Base Station Sleep and Spectrum Allocation in Heterogeneous Ultra-dense Networks

Abstract

To meet the exponential increasing high data rate demand of mobile users, heterogeneous ultra-dense networks (UDN) is widely seen as an essential technology to provide high-rate transmissions to nearby mobile users. However, the dense and random deployment of small base stations (SBSs) overlaid by macro base stations and their uncoordinated operation lead to important questions about the power consumption and aggressive frequency reuse of heterogeneous UDN. For the problem of huge power consumption and spectrum resource tension in heterogeneous UDN, a joint strategy of SBSs sleep and spectrum allocation is proposed. By using stochastic geometry, the coverage probabilities of base stations and the average ergodic rates of mobile users are derived in each tier and the whole network. In addition, we formulate the coverage probability maximization and power consumption minimization problems, and determine the optimal operating regimes for SBSs, and as well as spectrum allocation. The numerical results show that the SBSs sleep and spectrum allocation can reduce the power consumption and interference of the whole network.

Appendices

Appendix A

The coverage probability of the SBSs is given by

$$\begin{aligned} {{p}_{s}}=\eta \cdot p_{s}^{{\text {shared}}}+\left( 1-\eta \right) \cdot p_{s}^{{\text {unshared}}} \end{aligned}$$

(23)

where $p_{s}^{{\text {shared}}}$ and $p_{s}^{{\text {unshared}}}$ are the coverage probabilities of SBSs in shared spectrum resource and unshared spectrum resource.

The coverage probability of SBSs in shared spectrum resource is given by

$$\begin{aligned} p_s^{{\mathrm{shared}} }\left( T \right)=& {} {{\mathbb {E}} _r}\left[ {{\mathbb {P}} \left[ {\left. {{{\hbox {SINR}}_s}> T} \right| r} \right] } \right] \nonumber \\=& {} \int _{r> 0} {{\mathbb {P}} \left[ {{\hbox {SINR}_s}> T} \right] } {f_{{r_s}}}\left( r \right) dr\nonumber \\=& {} \int _{r> 0} {{\mathbb {P}} \left[ {\frac{{{P_s}h{r^{ - \alpha }}}}{{{\sigma ^2} + {I_{s,s}} + {I_{s,m}}}}> T} \right] } {f_{{r_s}}}\left( r \right) dr\nonumber \\=& {} 2\pi {p_a}{\lambda _s}\int _{r> 0} {{\mathbb {P}} \left[ {h > \frac{{T{r^\alpha }}}{{{P_s}}}\left( {{\sigma ^2} + {I_{s,s}} + {I_{s,m}}} \right) } \right] \cdot {e^{ - \pi {r^2}{\lambda _s}}}} rdr \end{aligned}$$

(24)

Using the fact that ${{h}_{s}}\sim \exp \left( 1 \right)$, the coverage probability can be expressed as

$$\begin{aligned} {\mathbb {P}} \left[ {h> \frac{{T{r^\alpha }}}{{{P_s}}}\left( {{\sigma ^2} + {I_{s,s}} + {I_{s,m}}} \right) } \right]& {}= {{\mathbb {E}}_{{I_{s,s}},{I_{s,m}}}}\left[ {{\mathbb {P}} \left[ {\left. {h > \frac{{T{r^\alpha }}}{{{P_s}}}\left( {{\sigma ^2} + {I_{s,s}} + {I_{s,m}}} \right) } \right| {I_{s,s}},{I_{s,m}}} \right] } \right] \nonumber \\& {}= {{\mathbb {E}} _{{I_{s,s}},{I_{s,m}}}}\left[ {\exp \left( { - \frac{c}{{{P_s}}}T{r^\alpha }\left( {{\sigma ^2} + {I_{s,s}} + {I_{s,m}}} \right) } \right) } \right] \nonumber \\& {}= {e^{ - \frac{1}{{{P_s}}}T{r^\alpha }{\sigma ^2}}}{\mathcal{L}_{{I_{s,s}}}}\left( {\frac{1}{{{P_s}}}T{r^\alpha }} \right) {\mathcal{L}_{{I_{s,m}}}}\left( {\frac{1}{{{P_s}}}T{r^\alpha }} \right) \end{aligned}$$

(25)

where ${{\mathcal {L}}_{{{I}_{m,m}}}}\left( s \right)$ and ${{\mathcal {L}}_{{{I}_{s,m}}}}\left( s \right)$ are the Laplace transform of random variable ${{I}_{s,s}}$ and ${{I}_{s,m}}$, respectively. Using the definition of the Laplace transform yields

$$\begin{aligned} {\mathcal{L}_{{I_{s,s}}}}\left( s \right) & {}= {{\mathbb {E}} _{{\varPhi _s},\left\{ {{g_{s,i}}} \right\} }}\left[ {\prod \limits _{i \in {\varPhi _s}\backslash \left\{ {{b_{so}}} \right\} } {{{\mathbb {E}}_{{g_{s,i}}}}\left[ {\exp \left( { - s{P_s}{g_{{{s,i}} }}R_{{{s,i}} }^{ - \alpha }} \right) } \right] } } \right] \nonumber \\& {}= {{\mathbb {E}} _{{\varPhi _s}}}\left[ {\prod \limits _{i \in {\varPhi _s}\backslash \left\{ {{b_{so}}} \right\} } {\frac{1}{{1 + s{P_s}R_{{{s,i}} }^{ - \alpha }}}} } \right] \nonumber \\ & {}=\exp \left( { - 2\pi {p_a}{\lambda _s}\int _r^\infty {\left( {1 - \frac{1}{{1 + s{P_s}{v^{ - \alpha }}}}} \right) vdv} } \right) \end{aligned}$$

(26)

Because ${{g}_{s,i}}\sim \exp \left( 1 \right)$, the Laplace transform of random variable ${{I}_{s,s}}$ is obtained. The integration limits are from r to $\infty$ since the closest interferer is at least at a distance r. Furthermore, plugging into (26) gives Laplace transform of random variable ${{I}_{s,m}}$ in (27).

$$\begin{aligned} {{\mathcal {L}}_{{{I}_{s,s}}}}\left( T{{r}^{\alpha }}/{{P}_{s}} \right) =\exp \left( -\pi {{p}_{a}}{{\lambda }_{s}}{{r}^{2}}\rho \left( T,\alpha \right) \right) \end{aligned}$$

(27)

where $\rho \left( T,\alpha \right) ={{T}^{2/\alpha }}\int _{{{T}^{-2/\alpha }}}^{\infty }{\frac{1}{1+{{u}^{\alpha /2}}}}du$.

Similarly, using the same approach as in (26), the Laplace transform yields

$$\begin{aligned} {\mathcal{L}_{{I_{s,m}}}}\left( s \right)& {}= {{\mathbb {E}}_{{\varPhi _m},\left\{ {{g_{m,i}}} \right\} }}\left[ {\prod \limits _{i \in {\varPhi _m}} {{{\mathbb {E}}_{{g_{m,i}}}}\left[ {\exp \left( { - s{P_m}{g_{m,i}}R_{m,i}^{ - \alpha }} \right) } \right] } } \right] \nonumber \\& {}= {{\mathbb {E}} _{{\varPhi _m}}}\left[ {\prod \limits _{i \in {\varPhi _m}} {\frac{c}{{c + s{P_m}R_{m,i}^{ - \alpha }}}} } \right] \nonumber \\ & {}= \exp \left( { - 2\pi {\lambda _m}\int _0^\infty {\left( {1 - \frac{c}{{c + s{P_m}{v^{ - \alpha }}}}} \right) vdv} } \right) \end{aligned}$$

(28)

Because ${{g}_{m,i}}\sim \exp \left( c \right)$, the Laplace transform of random variable ${{I}_{s,m}}$ is obtained. Furthermore, plugging $s=\left( T{{r}^{-\alpha }} \right) /{{P}_{s}}$ into (28) gives Laplace transform of random variable ${{I}_{s,m}}$ in (29).

$$\begin{aligned} {\mathcal{L}_{{I_{s,m}}}}\left( {T{r^\alpha }/{P_s}} \right)& {}= \exp \left( { - 2\pi {\lambda _m}\int _0^\infty {\frac{{\frac{{{P_m}}}{{c{P_s}}}T}}{{\frac{{{P_m}}}{{c{P_s}}}T + {{\left( {v/r} \right) }^\alpha }}}vdv} } \right) \nonumber \\& {}= \exp \left( { - \pi {\lambda _m}{r^2}{{\left( {T{P_m}/\left( {c{P_s}} \right) } \right) }^{2/\alpha }}\int _0^\infty {\frac{1}{{1 + {u^{2/\alpha }}}}du} } \right) \end{aligned}$$

(29)

Because $\int _{0}^{\infty }{\frac{1}{1{+}{{u}^{\alpha /2}}}}du=1/\left( \alpha /2 \right) {\Gamma } \left( 2/\alpha \right) {\Gamma } \left( 1-2/\alpha \right) =\frac{2\pi }{\alpha \sin \left( 2\pi /\alpha \right) }$, the Laplace transform of random variable ${{I}_{s,m}}$ is further simplified to

$$\begin{aligned} {{\mathcal {L}}_{{{I}_{s,m}}}}\left( T{{r}^{\alpha }}/{{P}_{s}} \right) =\exp \left( -\pi {{\lambda }_{m}}{{r}^{2}}{{\left( T{{P}_{m}}/\left( c{{P}_{s}} \right) \right) }^{2/\alpha }}C\left( \alpha \right) \right) \end{aligned}$$

(30)

where $C\left( \alpha \right) =\frac{2\pi }{\alpha \sin \left( 2\pi /\alpha \right) }$.

For small cell mobile users in the non-shared spectrum resource, the set of interferers is all the other SBSs. This implies ${{I}_{s,m}}=0$. Therefore, the coverage probability of SBSs is given by

$$\begin{aligned} p_s^{{\mathrm{unshared}} }\left( T \right)& {}= {{\mathbb {E}}_r}\left[ {{\mathbb {P}} \left[ {\left. {\mathrm{{SIN}}{\mathrm{{R}}_s}> T} \right| r} \right] } \right] \nonumber \\& {}= 2\pi {p_a}{\lambda _s}\int _{r> 0} {{\mathbb {P}} \left[ {h> \frac{{T{r^\alpha }}}{{{P_s}}}\left( {{\sigma ^2} + {I_{s,s}}} \right) } \right] \cdot {e^{ - \pi {r^2}{p_a}{\lambda _s}}}} rdr\nonumber \\& {}= 2\pi {p_a}{\lambda _s}\int _{r > 0} {{e^{ - \pi {r^2}{p_a}{\lambda _s}\left( {1 + \rho \left( {T,\alpha } \right) } \right) }}{e^{ - T{r^\alpha }{\sigma ^2}/{P_s}}}} rdr \end{aligned}$$

(31)

The proof of the coverage probability of MBSs is similar to SBSs.

Appendix B

The average ergodic rate of the macrocell mobile user is given by

$$\begin{aligned} {R_m}& {}= {\mathbb {E}} \left[ {\ln \left( {1 + \mathrm{{SI}}{\mathrm{{R}}_m}} \right) } \right] \nonumber \\& {}= \int _{r> 0}{{\mathbb {E}} \left( {\ln \left( {1 + \frac{{{P_m}h{r^{ - \alpha }}}}{{{I_{m,m}} + {I_{m,s}}}}} \right) } \right) } {f_{{r_m}}}\left( r \right) dr\nonumber \\& {}= 2\pi {\lambda _m}\int _{r> 0} {\int _{t> 0} {{\mathbb {P}} \left[ {\ln \left( {1 + \frac{{{P_m}h{r^{ - \alpha }}}}{{{I_{m,m}} + {I_{m,s}}}}} \right)> t} \right] } \cdot {e^{ - \pi {r^2}{\lambda _m}}}dt} rdr\nonumber \\& {}= 2\pi {\lambda _m}\int _{r> 0} {\int _{t> 0} {{\mathbb {P}} \left[ {h> \frac{{\left( {{e^t} - 1} \right) {r^\alpha }}}{{{P_m}}}\left( {{I_{m,m}} + {I_{m,s}}} \right) } \right] \cdot {e^{ - \pi {r^2}{\lambda _m}}}dt} } rdr\nonumber \\& {}= 2\pi {\lambda _m}\int _{r> 0} {\int _{t > 0} {{\mathcal{L}_{{I_{m,m}}}}\left( { - c\frac{{{r^\alpha }}}{{{P_m}}}\left( {{e^t} - 1} \right) } \right) {\mathcal{L}_{{I_{m,s}}}}\left( { - c\frac{{{r^\alpha }}}{{{P_m}}}\left( {{e^t} - 1} \right) } \right) \cdot {e^{ - \pi {r^2}{\lambda _m}}}} } dtrdr \end{aligned}$$

(32)

Similar to ${{\mathcal {L}}_{{{I}_{s,s}}}}$ and ${{\mathcal {L}}_{{{I}_{s,m}}}}$, the Laplace transform of random variable ${{I}_{m,m}}$ and ${{\mathcal {L}}_{{{I}_{m,s}}}}$ are given by

$$\begin{aligned} {{\mathcal {L}}_{{{I}_{m,m}}}}\left( c\left( {{e}^{t}}-1 \right) {{r}^{\alpha }}/{{P}_{m}} \right)= & {} \exp \left( -\pi {{\lambda }_{m}}{{r}^{2}}\rho \left( {{e}^{t}}-1,\alpha \right) \right) \end{aligned}$$

(33)

$$\begin{aligned} {{\mathcal {L}}_{{{I}_{m,s}}}}\left( c\left( {{e}^{t}}-1 \right) {{r}^{\alpha }}/{{P}_{m}} \right)= & {} \exp \left( -\pi {{\lambda }_{s}}{{r}^{2}}{{\left( c\left( {{e}^{t}}-1 \right) \eta {{P}_{s}}/{{P}_{m}} \right) }^{1/\alpha }}C\left( \alpha \right) \right) \end{aligned}$$

(34)

The proof of the average ergodic rate of the small cell mobile user is similar to macrocell mobile user.