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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


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