.\;\,\;\;{C_1} \sim {C_3}\;{\rm{in}}\;(4)\\ \quad \quad F(\mathit{\boldsymbol{P}}) - G({\mathit{\boldsymbol{P}}^{(\lambda )}}) - \langle \nabla G({\mathit{\boldsymbol{P}}^{(\lambda )}}),\mathit{\boldsymbol{P}} - {\mathit{\boldsymbol{P}}^{(\lambda )}}\rangle \ge \beta _m^{(\lambda )};\\ \;\;\;\;\;\;\;\;\;\;\;\forall m \in M \end{array}$ (17)

 ${R_1}({\mathit{\boldsymbol{P}}_{{\rm{opt}}}}) = {R_2}({\mathit{\boldsymbol{P}}_{{\rm{opt}}}}) = ... = {R_m}({\mathit{\boldsymbol{P}}_{{\rm{opt}}}})$ (18)

1) 将λκt的初始值置为0, 将P(0)代入方程 (16) 中求解, P(1)=P*

2) 判断t是否达到门限值, 如果达到门限值，则转到4);否则将最优值P(1)分别代入下列各式中:

 $\begin{array}{l} r_{\min }^* = \mathop {\min }\limits_{i = 1, 2, ..., m} {R_i}({\mathit{\boldsymbol{P}}^{(1)}})\\ r_m^{(\kappa )} = \mathop {\max }\limits_{i = 1, 2, ..., m} {R_i}({\mathit{\boldsymbol{P}}^{(1)}}) \end{array}$

3) 求凸优化问题 (17) 的解P*，如果无解, κ=κ+1, βm(κ)=(βm(κ-1)+rmin*)/2，迭代直至求得解P*t=t+1, P(t)=P*, 返回2)。

4) 用优化后的功率值P(t)求解方程 (16), λ=λ+1, P(λ)=P*, 计算$r_{\min }^{(\lambda )} = \mathop {\min }\limits_{i = 1, 2, ..., m} {R_i}({\mathit{\boldsymbol{P}}^{(\lambda )}})$, 迭代求解 (16) 直到$\left| {r_{\min }^{(\lambda )}-r_{\min }^{(\lambda-1)}} \right| \le \varepsilon$

3 仿真实验和性能分析 3.1 系统参数

 ${\mathit{\boldsymbol{H}}_1} = \left[ {\begin{array}{*{20}{c}} {0.4310} & {0.0002} & {0.2605} & {0.0039}\\ {0.0002} & {0.3018} & {0.0008} & {0.0054}\\ {0.0129} & {0.0005} & {0.4266} & {0.1007}\\ {0.0011} & {0.0031} & {0.0099} & {0.0634} \end{array}} \right]$ (19)
 ${\mathit{\boldsymbol{H}}_2} = \left[ {\begin{array}{*{20}{c}} {0.1476} & {0.0105} & {0.0018} & {0.0402}\\ {0.0034} & {0.1784} & {0.0013} & {0.2472}\\ {0.0014} & {0.0017} & {0.3164} & {0.0046}\\ {0.0048} & {0.4526} & {0.0012} & {0.6290} \end{array}} \right]$ (20)

3.2 优化后的用户发送功率和速率

DUE用户在共享一个信道H1时, 应用上述迭代算法解问题 (4), 初始化功率为用户功率的最大值, 仿真结果如图 1所示。由图 1可知, 本文算法在6次迭代后蜂窝用户的速率为3.0 bps/Hz, DUE速率收敛于2.085 4 bps/Hz, 优化后各个用户 (CUE1、DUE1、DUE2和DUE3) 的功率值分别为5.779 8 mW, 3.447 6 mW, 18.998 6 mW, 99.998 5 mW。

 图 1 共享信道H1时优化的用户速率 Figure 1 Optimized data rate of each link when DUE sharing channel H1

DUE用户在共享两个信道 (H1H2) 时, 应用上述迭代算法解问题 (4), 初始化DUE在各个信道功率相等, 且DUE功率之和为用户功率的最大值, 其中DUE在共享两个信道时的用户速率仿真结果如图 2所示。可以得出, 在85次迭代后蜂窝用户的速率为3.0 bps/Hz, DUE速率收敛于7.813 9 bps/Hz, 在信道H1上各个用户 (CUE1、DUE1、DUE2和DUE3) 优化后的功率值分别为1.289 2 mW, 99.395 2 mW, 0.944 2 mW, 42.928 2 mW；在信道H2上各个用户 (CUE2、DUE1、DUE2和DUE3) 的优化后的功率值分别为12.683 2 mW, 4.655 0E-10 mW, 36.730 9 mW, 44.981 6 mW。

 图 2 共享信道H1和H2时优化的用户速率 Figure 2 Optimized data rate of each link when DUE sharing channel H1 and H2
3.3 算法收敛速度对比

 图 3 共享信道H1不同t门限下最小D2D用户速率收敛对比 Figure 3 Comparison of minimal D2D user rate convergence when D2D sharing channel H1 at different thresholds t
 图 4 共享信道H1和H2不同t门限下最小D2D用户速率收敛对比 Figure 4 Comparison of minimum D2D user rate convergence when D2D sharing channel H1 and H2 at different thresholds t
4 结语

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