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 重庆大学学报  2020, Vol. 43 Issue (1): 100-112  DOI: 10.11835/j.issn.1000-582X.2020.01.011 RIS（文献管理工具） 0

### 引用本文

HU Guangtao, TANG Lun. Queue-aware online power allocation strategy in wireless backhaul networks[J]. Journal of Chongqing University, 2020, 43(1): 100-112. DOI: 10.11835/j.issn.1000-582X.2020.01.011.

### 文章历史

Queue-aware online power allocation strategy in wireless backhaul networks
HU Guangtao , TANG Lun
College of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, P. R. China
Abstract: Wireless backhaul technology is one of the promising solutions for next-generation mobile communication networks due to its merits of significantly reducing operator's cost, providing user equipment (UE) with fundamental flexibility and improving the network overall spectrum efficiency. By leveraging the Lyapunov stochastic optimization framework and convex optimization theory, this paper proposes a queue-aware power allocation algorithm for in-band full-duplex wireless backhaul networks. Specifically, in each discrete resource scheduling time slot, the algorithm dynamically allocates power for each user's downlink backhaul and access links by comprehensively taking channel state information (CSI) and queue state information (QSI) into consideration, so as to maximize the network average total spectrum efficiency (SE) while ensuring network stability and meeting the quality of service (QoS) requirements of users. In addition, the theoretical analysis and simulation results reveal that the proposed algorithm can flexibly strike a balance between SE and delay by simply tuning an introduced control parameter.
Keywords: wireless backhaul network    resource allocation    queue stability    Lyapunov

1 系统模型 1.1 系统场景

 图 1 带内全双工无线回传网络下行传输场景 Fig. 1 Downlink transmission scenario in in-band full-duplex wireless backhaul networks

MBS服务K个单天线用户(MUE, macro user equipment), 令K={1, 2, …, K}表示MUE集合, 每个SBS服务一个单天线用户SUE, SBS/SUE集合用N={1, 2, …, N}表示。考虑大规模MIMO系统(K+NM), MBS利用波束赋形技术使用相同的频率资源同时给多个用户传送数据。为了获得大规模MIMO天线的性能增益, 信道状态信息(CSI, channel state information)须在MBS发射端是已知的, 因此假设所考虑的无线异构自回传网络采用时分双工(TDD, time division duplex)协议。根据信道互易性原理, MBS可以利用其用户发送的上行导频信号来估计其下行链路。

1.2 信道模型和用户速率

 图 2 带内全双工无线回传网络下行链路干扰示意图 Fig. 2 Downlink interference graph in in-band full-duplex wireless backhaul networks

 $y_k^m\left( t \right) = \sqrt {p_k^m\left( t \right)/\beta _k^m\left( t \right)} h_k^m\left( t \right)w_k^m\left( t \right)s_k^m\left( t \right) +\\ \sum\limits_{n = 1}^N {\sqrt {p_n^s\left( t \right)/\beta _{n,k}^s\left( t \right)} } h_{n,k}^s\left( t \right)s_n^s\left( t \right) + r_k^m\left( t \right),$ (1)

 $\varepsilon _k^m\left( t \right) = \frac{{p_k^m\left( t \right)/\beta _k^m\left( t \right){{\left\| {h_k^m\left( t \right)w_k^m\left( t \right)} \right\|}^2}}}{{\sum\limits_{n \in N} {p_n^s\left( t \right)/\beta _{n,k}^s\left( t \right){{\left| {h_{n,k}^s\left( t \right)} \right|}^2} + {\sigma ^2}} }},$ (2)
 $R_k^m\left( t \right) = \log \left( {1 + \varepsilon _k^m\left( t \right)} \right),$ (3)

 $\begin{array}{*{20}{c}} {y_n^b\left( t \right) = \sqrt {p_n^b\left( t \right)/\beta _n^b\left( t \right)} h_n^b\left( t \right)w_n^b\left( t \right)s_n^b\left( t \right) + \sum\limits_{n' \in N,n' \ne n} {\sqrt {p_{n'}^s\left( t \right)/\beta _{n',n}^b\left( t \right)} h_{n',n}^b\left( t \right)s_{n'}^s\left( t \right)} + }\\ {\sqrt {\gamma p_n^s\left( t \right)} s_n^s\left( t \right) + r_n^b\left( t \right),} \end{array}$ (4)

 $\begin{array}{*{20}{c}} {y_n^s\left( t \right) = \sqrt {p_n^s\left( t \right)/\beta _n^s\left( t \right)} h_n^s\left( t \right)w_n^s\left( t \right)s_n^s\left( t \right) +\\ \sum\limits_{n' \in N,n' \ne n} {\sqrt {p_{n'}^s\left( t \right)/\beta _{n',n}^s\left( t \right)} h_{n',n}^s\left( t \right)s_{n'}^s\left( t \right)} + r_n^s\left( t \right) + }\\ {\sum\limits_{k \in K} {\sqrt {p_k^m\left( t \right)/\beta _n^m\left( t \right)} } h_n^m\left( t \right)w_k^m\left( t \right)s_k^m\left( t \right) + \sum\limits_{n' \in N} {\sqrt {p_{n'}^b\left( t \right)/\beta _n^m\left( t \right)} } h_n^m\left( t \right)w_{n'}^b\left( t \right)s_{n'}^b\left( t \right),} \end{array}$ (5)

1.3 队列模型

 $Q_k^m\left( {t + 1} \right) = {\left[ {Q_k^m\left( t \right) + \alpha _k^m\left( t \right) - R_k^m\left( t \right)\tau } \right]^ + },$ (6)
 ${Q_n}\left( {t + 1} \right) = {\left[ {{Q_n}\left( t \right) + {\alpha _n}\left( t \right) - R_n^b\left( t \right)\tau } \right]^ + },$ (7)

 ${D_n}\left( {t + 1} \right) = {\left[ {{D_n}\left( t \right) + R_n^b\left( t \right)\tau - R_n^s\left( t \right)\tau } \right]^ + },$ (8)

 $\bar Q = \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\sum\limits_{t = 0}^{T - 1} {E\left[ {\mathit{\boldsymbol{Q}}\left( t \right)} \right]} < \infty ,$ (9)

1.4 问题描述

 $R\left( t \right) = \sum\limits_{k \in K} {R_k^m\left( t \right)} + \sum\limits_{n \in N} {R_n^s\left( t \right)} ,$ (10)

P(t)=[pkm(t), ∀kK, pnb(t), pns(t), nN]表示时隙t内网络传输功率向量。在所考虑的带内全双工无线回传网络中, 功率分配是一个关键的问题, 其不仅决定了网络中的干扰和能耗水平, 又因MBS需分配额外的功率以支持SUE的回传链路传输, 因此, 其还影响了网络的频谱效率。主要探究无线回传网络中最优的功率分配策略, 以实现在满足各用户QoS需求和系统稳定性的前提下, 最大化网络平均和谱效, 问题具体描述如下

 $\begin{array}{l} \mathop {\max }\limits_P \bar R\\ {\rm{s}}.{\rm{t}}.\\ C1.\bar Q_k^m < \infty \forall k \in K\\ \;\;{{\bar D}_n} < \infty ,{{\bar Q}_n} < \infty \forall n \in N;\\ C2.\sum\limits_{k \in K} {p_k^m\left( t \right)} + \sum\limits_{n \in N} {p_n^b\left( t \right)} \le p_{\max }^m;\\ C3.p_n^s\left( t \right) \le p_{\max }^s\left( t \right)\forall n \in N;\\ C4.p_k^m\left( t \right) \ge 0\forall k \in K\\ p_n^b\left( t \right) \ge 0,p_n^s\left( t \right) \ge 0\forall n \in N;\\ C5.R_k^m\left( t \right) \ge R_k^{mthr}\forall k \in K;\\ C6.R_k^s\left( t \right) \ge R_n^{sthr}\forall n \in N。\end{array}$ (11)

2 问题转化与算法提出

2.1 问题转化

 $L\left( t \right) = \frac{1}{2}\left( {\sum\limits_{k \in K} {{{\left( {Q_k^m\left( t \right)} \right)}^2}} + \sum\limits_{n \in N} {{{\left( {{Q_n}\left( t \right)} \right)}^2}} + \sum\limits_{n \in N} {{{\left( {{D_n}\left( t \right)} \right)}^2}} } \right),$ (12)

 $\Delta \left( t \right) = E\left[ {L\left( {t + 1} \right) - L\left( t \right)\mathit{\boldsymbol{Q}}\left( t \right),\mathit{\boldsymbol{D}}\left( t \right)} \right],$ (13)

Lyapunov偏移与加权代价函数之和可表示为Δ(t)－VE[R(t)|Q(t), D(t)], 其中, V是一个非负实数, 用于调节网络稳定性和网络平均和谱效间的相对重要程度。

 $\begin{array}{*{20}{c}} {\Delta \left( t \right) - VE\left[ {R\left( t \right)|\mathit{\boldsymbol{Q}}\left( t \right),\mathit{\boldsymbol{D}}\left( t \right)} \right] \le B +\\ E\left[ {\sum\limits_{k \in K} {Q_k^m} \left( t \right)\left( {\lambda _k^m - R_k^m\left( t \right)\tau } \right) + \sum\limits_{n \in N} {{Q_n}} \left( t \right)\left( {{\lambda _n} - R_n^b\left( t \right)\tau } \right)} \right.}\\ {\left. { + \sum\limits_{n \in N} {{D_n}} \left( t \right)\left( {R_n^b\left( t \right)\tau - R_n^s\left( t \right)\tau } \right)} \right] - VE\left[ {R\left( t \right)|\mathit{\boldsymbol{Q}}\left( t \right),\mathit{\boldsymbol{D}}\left( t \right)} \right],} \end{array}$ (14)

 $\begin{array}{l} \mathop {\min }\limits_{P\left( t \right)} f\left( t \right) = - \sum\limits_{k \in K} {\left( {Q_k^m\left( t \right)\tau + V} \right)R_k^m\left( t \right)} - \sum\limits_{n \in N} {\left( {{Q_n}\left( t \right) - {D_n}\left( t \right)} \right)\tau R_n^b\left( t \right)} - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{n \in N} {\left( {{D_n}\left( t \right)\tau + V} \right)R_n^s\left( t \right)} ,\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{s}}.\;{\rm{t}}.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C2 \cdot \sum\limits_{k \in K} {p_k^m} \left( t \right) + \sum\limits_{n \in N} {p_n^b} \left( t \right) \le p_{\max }^m;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C3 \cdot p_n^s\left( t \right) \le p_{\max }^s\forall n \in N;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C4 \cdot p_k^m\left( t \right) \ge 0\forall k \in K\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;p_n^b\left( t \right) \ge 0,p_n^s\left( t \right) \ge 0\forall n \in N;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C5 \cdot R_k^m\left( t \right) \ge R_k^{mthr}\forall k \in K;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C6.R_n^s\left( t \right) \ge R_n^{sthr}\forall n \in N。\end{array}$ (15)
2.2 基于队列感知的无线接入与回传联合功率分配算法

 $c\log z + \mu \le \log \left( {1 + z} \right),$ (16)

 $c = \frac{{{z_0}}}{{1 + {z_0}}},$ (17)
 $\mu = \log \left( {1 + {z_0}} \right) - \frac{{{z_0}}}{{1 + {z_0}}}\log {z_0},$ (18)

 $\begin{array}{*{20}{c}} {\mathop {R_k^m}\limits^ \gg = c_k^m\log \left( {\varepsilon _k^m} \right) + \mu _k^m = }\\ {c_k^m\left\{ {a\tilde p_k^m + \log \left( {1/\beta _k^m{{\left\| {h_k^mw_k^m} \right\|}^2}} \right) - \log \left( {\sum\limits_{n \in N} {{e^{\tilde p_n^s}}} \left( {1/\beta _{n,k}^s} \right){{\left| {h_{n,k}^s} \right|}^2} + {\sigma ^2}} \right)} \right\} + \mu _k^m,} \end{array}$ (19)

 $\begin{array}{l} \mathop {\min }\limits_{\tilde P} \mathop f\limits^ \gg = - \sum\limits_{k \in K} {\left( {Q_k^m\tau + V} \right)} \mathop {R_k^m}\limits^ \gg - \sum\limits_{n \in N} {\left( {{Q_n} - {D_n}} \right)} \tau \mathop {R_n^b}\limits^ \gg - \sum\limits_{n \in N} {\left( {{D_n}\tau + V} \right)} \mathop {R_n^s}\limits^ \gg ,\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{s}}.\;{\rm{t}}.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C2 \cdot \sum\limits_{k \in K} {{e^{\tilde p_k^n}}} + \sum\limits_{n \in N} {{e^{\tilde p_h^b}}} \le p_{\max }^m,\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C3.{e^{\tilde p_n^s}} \le p_{\max }^s\forall n \in N,\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C5 \cdot \mathop {R_k^m}\limits^ \gg \ge R_k^{mthr}\forall k \in K,\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;C6 \cdot \mathop {R_n^s}\limits^ \gg \ge R_k^{sthr}\forall n \in N, \end{array}$ (20)

 ${\left[ {{z_0}} \right]^i} = {\left[ \varepsilon \right]^i}\forall \varepsilon _k^m,\varepsilon _n^b,\varepsilon _n^s,$ (21)

 $\begin{array}{*{20}{c}} {\mathop {L\left( {\tilde P,\alpha ,\beta ,\eta ,v} \right)}\limits_{{\alpha _n} \ge 0,\beta \ge 0,{\eta _k} \ge 0,{v_n} \ge 0} = \mathop f\limits^ \gg + \sum\limits_{n \in N} {{\alpha _n}\left( {{e^{\tilde p_n^s}} - p_{\max }^s} \right)} + \beta \left( {\sum\limits_{k \in K} {{e^{\tilde p_k^m}}} + \sum\limits_{n \in N} {{e^{\tilde p_h^b}}} - p_{\max }^m} \right) + }\\ {\sum\limits_{k \in K} {{\eta _k}} \left( {R_k^{{\rm{mthr}}} - \mathop {R_k^m}\limits^ \gg } \right) + \sum\limits_{n \in N} {{v_n}} \left( {R_n^{{\rm{sthr}}} - \mathop {R_n^s}\limits^ \gg } \right),} \end{array}$ (22)

 $\mathop {\min }\limits_{\mathit{\boldsymbol{\tilde P}}} \mathop {\max }\limits_{\alpha ,\beta ,\eta ,v} L\left( {\mathit{\boldsymbol{\tilde P}},\alpha ,\beta ,\eta ,v} \right),$ (23)

 $\frac{{\partial L\left( {\mathit{\boldsymbol{\tilde P}},\alpha ,\beta ,\eta ,v} \right)}}{{\partial \mathit{\boldsymbol{\tilde P}}_k^m}} = - \left( {Q_k^m\tau + V + {\eta _k}} \right) \cdot c_k^ma - \sum\limits_{n \in N} {\left( {{D_n}\tau + V + {v_n}} \right)} \frac{{\partial \mathop {R_n^s}\limits^ \gg }}{{\partial \mathit{\boldsymbol{\tilde P}}_k^m}} + \beta {e^{\mathit{\boldsymbol{\tilde P}}_k^m}},$ (24)
 $\frac{{\partial L\left( {\mathit{\boldsymbol{\tilde P}},\alpha ,\beta ,\eta ,v} \right)}}{{\partial \mathit{\boldsymbol{\tilde P}}_{n'}^b}} = - \left( {{Q_{n'}} - {D_{n'}}} \right)\tau c_{n'}^ba - \sum\limits_{n \in N} {\left( {{D_n}\tau + V + {v_n}} \right)} \frac{{\partial \mathop {R_n^s}\limits^ \gg }}{{\partial \mathit{\boldsymbol{\tilde P}}_{n'}^b}} + \beta {e^{\mathit{\boldsymbol{\tilde P}}_{n'}^b}},$ (25)
 $\begin{array}{*{20}{c}} {\frac{{\partial L\left( {\mathit{\boldsymbol{\tilde P}},\alpha ,\beta ,\eta ,v} \right)}}{{\partial \mathit{\boldsymbol{\tilde P}}_{n'}^s}} = - \left( {{D_{n'}}\tau + V + {v_n}} \right)c_{n'}^sa - \sum\limits_{k \in K} {\left( {Q_k^m\tau + V + {\eta _k}} \right)} \frac{{\partial \mathop {R_k^m}\limits^ \gg }}{{\partial \mathit{\boldsymbol{\tilde P}}_{n'}^s}} - }\\ { - \sum\limits_{n \in N} {\left( {{Q_n} - {D_n}} \right)\tau \frac{{\partial \mathop {R_n^b}\limits^ \gg }}{{\partial \mathit{\boldsymbol{\tilde P}}_{n'}^s}} + {\alpha _{n'}}{e^{\mathit{\boldsymbol{\tilde P}}_{n'}^s}}} 。} \end{array}$ (26)

 ${l^{{i_{2 + 1}}}} = {\left[ {{l^{{i_2}}} + \delta \frac{{\partial L\left( {\mathit{\boldsymbol{\tilde P}},\alpha ,\beta ,\eta ,v} \right)}}{{\partial {l^{{i_2}}}}}} \right]^ + },l = \left\{ {{\alpha _n},\beta ,{\eta _k},{v_n}} \right\},$ (27)

3 性能分析

a) 网络平均队列长度

 $\mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\sum\limits_{t = 0}^{T - 1} {\left[ {\sum\limits_{k \in K} {Q_k^m} \left( t \right) + \sum\limits_{n \in N} {{Q_n}} \left( t \right) + \sum\limits_{n \in N} {{D_n}} \left( t \right)} \right]} \le \frac{{B + V{R^{\max }}}}{\varepsilon },$ (28)

b) 网络平均和谱效

 $\mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\sum\limits_{t = 0}^{T - 1} {R\left( t \right)} \ge {{\bar R}^{opt}} - \frac{B}{V},$ (29)

 $E\left[ {{R^ * }\left( t \right)} \right] \ge {{\bar R}^{opt}} - \varphi ,$ (30)
 $E\left[ {R_n^{s * }\left( t \right)\tau } \right] \ge E\left[ {R_n^{b * }\left( t \right)\tau } \right] \ge \left( {{\lambda _n} + \varepsilon } \right),$ (31)
 $E\left[ {R_k^{m * }\left( t \right)\tau } \right] \ge \left( {\lambda _k^m + \varepsilon } \right),$ (32)

 $\begin{array}{*{20}{c}} {\Delta \left( t \right) - VE\left[ {R\left( t \right)|Q\left( t \right),\mathit{\boldsymbol{D}}\left( t \right)} \right] \le B +\\ E\left[ {\sum\limits_{k \in K} {Q_k^m} \left( t \right)\left( {\lambda _k^m - R_k^{m * }\left( t \right)\tau } \right) + \sum\limits_{n \in N} {{Q_n}} \left( t \right)\left( {{\lambda _n} - R_n^{b * }\left( t \right)\tau } \right)} \right.}\\ {\left. { + \sum\limits_{n \in N} {{D_n}} \left( t \right)\left( {R_n^{b * }\left( t \right)\tau - R_n^{s * }\left( t \right)\tau } \right)} \right] - VE\left[ {{R^ * }\left( t \right)|\mathit{\boldsymbol{Q}}\left( t \right),\mathit{\boldsymbol{D}}\left( t \right)} \right],} \end{array}$ (33)

 $\begin{array}{*{20}{c}} {\Delta \left( t \right) - VE\left[ {R\left( t \right)|\mathit{\boldsymbol{Q}}\left( t \right),\mathit{\boldsymbol{D}}\left( t \right)} \right] \le B - \varepsilon \sum\limits_{k \in K} {Q_k^m\left( t \right)} - \varepsilon \sum\limits_{n \in N} {{Q_n}\left( t \right)} - }\\ {c\sum\limits_{n \in N} {{D_n}\left( t \right)} - VE\left[ {{R^ * }\left( t \right)|\mathit{\boldsymbol{Q}}\left( t \right),\mathit{\boldsymbol{D}}\left( t \right)} \right],} \end{array}$ (34)

 $\Delta \left( t \right) \le B - \varepsilon \sum\limits_{k \in K} {Q_k^m} \left( t \right) - \varepsilon \sum\limits_{n \in N} {{Q_n}} \left( t \right) + V{R^{\max }},$ (35)

 $\mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\sum\limits_{t = 0}^{T - 1} {\left( {\sum\limits_{k \in K} {Q_k^m\left( t \right)} + \sum\limits_{n \in N} {{Q_n}\left( t \right)} } \right)} \le \frac{{B + V{R^{\max }}}}{\varepsilon },$ (36)

 $\mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\sum\limits_{t = 0}^{T - 1} {\sum\limits_{n \in N} {{D_n}\left( t \right)} } \le \frac{{B + V{R^{\max }}}}{c},$ (37)

 $\begin{array}{*{20}{c}} {\Delta \left( t \right) \le B + VE\left[ {R\left( t \right)\left| {\mathit{\boldsymbol{Q}}\left( t \right),\mathit{\boldsymbol{D}}\left( t \right)} \right.} \right] - VE\left[ {{R^ * }\left( t \right)\left| {\mathit{\boldsymbol{Q}}\left( t \right),\mathit{\boldsymbol{D}}\left( t \right)} \right.} \right]}\\ { \le B + VE\left[ {R\left( t \right)\left| {\mathit{\boldsymbol{Q}}\left( t \right),\mathit{\boldsymbol{D}}\left( t \right)} \right.} \right] - V{{\bar R}^{opt}},} \end{array}$ (38)

 $\mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\sum\limits_{t = 0}^{T - 1} {R\left( t \right)} \ge {{\bar R}^{opt}} - \frac{B}{V},$ (39)

4 仿真结果与分析

 图 3 平均队列长度 Fig. 3 Time-varying graph of average queue length

 图 4 平均谱效 Fig. 4 Time-varying graph of average spectrum efficiency

 图 5 平均队列长度vs控制参量V Fig. 5 Average queue length vs control parameter V
 图 6 平均网络和谱效vs控制参量V Fig. 6 Average network and spectrum efficiency vs control parameter V

 图 7 时延vs平均业务到达率 Fig. 7 Delay vs average service arrival rate

 图 8 平均网络和谱效vs平均业务到达率 Fig. 8 Average network and spectrum efficiency vs average service arrival rate
5 结语

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