Performance analysis of two tier HetNets with massive MIMO enabled wireless backhauling

Abstract

Dense deployment of small cells escalates the network capacity, making it a candidate technology for the future wireless network. However, backhauling of such a large number of small cells becomes a major challenge in its successful deployment. In this paper, a framework for massive MIMO enabled wireless backhauling in a two-tier heterogeneous network is developed. The framework considers massive MIMO enabled macro base station and in-band full-duplex (IBFD) small cell base station. Massive MIMO system provides high spectral efficiency and high energy efficiency. Besides, due to IBFD capability, the small cells can utilize the same frequency band to communicate over the access link and the backhaul link simultaneously. Tools from the stochastic geometry are utilized to model and derive the analytical expressions of coverage probability and area spectral efficiency (ASE). Furthermore, a trade-off between ASE and coverage probability is observed. All the analytical results are verified by simulation. Simulation results show that massive MIMO and IBFD communication may be useful technologies from the backhauling perspective in the dense small cell network. It is observed that the proposed framework provides 42.30% improvement in the coverage performance over the other existing framework for rate threshold of 1 bps/Hz. Furthermore, optimal performance for the proposed model can be achieved by tuning of bandwidth allocation factor.

Appendices

Appendix 1: Proof of Lemma 1

Let q denotes the index of the serving tier. Also, the index of the MBS and SBS tier are represented as m and s, respectively. Accordingly, the probability that user is served by MBS is

$${\varPi _m} = P\left[ {q = m} \right],$$

(40)

$$= {E_{{T_m}}}\left[ {P\left[ {{P_m}^\prime \left( {{T_m}} \right) > {P_s}^\prime } \right] } \right],$$

(41)

$$\mathop{=}\limits ^{\left( a \right) } {E_{{T_m}}}\left[ {P\left[ {{T_s} > {{\left( {\frac{{L{P_s}}}{{\left( {N - L - 1} \right) {P_m}}}} \right) }^{1/\delta }}{T_m}} \right] } \right],$$

(42)

$$\mathop{=}\limits ^{\left( b \right) } \int \limits _0^\infty {P\left[ {{T_s} > {{\left( {\frac{{L{P_s}}}{{\left( {N - L - 1} \right) {P_m}}}} \right) }^{1/\delta }}{T_m}} \right] } {f_{{T_m}}}\left( t \right) dt,$$

(43)

where (a) is obtained by substituting (2) and (3) in (40) and (b) can be obtained by averaging $P(\bullet )$ in (41) over the distribution of $T_{m}$. Moreover, $P\left[ {{T_s} > {{\left( {\frac{{L{P_s}}}{{\left( {N - L - 1} \right) {P_m}}}} \right) }^{1/\delta }}{T_m}} \right]$ and ${f_{{T_m}}}\left( t \right)$ can be derived by the application of the null probability of a 2-D Poisson process with density $\lambda$ in an area A is $exp(\lambda A)$. Following this

$$P\left[ {{T_s} > {{\left( {\frac{{L{P_s}}}{{\left( {N - L - 1} \right) {P_m}}}} \right) }^{1/\delta }}{T_m}} \right],$$

(44)

$$\begin{aligned}&= P[no\;BS\;closer\;than\;{\left( {\frac{{L{P_s}}}{{\left( {N - L - 1} \right) {P_m}}}} \right) ^{1/\delta }}\nonumber \\&\quad \quad {T_m}\;in\;MBS\;tier] \end{aligned}$$

(45)

$$= \exp \left( { - \pi {\lambda _s}{{\left( {\frac{{L{P_s}}}{{\left( {N - L - 1} \right) {P_m}}}} \right) }^{1/\delta }}{t^2}} \right),$$

(46)

and

$${f_{{T_m}}}\left( t \right) = 1 - \frac{d}{{dt}}P\left[ {{T_m} > t} \right],$$

(47)

$$= 2\pi {\lambda _m}t\,{e^{ - \pi {\lambda _m}{t^2}}}.$$

(48)

Appendix 2: Proof of Theorem 1

Proceeding with (21), downlink rate coverage probability of MBS to user link can be derived as

$${C_m} = {E_{{X_{m,u}}}}P\left[ {\left. {SIN{R_{a,m}} > {\tau _{a,m}}} \right| {X_{m,u}} = {x_{m,u}}} \right],$$

(49)

$${C_m} = \int \limits _{{x_{m,u}}> 0} {P\left[ {\left. {SIN{R_{a,m}} > {\tau _{a,m}}} \right| {X_{m,u}} = {x_{m,u}}} \right] }\times {f_{{X_{m,u}}}}({x_{m,u}})\,d{x_{m,u}}.$$

(50)

After substituting the values of $SINR_{a,m}$ and ${f_{{X_{m,u}}}}({x_{m,u}})$ from (11) and (6), (50) can be modified as

$$\begin{aligned} {C_m} &= \frac{{2\pi {\lambda _m}}}{{{\varPi _m}}}\int \limits _{{x_{m,u}} > 0} {{F_{{I_1}}}\left[ {\frac{{{P_m}}}{{{\tau _{a,m}}\,x_{m,u}^\delta }} - {N_0}} \right] } \;{x_{m,u}}\nonumber \\&\quad \times {e^{\left[ { - \pi {\lambda _m}x_{m,u}^2 - \pi {\lambda _s}(\frac{{L{P_s}}}{{(N - L - 1){P_m}}}x_{m,u}^2)} \right] }}\,d{x_{m,u}}, \end{aligned}$$

(51)

where ${{F_{{I_1}}[\bullet ]}}$ is the CDF of the aggregate interference, and Gil-Pelaez theorem is applied to evaluate the aforementioned CDF.

$$\begin{aligned}&{F_{{I_1}}}\left[ {\frac{{{P_m}}}{{{\tau _{a,m}}\,x_{m,u}^\delta }} - {N_0}} \right] = \frac{1}{2} - \frac{1}{\pi }\nonumber \\&\quad \times \int \limits _0^\infty {{\mathop{Im}\nolimits } \left( {{L_{{I_1}}}( - j\omega )\,{e^{ - j\omega \left( {\frac{{{P_m}}}{{{\tau _{a,m}}x_{m,u}^\delta }} - {N_0}} \right) }}} \right) } \frac{{d\omega }}{\omega }, \end{aligned}$$

(52)

$$\begin{aligned}&= \frac{1}{2} - \frac{1}{\pi }\int \limits _0^\infty {\mathop{Im}\nolimits }\biggl ( {L_{{I_{m,{u_m}}}}}( - j\omega ){L_{{I_{s,{u_m}}}}}( - j\omega )\nonumber \\&\quad \times e^{ - j\omega \left( {\frac{{{P_m}}}{{{\tau _{a,m}}x_{m,u}^\delta }} - {N_0}} \right) } \biggr )\frac{{d\omega }}{\omega }, \end{aligned}$$

(53)

herein ${L_{{I_{m,{u_m}}}}}(s)$ and ${L_{{I_{s,{u_m}}}}}(s)$ are the Laplace transform of the interferences ${{I_{m,{u_m}}}}$ and ${{I_{s,{u_m}}}}$, respectively and can be derived as follows

$${L_{{I_{m,{u_m}}}}}(s) = {E_{{I_{m,{u_m}}}}}\left[ {{e^{ - s{I_{m,{u_m}}}}}} \right],$$

(54)

$$= E\left[ {s\sum \limits _{i \in {\varPhi _x}/m} {{P_m} {{\left| {{X_{i,u}}} \right| }^{ - \delta }}} } \right],$$

(55)

$$\underline{\underline{\left( a \right) \,}} \, {E_{{\varPhi _x}}}\left[ {\prod \limits _{i \in {\varPhi _x}/m} {{E_g}} \left\{ {\exp \left( {{P_m} {{\left| {{x_{i,u}}} \right| }^{ - \delta }}} \right) } \right\} } \right],$$

(56)

$$\underline{\underline{\left( b \right) \,}} \,\exp \left( { - 2\pi {\lambda _m}\int \limits _{{x_{m,u}}}^\infty {\left[ {1 - \exp \left( {s{P_m} {x_{i,u}}^{ - \delta }} \right) } \right] } {x_{i,u}} d{x_{i,u}}} \right),$$

(57)

$$\underline{\underline{\left( c \right) \,}} \,\exp \left[ { - \pi {\lambda _m}\left( {\frac{{\left| \!{\overline{\, {1 - \frac{2}{\delta }} \,}} \right. + \frac{2}{\delta }\left| \!{\overline{\, {\frac{{ - 2}}{\delta },\frac{{s{P_m}}}{{x_{m,u}^\delta }}} \,}} \right. }}{{{{\left( {s{P_m}} \right) }^{-2/\delta }}}} - x_{m,u}^2} \right) } \right],$$

(58)

where (a) is due to the independence of interfering links, (b) follows the probability generating functional (PGFL) of PPP, and (c) is achieved by solving the integration.

Thereafter, ${L_{{I_{s,{u_m}}}}}(s)$ can be calculated as

$${L_{{I_{s,{u_m}}}}}(s) = {E_{{\varPhi _y}}}\left[ {\prod \limits _{j \in {\varPhi _y}} {{E_g}} \left\{ {\exp \left( {s{P_s} {g_{j,u}}{{\left| {{X_{j,u}}} \right| }^{ - \delta }}} \right) } \right\} } \right],$$

(59)

by the PGFL of PPP

$$\begin{aligned} & = \exp \Biggl ( - 2\pi {\lambda _s}\int \limits _{{0}}^\infty {\left[ {1 - {E_g}\left\{ {\exp \left( {s{P_s} {g_{j,u}}{{\left| {{x_{j,u}}} \right| }^{ - \delta }}} \right) } \right\} } \right] }\nonumber \\&\quad \times \,{x_{j,u}}\,d{x_{j,u}}\Biggr ), \end{aligned}$$

(60)

$${L_{{I_{s,{u_m}}}}}\left( s \right) = \mathop{\lim }\limits _{{x_{s,u}} \rightarrow 0} {L'_{{I_{s,{u_m}}}}}\left( s \right),$$

(61)

$$\begin{aligned} {L'_{{I_{s,{u_m}}}}}\left( s \right) &= \exp \Biggl ( - 2\pi {\lambda _s}\int \limits _{{x_{s,u}}}^\infty {\left[ {1 - {E_g}\left\{ {\exp \left( {s{P_s} {g_{j,u}}{{\left| {{x_{j,u}}} \right| }^{ - \delta }}} \right) } \right\} } \right] }\nonumber \\&\quad \times \,{x_{j,u}}\,d{x_{j,u}}\Biggr ), \end{aligned}$$

(62)

$$= \exp \left( { - 2\pi {\lambda _s}\int \limits _{{x_{s,u}}}^\infty {\left[ {1 - \frac{1}{{1 + s{P_s} {g_{j,u}} {x_{j,u}}^{ - \delta } }}} \right] } \,{x_{j,u}}\,d{x_{j,u}}} \right),$$

(63)

$$= \exp \left( { - \frac{{2\pi {\lambda _s}{P_s}s}}{{\delta - 2}}\,x_{s,u}^{2 - \delta }{}_2{F_1}\left[ {1,1 - \frac{2}{\delta },2 - \frac{2}{\delta },\frac{{s{P_s}}}{{x_{s,u}^\delta }}} \right] } \right).$$

(64)

Now, from (60)

$$\begin{aligned} {L_{{I_{s,{u_m}}}}}\left( s \right) &= \mathop{\lim }\limits _{{x_{s,u}} \rightarrow 0} \exp \left( - \frac{{2\pi {\lambda _s}{P_s}s}}{{\delta - 2}}\,x_{s,u}^{2 - \delta }\nonumber \right. \\&\quad \left. {}_2{F_1}\left[ {1,1 - \frac{2}{\delta },2 - \frac{2}{\delta },\frac{{s{P_s}}}{{x_{s,u}^\delta }}} \right] \right) . \end{aligned}$$

(65)

By using the Kumar’s formula

$${}_2{F_1}\left[ {a,b,c,z} \right] = {\left( {1 - z} \right) ^{ - b}}{}_2{F_1}\left[ {c - a,b,c,\frac{z}{{z - 1}}} \right],$$

(66)

$$\begin{aligned} {L_{{I_{s,{u_m}}}}}\left( s \right)& = \mathop{\lim }\limits _{{x_{s,u}} \rightarrow 0} \exp \biggl (- \frac{{2\pi {\lambda _s}{P_s}s}}{{\delta - 2}} \biggl ({x_{s,u}^\delta - s{P_s}} \biggr )^{2-\delta /\delta }\nonumber \\&\quad {}_2{F_1}\biggl [{1 - \frac{2}{\delta },1 - \frac{2}{\delta },2 - \frac{2}{\delta },\frac{s P_{s}}{s P_{s}-x_{s,u}^\delta }}\biggr ] \biggr ), \end{aligned}$$

(67)

$${L_{{I_{s,{u_m}}}}}\left( s \right) = \exp \left( {\frac{{2\pi {\lambda _s}}}{{\delta - 2}}\,{{\left( { - s{P_s}} \right) }^{2/\delta }}\left| \!{\overline{\, {2 - \frac{2}{\delta }} \,}} \right. \,\left| \!{\overline{\, {\frac{2}{\delta }} \,}} \right. } \right).$$

(68)

Appendix 3: Proof of Theorem 2

Steps for deriving the coverage probability when a user is connected with SBS are as follows

$$C_{p} = P\left[ {SIN{R_{a,s}} > {\tau _{a,s}}\left| {{X_{s,u}} = {x_{s,u}}} \right. } \right],$$

(69)

$$\begin{aligned} {C_p}& = \int \limits _0^\infty {\frac{{2\pi {\lambda _s}}}{{{\varPi _s}}} P\left[ {\frac{{{P_s} {g_{s,u}}{{\left| {{X_{s,u}}} \right| }^{ - \delta }}}}{{{{\mathop{I}\nolimits } _{m,{u_s}}} + {I_{s,{u_s}}} + {N_0}}} > {\tau _{a,s}}} \right] }\nonumber \\&\quad\times {e^{\left[ { - \pi {\lambda _s}x_{s,u}^2 - \pi {\lambda _m}(\frac{{(N - L - 1){P_m}}}{{L{P_s}}}x_{s,u}^2)} \right] }} {x_{s,u}} d{x_{s,u}}, \end{aligned}$$

(70)

where

$$P\left[ {\frac{{{P_s} {g_{s,u}}{{\left| {{X_{s,u}}} \right| }^{ - \delta }}}}{{{{\mathop{I}\nolimits } _{m,{u_s}}} + {I_{s,{u_s}}} + {N_0}}} > {\tau _{a,s}}} \right]$$

(71)

$$= P\left[ {{g_{s,u}} > \frac{{{\tau _{a,s}}\left( {{{\mathop{I}\nolimits } _{m,{u_s}}} + {I_{s,{u_s}}} + {N_0}} \right) }}{{{P_s}{{\left| {{X_{s,u}}} \right| }^{ - \delta }}}}} \right],$$

(72)

and $P(\bullet )$ is derived below. Since channel is Rayleigh distributed with unit mean, (71) can be simplified as

$$= {L_{{I_{s,{u_s}}}}}\left( s \right) {L_{{I_{m,{u_s}}}}}\left( s \right) {e^{ - \frac{{{\tau _{a,s}}x_{s,u}^\delta {N_0}}}{{{P_s}}}\,}},$$

(73)

where ${L_{{I_{s,{u_s}}}}}\left( s \right)$ and ${L_{{I_{m,{u_s}}}}}\left( s \right)$ are the Laplace transform of the interferences $I_{s,u_{s}}$ and $I_{m,u_{s}}$, respectively for $s = \frac{{{\tau _{a,s}}x_{s,u}^\delta }}{{{P_s}}}$. These can be calculated with the same procedure which is explained in Theorem 1 and are given as follows

$${L_{{I_{m,{u_s}}}}}\left( s \right) = {E_{{\varPhi _x}}}\left[ {\prod \limits _{i \in {\varPhi _x}} {\exp \left( { - s{P_m}x_{i,u}^{ - \delta }} \right) } } \right],$$

(74)

$$= \exp \left( {2\pi {\lambda _m}\int \limits _0^\infty {\left( {1 - \exp \left[ { - s{P_m}x_{i,u}^{ - \delta }} \right] } \right) {x_{i,u}}d{x_{i,u}}} } \right),$$

(75)

$$= \exp \left( { - \pi {\lambda _m}{{\left( {s{P_m}} \right) }^{2/\delta }}\left| \!{\overline{\, {1 - \frac{2}{\delta }} \,}} \right. } \right),$$

(76)

$${L_{{I_{s,{u_s}}}}}\left( s \right) = {E_{{\varPhi _y}}}\left[ {\prod \limits _{j \in {\varPhi _{y/s}}} {\exp \left( { - s{P_s}{g_{j,u}}x_{j,u}^{ - \delta }} \right) } } \right],$$

(77)

$$= \exp \left( { - 2\pi {\lambda _s}\int \limits _{{x_{s,u}}}^\infty {\left( {1 - {1 \over {1 + s{P_s}{g_{j,u}}x_{j,u}^{ - \delta }}}} \right) } {x_{j,u}}\,d{x_{j,u}}} \right),$$

(78)

$$= \exp \left( { - {{2\pi {\lambda _s}{P_s}s} \over {\delta - 2}}\,x_{s,u}^{2 - \delta }{}_2{F_1}\left[ {1,1 - {2 \over \delta },2 - {2 \over \delta },{{s{P_s}} \over {x_{s,u}^\delta }}} \right] } \right).$$

(79)

Appendix 4: Proof of Theorem 3

Coverage probability of backhaul link (i.e., SBS associated with the MBS) can be derived analogous to the proof of Theorem 1. Moreover, key steps therein are mentioned below.

$$C_{b} = P\left[ {SIN{R_b} > {\tau _b}\left| {{X_{m,s}} = {x_{m,s}}} \right. } \right],$$

(80)

$$\begin{aligned} {C_b}& = 2\pi {\lambda _m}\int \limits _{{x_{m,s}} > 0} {{F_{{I_2}}}\left[ {\frac{{{P_m}}}{{{\tau _{b}}\,x_{m,s}^\delta }} - \alpha {P_s}} \right] } \;{x_{m,s}}\nonumber \\&\quad \times {e^{ - \pi {\lambda _m}x_{m,s}^2}}\,d{x_{m,s}}, \end{aligned}$$

(81)

where ${{F_{{I_2}}(\bullet )}}$ is the CDF of the aggregate interference, and Gil-Pelaez theorem is applied to evaluate the aforementioned CDF.

$$\begin{aligned}&{F_{{I_2}}}\left[ {\frac{{{P_m}}}{{{\tau _b}\,x_{m,s}^\delta }} - \alpha P_{s}} \right] = \frac{1}{2} - \frac{1}{\pi }\nonumber \\&\quad \times \int \limits _0^\infty {{\mathop{Im}\nolimits } \left( {{L_{{I_2}}}( - j\omega )\,{e^{ - j\omega \left( {\frac{{{P_m}}}{{{\tau _b}x_{m,s}^\delta }} - \alpha {P_s}} \right) }}} \right) } \frac{{d\omega }}{\omega }, \end{aligned}$$

(82)

$$\begin{aligned}&{F_{{I_1}}}\left[ {\frac{{{P_m}}}{{{\tau _b}\,x_{m,s}^\delta }} - \alpha {P_s}} \right] = \frac{1}{2} - \frac{1}{\pi }\nonumber \\&\quad \times \int \limits _0^\infty {{\mathop{Im}\nolimits } \left( {{L_{{I_{m,s}}}}( - j\omega ){L_{{I_{s,s}}}}( - j\omega )\,{e^{ - j\omega \left( {\frac{{{P_m}}}{{{\tau _b}x_{m,s}^\delta }} - \alpha {P_s}} \right) }}} \right) } \frac{{d\omega }}{\omega }, \end{aligned}$$

(83)

where ${L_{{I_{m,s}}}}(s)$ and ${L_{{I_{s,s}}}}(s)$ are the Laplace transform of the interferences $I_{m,s}$ and $I_{s,s}$, respectively and can be evaluated as

$${L_{{I_{m,s}}}}(s) = {E_{{I_{m,s}}}}\left[ {{e^{ - s{I_{m,s}}}}} \right],$$

(84)

$$= E\left[ {s\sum \limits _{i \in {\varPhi _x}/m} {{P_m} {{\left| {{X_{i,s}}} \right| }^{ - \delta }}} } \right],$$

(85)

$$= \,\exp \left[ { - \pi {\lambda _m}\left( {\frac{{\left| \!{\overline{\, {1 - \frac{2}{\delta }} \,}} \right. + \frac{2}{\delta }\left| \!{\overline{\, {\frac{{ - 2}}{\delta },\frac{{s{P_m}}}{{x_{m,s}^\delta }}} \,}} \right. }}{{{{\left( {s{P_m}} \right) }^{-2/\delta }}}} - x_{m,s}^2} \right) } \right],$$

(86)

$${L_{s,s}}\left( s \right) = \,{E_{{I_{s,s}}}}\left[ {{e^{ - s{I_{s,s}}}}} \right],$$

(87)

$$= E\left[ {s\sum \limits _{j \in {\varPhi _y}/s} {{P_s} \,{g_{j,s}}{{\left| {{X_{j,s}}} \right| }^{ - \delta }}} } \right],$$

(88)

$$= \exp \left( { - \frac{{2\pi {\lambda _s}{P_s}s}}{{\delta - 2}}\,x_{m,s}^{2 - \delta }{}_2{F_1}\left[ {1,1 - \frac{2}{\delta },2 - \frac{2}{\delta },\frac{{s{P_s}}}{{x_{m,s}^\delta }}} \right] } \right).$$

(89)