Optimal Switching Strategy for Heterogeneous Energy Supplying Energy-efficient Two-tier Femtocell Networks

Abstract

Energy harvesting and sleeping strategy are the important methods to deal with the increasing grid power consumption problem. However, power supply solely from energy harvesting modules would be more likely to cause energy outage of base station (BS), and the load of sleeping BS would lead to coverage hole problem, which directly affect user’s quality of service (QoS). In this paper, to achieve the trade-off between grid power consumption and user’s QoS for each tier, optimal joint of ON/OFF mode of a BS and power source switching strategies are designed for two-tier femtocell networks, which are powered by the cooperation of the harvested energy and grid power. Then, exploiting stochastic geometry and random walk theory, BS availability and availability region are characterized with the proposed strategies. Moreover, the relationships among BS availability, battery capacity of grid power and cut-off energy level are exploited to verify the proposed strategies. By numerical simulations, we find that when energy storage capacity achieves a certain value, the effect of grid power battery capacity on availability almost vanishes especially in femto tier. So these redundant grid power can be saved when the stored energy in battery meets current traffic requirement.

Appendices

Appendix 1

From Eq. (3) we can obtain that ${\rho _m} = \frac{1}{{1 + {\varphi _{1m}} + {\varphi _{2m}}}}$, where ${\varphi _{1m}} = \frac{{{N_0}}}{{{\mu '_m}( {E[ {J_{{k_{11}}}^m} ] + E[ {J_{{k_{12}}}^m} ]} )}}$, ${\varphi _{2m}} = \frac{{{N_m} - {N_0}}}{{{\mu _m}( {E[ {J_{{k_{11}}}^m} ] + E[ {J_{{k_{12}}}^m} ]} )}}$. So the variation trend of ${\rho _m}$ depends on ${\varphi _{1m}}$ and ${\varphi _{2m}}$. Without loss of generality, we present the derivations of ${\varphi _{1m}}$ and ${\varphi _{2m}}$ in (16).

$$\begin{aligned} \varphi ' _{1m}= & \frac{{ - {N_0}{{ {E'\left[ {J_{{k_{11}}}^m} \right] } } }}}{{{{\mu '}_m}{{\left( {E\left[ {J_{{k_{11}}}^m} \right] + E\left[ {J_{{k_{12}}}^m} \right] } \right) }^2}}},\nonumber \\ \varphi ' _{2m}= & \frac{{E\left[ {J_{{k_{11}}}^m} \right] + E\left[ {J_{{k_{12}}}^m} \right] - \left( {{N_m} - {N_0}} \right) {{E'\left[ {J_{{k_{11}}}^m} \right] }}}}{{{\mu _m}{{\left( {E\left[ {J_{{k_{11}}}^m} \right] + E\left[ {J_{{k_{12}}}^m} \right] } \right) }^2}}}. \end{aligned}$$

(16)

Then for the situation of $E'[ {J_{{k_{11}}}^m} ]<0$ and $E[ {J_{{k_{11}}}^m} ] + E[ {J_{{k_{12}}}^m} ] > ( {{N_m} - {N_0}} ){{E'[ {J_{{k_{11}}}^m} ] }}$, ${\varphi _{1m}}$ and ${\varphi _{2m}}$ are increasing functions of $N_m$, so ${\rho _m}$ decreases with $N_m$. For the situation of $E'[ {J_{{k_{11}}}^m} ]> 0$ and $E[ {J_{{k_{11}}}^m} ] + E[ {J_{{k_{12}}}^m} ] < ( {{N_m} - {N_0}} ){{E'[ {J_{{k_{11}}}^m} ] }}$, ${\varphi _{1m}}$ and ${\varphi _{2m}}$ are decreasing functions of $N_m$, so ${\rho _m}$ increases with $N_m$. For the situation of $E'[ {J_{{k_{11}}}^m} ]>0$ and $E[ {J_{{k_{11}}}^m} ] + E[ {J_{{k_{12}}}^m} ] > ( {{N_m} - {N_0}} ){{E'[ {J_{{k_{11}}}^m} ] }}$, the monotonicity of ${\rho _m}$ will depend on $\left| \frac{{\varphi ' _{1m}}}{{\varphi ' _{2m}}} \right| = \frac{{{N_0}{\mu _m}E'\left[ {J_{{k_{11}}}^m} \right] }}{{{{\mu '}_m}\left[ {E\left[ {J_{{k_{11}}}^m} \right] + E\left[ {J_{{k_{22}}}^m} \right] - \left( {{N_m} - {N_0}} \right) E'\left[ {J_{{k_{11}}}^m} \right] } \right] }}$. When $\left| \frac{{\varphi ' _{1m}}}{{\varphi ' _{2m}}} \right| >1$, ${\rho _m}$ will be an increasing function of $N_m$. Otherwise, ${\rho _m}$ will decrease with $N_m$. For the situation of $E'[ {J_{{k_{11}}}^m} ]<0$ and $E[ {J_{{k_{11}}}^m} ] + E[ {J_{{k_{12}}}^m} ] < ( {{N_m} - {N_0}} ){{E'[ {J_{{k_{11}}}^m} ] }}$, when $\left| \frac{{\varphi ' _{1m}}}{{\varphi ' _{2m}}} \right| >1$, ${\rho _m}$ will be a decreasing function of $N_m$. Otherwise, ${\rho _m}$ will increase with $N_m$.

Appendix 2

Set $g( {{k_1}} ) = \frac{{{\mu _f}E( {J_{{k_1}}^f} )}}{{{N_f}}}$, $g( {{k_{11}}}) =\frac{{{\mu _f}E( {J_{{k_{11}}}^f} )}}{{{N_f}}}$, $g( {{k_{12}}} ) = \frac{{{\mu _f}E( {J_{{k_{12}}}^f} )}}{{{N_f}}}$, $g( {{k_{13}}} ) = \frac{{{\mu _f}E( {J_{{k_{13}}}^f} )}}{{{N_f}}}$, $g( k ) = g( {{k_1}} ) + \frac{{1 - p}}{p}[ {g( {{k_{11}}} ) + g( {{k_{12}}} ) + g( {{k_{13}}} )} ]$, then (7) can be transformed into ${\rho _f} = \frac{1}{{1 + \frac{1}{{g\left( k \right) }}}}$. Because the value of $\frac{{1 - p}}{p}$ is nonnegative, then to determine the value of $g'\left( {{k}} \right)$, we need to ensure the value of $g'\left( {{k_1}} \right)$, ${g'\left( {{k_{11}}} \right) }$, ${g'\left( {{k_{12}}} \right) }$ and ${g'\left( {{k_{13}}} \right) }$. Obviously, following the Remark 1 introduced in Sect. 3, function $g\left( {{k_1}} \right)$ will increase with the increase in ${N_f}$. Moreover, from (10) we known that $g\left( {{k_{11}}} \right)$ (or $g\left( {{k_{13}}} \right)$) decreases with the increase in ${N_f}$, and $\frac{{{\mu _f}E[ {J_{{k_{12}}}^f} ]}}{{{N_f} - {N'_0}}}$ decreases with the increase in ${N_f}$ can also be concluded from the Remark 1. The relationship between $g\left( {{k_{12}}} \right)$ and ${N_f}$ is analyzed in the following. The derivations of $g\left( {{k_{12}}} \right)$ is presented as

$$\begin{aligned} g'\left( {{k_{12}}} \right)=\,& {\left( {\frac{{{\mu _f}E\left[ {J_{{k_{12}}}^f} \right] }}{{{N_f} - {N'_0}}}\frac{{{N_f} - {N'_0}}}{{{N_f}}}} \right) ^\prime }\nonumber \\=\,& {\left( {\frac{{{\mu _f}E\left[ {J_{{k_{12}}}^f} \right] }}{{{N_f} - {N'_0}}}} \right) ^\prime }\frac{{{N_f} - {N'_0}}}{{{N_f}}} + \frac{{{\mu _f}E\left[ {J_{{k_{12}}}^f} \right] }}{{{N_f} - {N'_0}}}\frac{{{N'_0}}}{{{N_f^2}}}\nonumber \\=\, & {\mu _f} {\frac{{{N _f}E'\left[ {J_{{k_{12}}}^f} \right] -E\left[ {J_{{k_{12}}}^f} \right] }}{{{N_f^2}}}} < \frac{{{\mu _f}E\left[ {J_{{k_{12}}}^f} \right] }}{{{N_f} - {N'_0}}}\frac{{{N'_0}}}{{{N_f}}}. \end{aligned}$$

(17)

For the case of $g'\left( {{k_{12}}} \right) <0$ $({N_f}E'[ {J_{{k_{12}}}^f} ] < E[ {J_{{k_{12}}}^f} ])$, when satisfy $g'\left( {{k_1}} \right) > \frac{{1 - p}}{p}[g'\left( {{k_{11}}} \right) + g'\left( {{k_{12}}} \right) + g'\left( {{k_{13}}} \right) ]$, i.e., $p[E'(J_{{k_1}}^f){N_f} - E(J_{{k_1}}^f)] > (1 - p)[E'(J_{{k_{12}}}^f){N_f} - E(J_{{k_{12}}}^f) - 2E(J_{{k_{11}}}^f)]$, $g\left( {{k}} \right)$ is an increasing function of $N_f$, and we can also obtain that $\rho _f$ is an increasing function of $N_f$. On the contrary, when $g'\left( {{k_1}} \right) < \frac{{1 - p}}{p}[g'\left( {{k_{11}}} \right) + g'\left( {{k_{12}}} \right) + g'\left( {{k_{13}}} \right) ]$, i.e., $p[E'(J_{{k_1}}^f){N_f} - E(J_{{k_1}}^f)] < (1 - p)[E'(J_{{k_{12}}}^f) {N_f}- E(J_{{k_{12}}}^f) - 2E(J_{{k_{11}}}^f)]$, $\rho _f$ is a decreasing function of $N_f$.

For the case of $0< g'\left( {{k_{12}}} \right)$ $\left( {N_f}E'\left[ {J_{{k_{12}}}^f} \right] > E\left[ {J_{{k_{12}}}^f} \right] \right)$, when $g'\left( {{k_1}} \right) +\frac{{1 - p}}{p}g'\left( {{k_{12}}} \right) > \frac{{1 - p}}{p}[g'\left( {{k_{11}}} \right) + g'\left( {{k_{12}}} \right) + g'\left( {{k_{13}}} \right) ]$, i.e., $p[E'(J_{{k_1}}^f){N_f} - E(J_{{k_1}}^f)] + (1 - p)[E'(J_{{k_{12}}}^f){N_f} - E(J_{{k_{12}}}^f)] > - 2(1 - p)E(J_{{k_{11}}}^f)$, $\rho _f$ increases with the increase in $N_f$. On the contrary, when $g'\left( {{k_1}} \right) + \frac{{1 - p}}{p} g'\left( {{k_{12}}} \right) < \frac{{1 - p}}{p}[g'\left( {{k_{11}}} \right) + g'\left( {{k_{13}}} \right) ]$, i.e., $p[E'(J_{{k_1}}^f){N_f} - E(J_{{k_1}}^f)] + (1 - p)[E'(J_{{k_{12}}}^f){N_f} - E(J_{{k_{12}}}^f)] < - 2(1 - p)E(J_{{k_{11}}}^f)$, $\rho _f$ is a decreasing function of $N_f$.