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SCIENCE CHINA Information Sciences, Volume 62, Issue 4: 042303(2019) https://doi.org/10.1007/s11432-018-9494-9

BS sleeping strategy for energy-delay tradeoff in wireless-backhauling UDN

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  • ReceivedJan 25, 2018
  • AcceptedMay 29, 2018
  • PublishedFeb 22, 2019

Abstract

Ultra-dense network (UDN) has been recognized as a promising technology for 5G. Although turning off low-load base stations (BSs) can improve energy efficiency, it may cause degradation of delay performance. This makes energy-delay tradeoff (EDT) an important topic. In this paper, a theoretical framework for EDT, in wireless-backhauling UDN, is developed. First, we investigate association probabilities of UEs and transmission probabilities of BSs. Expressions for energy consumption and network packet delay are obtained and the impact that BS sleeping ratio has on energy consumption and packet delay are analyzed. Then, we formulate the EDT problem as a cost minimization problem to select the optimal set of sleeping small cells. To solve the EDT optimization problem, a locally optimal sleeping ratio for EDT is obtained using the dynamic gradient iteration algorithm and we prove that it can converge to the global optimal sleeping ratio. Then, queue-aware and channel-queue-aware sleeping strategies are proposed to find the optimal set of sleeping small cells according to the optimal sleeping ratio. We then see that the simulation and numerical results confirm the effectiveness of the proposed sleeping schemes.


Acknowledgment

This work was partially supported by National Major Project (Grant No. 2017ZX03001002-004), National Natural Science Foundation Project (Grant No. 61521061), 333 Program of Jiangsu (Grant No. BRA2017366), and Huawei Technologies Co., Ltd.


Supplement

Appendix

Analysis of the feasible region for $\theta$

According to (10) and (15), we have \begin{align}\theta &= 1 - {\lambda _s}^{ - 1}{( {{A_b}{P_{\rm st}}} )^{ - \frac{2}{\alpha }}}\left( { {\frac{\lambda{\lambda _u}P_{\rm mt}^{\frac{2}{\alpha }}}{{1 + {{\bar \xi }_m}( \theta )Z( \beta )}}\int_0^{\frac{l}{{{W_m}\log ( {1 + \beta } )}}} {\frac{{{y_m} - Z( \beta )}}{{1 + {{\bar \xi }_m}( \theta ){y_m}}}} {\rm d}t} - {\lambda _m}P_{\rm mt}^{\frac{2}{\alpha }}} \right) \\ &= 1 + \frac{{{\lambda _m}}}{{{\lambda _s}}}{\left( {\frac{{{P_{\rm mt}}}}{{{A_b}{P_{\rm mt}}}}} \right)^{\frac{2}{\alpha }}} - \frac{{\lambda {\lambda _u}\int_0^{\frac{l}{{{W_s}\log ( {1 + \beta } )}}} {\frac{{{y_s} - Z( \beta )}}{{1 + {{\bar \xi }_s}( \theta ){y_s}}}} {\rm d} t}}{{{\lambda _s}( {1 + {{\bar \xi }_s}( \theta )Z( \beta )} )}}. \tag{35} \end{align} We can observe that $\theta$ increases with the increase of $\bar~\xi_{m}(\theta)$ and $\bar~\xi_{s}(\theta)$, respectively, thus, $\bar~\xi_{m}(\theta)$ and $\bar~\xi_{s}(\theta)$ both are increasing function of $\theta$. Substituting $\bar~\xi_{m}(\theta)=0,~1$ into (A1), and $\bar~\xi_{s}(\theta)=0,1$ into (A1), (29) and (30) are derived, and where \begin{align}&{X_1} = \int_0^{\frac{l}{{{W_s}\log ( {1 + \beta } )}}} {( {{y_s} - Z( \beta )} )} {\rm d} t, \tag{36} \\ &{X_2} = \lambda {\lambda _u}{P_{\rm mt}}^{\frac{2}{\alpha }}\left( {\int_0^{\frac{l}{{{W_m}\log ( {1 + \beta } )}}} {( {{y_m} - Z( \beta )} )} {\rm d} t} \right), \tag{37} \\ &{Y_1} = {( {1 + Z( \beta )} )^{ - 1}}\int_0^{\frac{l}{{{W_s}\log ( {1 + \beta } )}}} {\frac{{{y_s} - Z( \beta )}}{{1 + {y_s}}}} {\rm d} t, \tag{38} \\ &{Y_2} = \lambda {\lambda _u}{P_{\rm mt}}^{\frac{2}{\alpha }}\left( {\frac{1}{{1 + Z( \beta )}}\int_0^{\frac{l}{{{W_m}\log ( {1 + \beta } )}}} {\frac{{{y_m} - Z( \beta )}}{{1 + {y_m}}}} {\rm d} t} \right). \tag{39} \end{align}

Analysis of cost function

\begin{align}&{Z_1} = \frac{1}{2}( {1 - {{\Pr }_{\rm SUE}}( \theta )} ){\bar D_{\rm Tm}}\frac{1}{{{{( {1 - {\xi _{{m}}}( \theta )} )}^2}}}\frac{{\partial {\xi _{{m}}}( \theta )}}{{\partial \theta }} - \frac{{\partial {{\Pr }_{\rm SUE}}( \theta )}}{{\partial \theta }}\left( {1{\rm{ + }}\frac{1}{{2( {1 - {\xi _m}( \theta )} )}}} \right){\bar D_{\rm Tm}}, \tag{40} \\ &{Z_2} = \frac{1}{2}{{\Pr}_{\rm SUE}}( \theta ){\bar D_{\rm Tsr}}\frac{1}{{{{( {1 - {\xi _{{s}}}( \theta )} )}^2}}}\frac{{\partial {\xi _{{s}}}( \theta )}}{{\partial \theta }} - \frac{{{\lambda _s}}}{{{\lambda _g}}}{{\Pr}_{\rm SUE}}( \theta ){\bar D_{\rm Tsb}}\left( {1{\rm{ + }}\frac{1}{{2( {1 - {\mu _g}( \theta )} )}}} \right), \tag{41} \\ &{Z_3} = \frac{{{\lambda _s}}}{{{\lambda _g}}}{\bar D_{\rm Tsb}}( {1 - \theta } )\frac{{\partial {{\Pr }_{\rm SUE}}( \theta )}}{{\partial \theta }}\left( {1{\rm{ + }}\frac{1}{{2( {1 - {\mu _g}( \theta )} )}}} \right) + \frac{1}{2}\frac{{{\lambda _s}}}{{{\lambda _g}}}{{\Pr}_{\rm SUE}}( \theta ){\bar D_{\rm Tsb}}( {1 - \theta } )\frac{1}{{{{( {1 - {\mu _g}( \theta )} )}^2}}}\frac{{\partial {\mu _g}( \theta )}}{{\partial \theta }}, \tag{42} \\ &{Z_4} = \frac{{\partial {{\Pr }_{\rm SUE}}( \theta )}}{{\partial \theta }}\left( {1{\rm{ + }}\frac{1}{{2( {1 - {\xi _{\rm{s}}}( \theta )} )}}} \right){\bar D_{\rm Tsr}}, \tag{43} \\ &\frac{{\partial {{\Pr }_{\rm SUE}}( \theta )}}{{\partial \theta }} =- \frac{{{\lambda _m}{\lambda _s}{{( {{A_b}{P_{\rm st}}{P_{\rm mt}}} )}^{\frac{2}{\alpha }}}}}{{{{\big( {( {1 - \theta } ){\lambda _s}{{( {{A_b}{P_{\rm st}}} )}^{\frac{2}{\alpha }}} + {\lambda _m}{P_{\rm mt}}^{\frac{2}{\alpha }}} \big)}^{2}}}}, \tag{44} \\ &\frac{{\partial {\xi _m}( \theta )}}{{\partial \theta }} = \frac{{{\lambda _s}}}{{\lambda {\lambda _u}}}{\left( {\frac{{{A_b}{P_{\rm st}}}}{{{P_{\rm mt}}}}} \right)^{\frac{2}{\alpha }}}{( {1 + {\xi _m}( \theta )Z( \beta )} )^{2}}\left( {Z( \beta )\int_0^{\frac{l}{{{W_m}\log ( {1 + \beta } )}}} {{q_m}( \theta ){\rm d}t} }\right. \\ & \left.+ ( {1 + {\xi _m}( \theta )Z( \beta )} )\int_0^{\frac{l}{{{W_m}\log ( {1 + \beta } )}}} {\frac{{{y_m}{q_m}( \theta )}}{{1 + {\xi _m}( \theta ){y_m}}}{\rm d}t} \right)^{-1}, \tag{45} \\ &\frac{{\partial {\xi _s}( \theta )}}{{\partial \theta }} = \frac{{{\lambda _s}}}{{\lambda {\lambda _u}}}{( {1 + {\xi _s}( \theta )Z( \beta )} )^{2}}\left( {Z( \beta )\int_0^{\frac{l}{{{W_s}\log ( {1 + \beta } )}}} {{q_s}( \theta ){\rm d}t} } { + ( {1 + {\xi _s}( \theta )Z( \beta )} )\int_0^{\frac{l}{{{W_s}\log ( {1 + \beta } )}}} {\frac{{{y_s}{q_s}( \theta )}}{{1 + {\xi _s}( \theta ){y_s}}}{\rm d}t} } \right)^{ - 1}, \tag{46} \\ &{q_s}( \theta ) = \frac{{{y_s} - Z( \beta )}}{{1 + {\xi _s}( \theta ){y_s}}}, {q_m}( \theta ) = \frac{{{y_m} - Z( \beta )}}{{1 + {\xi _m}( \theta ){y_m}}}. \tag{47} \end{align}

We can obtain ${{\partial~{{D}}(~\theta~)}~/~{\partial~\theta~}}~=~{Z_1}~+~{Z_2}~+~{Z_3}~+{Z_4}$, it can be observed that $Z_{1}$ and $Z_{2}$ increase with the increase of sleeping ratio $\theta$. And ${{\partial~{Z_3}}~/~{\partial~\theta~}}~>~0$, ${{\partial~{Z_4}}~/~{\partial~\theta~}}~>~0$, thus ${{\partial~\bar~D(~\theta~)}~/~{\partial~\theta~}}$ is an increasing function of $\theta$. On the other hand, it can be proven that system energy consumption is a decreasing function of sleeping ratio [17]. Therefore, the cost function is approximately convex for sleeping ratio in the feasible region [30].


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  • Figure 1

    (Color online) Network model for two-tier wireless-backhauling UDN.

  • Figure 2

    (Color online) Simulation and numerical results for mean packet delay vs. sleeping ratio $\theta$.

  • Figure 3

    (Color online) Numerical results for mean packet delay vs. gateway density $\lambda_{g}$.

  • Figure 4

    (Color online) Numerical results for system energy consumption vs. sleeping ratio $\theta$.

  • Figure 5

    (Color online) Energy consumption vs. mean network packet delay for different $\theta$.

  • Figure 6

    (Color online) Numerical results for cost function of EDT problem vs. BS sleeping ratio $\theta$.

  • Figure 7

    (Color online) Optimal sleeping ratio vs. small cell density for different weighting factor.

  • Figure 8

    (Color online) Optimal state set of small cells for different small cell density. (a) $\lambda_s=1.5~\times~10^{-5}$; (b) $\lambda_s=$ $2.0~\times~10^{-5}$; (c) $\lambda_s=2.5~\times~10^{-5}$; (d) $\lambda_s=3.0~\times~10^{-5}$.

  • Figure 9

    (Color online) System energy consumption vs. $\lambda_{s}$ with $\theta^{*}$ for different sleeping schemes.

  • Figure 10

    (Color online) Mean delay vs. small cell density with the optimal sleeping ratio for different sleeping schemes.

  •   

    Algorithm 1 Queue-aware sleeping strategy

    Input: SBS set ${\mathcal{B}_S}$, MBS set $\mathcal{B}_M$, UE set ${{\cal~U}}$, $\theta^{*}$, $~T$ $\Delta~t~$, $\forall~i~\in~{\cal~U}$, $\forall~j~\in~\{\mathcal{B}_S,~\mathcal{B}_M\}$, $\forall~k~\in~\{\mathcal{B}_S\}$. Output: Optimal state set of SBS $\mathcal{S}^{*}$.

    Initialize: all MBSs and SBSs are active, $\mathcal{S}=(1,1,1,\ldots,1)$ and $n=1$;

    Calculate $N_{\rm~off}$ according to (33);

    Select the serving BS ${~j^{*}~=~\mathop~{\arg~\max~\{{\rm~RSRP}_j\}~}\nolimits_{j~\in~\{~\mathcal{B}_S,\mathcal{B}_M\}~}~}$ for each UE $i$;

    Find the set of UEs ${{\cal~U}}_{j}$ that can be served by each BS $j$;

    for each $t\in~[1,~T]$ do

    Calculate transmission rate $R_{k}(t)$ update queue length according to (34), for small cell BS $k$;

    end for

    Calculate $\bar~Q_k=\frac{{\sum\nolimits_{t~=~1}^T~{{Q_k}(t)}~}}{T}$ for each small cell BS $k$;

    while $n\le~N_{\rm~off}$ then

    for each small cell BS $k$ do

    if $\mathcal{S}(1,k)=1$ and $k~=~\mathop~{\min~}\nolimits_{k~\in~\mathcal{B}_S}~\{~{{\bar~Q}_k}\}~$;

    $\mathcal{S}(1,k)=0$, assign UEs in ${{\cal~U}}_{k}$ to neighboring BSs;

    end if

    end for

    $n=n+1$;

    end while

  • Table 1   System parameters
    Parameter Value Parameter Value Parameter Value Parameter Value
    ${\lambda~_g}$ $5\times~10^{-6}$ ${P_{s0}}$ $4.8$ W ${W_b}$ $20$ MHz ${\lambda}$ 0.5 s$^{-1}$
    ${\lambda~_m}$ $1\times~10^{-5}$ ${P_{m0}}$ $10$ W $W_m$ $10$ MHz $\Delta~p_{m}$ $10$
    ${\lambda~_s}$ $5\times~10^{-5}$ ${P_S}$ 2.4 W $W_s$ $10$ MHz$\Delta~p_{s}$ $8$
    ${\lambda~_u}$ $2\times~10^{-4}$${P_G}$ 100 W $l$$0.1$ MB $\beta$ $5$
  •   

    Algorithm 2 Channel-queue-aware sleeping strategy

    Input: SBS set ${\mathcal{B}_S}$, MBS set $\mathcal{B}_M$, UE set ${{\cal~U}}$, $\theta^{*}$, $~T$ $\Delta~t~$, $\forall~i~\in~{\cal~U}$, $\forall~j~\in~\{\mathcal{B}_S,~\mathcal{B}_M\}$, $\forall~k~\in~\{\mathcal{B}_S\}$. Output: Optimal state set of SBS $\mathcal{S}^{*}$.

    Initialize: all MBSs and SBSs are active, $\mathcal{S}=(1,1,1,\ldots,1)$ and $n=1$;

    Calculate $N_{\rm~off}$ according to (33);

    Select the serving BS ${~j~^{*}=~\mathop~{\arg~\max~\{{\rm~RSRP}_j\}~}\nolimits_{j~\in~\{~\mathcal{B}_S,\mathcal{B}_M\}~}~}$ for each UE $i$;

    Find the set of UEs ${{\cal~U}}_{j}$ that can be served by each BS $j$;

    for each $t\in~[1,~T]$ do

    Calculate transmission rate $R_{k}(t)$ update queue length according to (34), for small cell BS $k$;

    end for

    Calculate $\bar~Q_k=\frac{{\sum\nolimits_{t~=~1}^T~{{Q_k}(t)}~}}{T}$ and $\bar~R_k=\frac{{\sum\nolimits_{t~=~1}^T~{{R_k}(t)}~}}{T}$ for each small cell BS $k$;

    while $n\le~N_{\rm~off}$ then

    for each small cell BS $k$ do

    if $\mathcal{S}(1,k)=1$ and $k~=~\mathop~{\min~}\nolimits_{k~\in~\mathcal{B}_S}~\{~{{\bar~Q}_k}\bar~R_k\}~$;

    $\mathcal{S}(1,k)=0$, assign UEs in ${{\cal~U}}_{k}$ to neighboring BSs;

    end if

    end for

    $n=n+1$;

    end while

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