Performance Analysis of CoMP in Ultra-Dense Networks with Limited Backhaul Capacity

Abstract

Coordinated multipoint (CoMP) transmission provides an effective way to mitigate inter-cell interference as well as improve network performance in ultra-dense networks (UDNs). However, applying coordination among access points (APs) requires high backhaul capacity (BC), which becomes a bottleneck for employing CoMP. Considering a downlink UDN with limited BC, this paper conducts performance analysis of two CoMP schemes including joint transmission (JT) and coordinated scheduling/beamforming (CS/CB) from three aspects: a typical user’s perspective, a typical AP’s perspective and a network perspective. Firstly, the coverage probability (CP) and ergodic capacity (EC) of a typical user are characterized. Secondly, per-AP backhaul consumption is explicitly quantified and the successful serving probability (SSP) of a typical AP is proposed to capture its robustness to limited BC. Additionally, we propound effective ergodic capacity (EEC) to consider performance gain and backhaul consumption simultaneously from a global perspective. Further, the approximated expressions of all the performance metrics are given so that the impact of cluster size and AP density can be overtly observed. Numerical results illustrate that JT outperforms CS/CB in the case of high BC threshold, small cluster size or high AP density and a comparative smaller size is preferable for both two schemes for practice.

Appendices

Appendix 1

Considering JT in downlink UDN, the CP of a typical user is

$$\begin{aligned} CP_{n}^{JT} \left( {\theta ,\lambda_{a} ,\alpha } \right) & = {\mathbb{P}}\left( {SINR_{n}^{JT} > \theta } \right) \\ & = {\mathbb{P}}\left( {\frac{{\sum\nolimits_{k = 1}^{n} {P_{t} h_{k} r_{k}^{ - \alpha } } }}{{\sum\nolimits_{m > n} {P_{t} h_{m} r_{m}^{ - \alpha } } + \sigma^{2} }} > \theta } \right) = {\mathbb{P}}\left( {\frac{{\sum\nolimits_{k = 1}^{n} {h_{k} r_{k}^{ - \alpha } } }}{{I + \frac{{\sigma^{2} }}{{P_{t} }}}} > \theta } \right) \\ \end{aligned}$$

(29)

where $I = \sum\nolimits_{m > n} {h_{m} r_{m}^{ - \alpha } }$.

Following the similar steps in [27] and referring to the derived conclusion of Laplace transform of random variable ${\mathcal{L}}_{I} \left( s \right)$, we obtain

$$\begin{aligned} CP_{n}^{JT} \left( {\theta ,\lambda_{a} ,\alpha } \right) & = {\mathbb{E}}_{r,I} \left( {\exp \left( { - \frac{{\theta \left( {I + \frac{{\sigma^{2} }}{{P_{t} }}} \right)}}{{\sum\nolimits_{k = 1}^{n} {r_{k}^{ - \alpha } } }}} \right)} \right) \\ & = {\mathbb{E}}_{r} \left( {{\mathcal{L}}_{I} \left( {\frac{\theta }{{\sum\nolimits_{k = 1}^{n} {r_{k}^{ - \alpha } } }}} \right)\exp \left( { - \frac{{\theta \sigma^{2} }}{{P_{t} \sum\nolimits_{k = 1}^{n} {r_{k}^{ - \alpha } } }}} \right)} \right) \\ \, = \int\limits_{{0 < r_{1} < \cdots < r_{n} < \infty }} {\exp \left( { - \frac{{\pi \lambda_{a} \theta^{{\frac{2}{\alpha }}} \int_{{\zeta_{cp}^{JT} }}^{\infty } {\frac{1}{{1 + \mu^{{\frac{\alpha }{2}}} }}d\mu } }}{{\left( {\sum\nolimits_{k = 1}^{n} {r_{k}^{ - \alpha } } } \right)^{{\frac{2}{\alpha }}} }}} \right)} \\ & \quad \times \exp \left( { - \frac{{\theta \sigma^{2} }}{{P_{t} \sum\nolimits_{k = 1}^{n} {r_{k}^{ - \alpha } } }}} \right)f_{R} \left( {r_{1} , \ldots ,r_{n} } \right)dr_{1} \ldots dr_{n} \\ \end{aligned}$$

(30)

where $\zeta_{cp}^{JT} = \theta^{{ - \frac{2}{\alpha }}} \left( {\sum\nolimits_{k = 1}^{n} {r_{k}^{ - \alpha } } } \right)^{{\frac{2}{\alpha }}} r_{n}^{2}$ and the joint distance distribution of $\left\{ {r_{1} ,r_{2} , \ldots ,r_{n} } \right\}$ is $f_{R} \left( {r_{1} ,r_{2} , \ldots ,r_{n} } \right) = e^{{ - \pi \lambda_{a} r_{n}^{2} }} \left( {2\pi \lambda_{a} } \right)^{n} r_{1} r_{2} \cdots r_{n} dr_{1} dr_{2} \cdots dr_{n}$ [30].

Appendix 2

According to the characteristic of HPPP, the average number of users in the coverage of $AP_{0}$ is $k_{JT} = \left\lceil {n\lambda_{u} /\lambda_{a} } \right\rceil$.

Then following the similar proof as Theorem 1, the BC of a typical AP, i.e., the EC provided by $AP_{0}$ to the users in its coverage is

$$\begin{aligned} BC_{n} \left( {\lambda_{a} ,\alpha } \right) & = {\mathbb{E}}_{l,I} \left[ {\ln \left( {1 + \frac{{\sum\nolimits_{i = 1}^{{k_{JT} }} {P_{t} g_{i} l_{i}^{ - \alpha } } }}{{\sum\nolimits_{m > n} {P_{t} h_{m} r_{m}^{ - \alpha } } + \sigma^{2} }}} \right)} \right] \\ & = \int\limits_{{0 < l_{1} < \cdots < l_{{k_{JT} }} < \infty }} {\int\limits_{t > 0} {\exp \left[ { - \frac{{\left( {e^{t} - 1} \right)\sigma^{2} }}{{P_{t} \sum\nolimits_{i = 1}^{{k_{JT} }} {l_{i}^{ - \alpha } } }}} \right]{\mathcal{L}}_{I} \left( {\frac{{e^{t} - 1}}{{\sum\nolimits_{i = 1}^{{k_{JT} }} {l_{i}^{ - \alpha } } }}} \right)dt} } \\ & \quad \times f_{L} \left( {l_{1} , \ldots ,l_{{k_{JT} }} } \right)dl_{1} \ldots dl_{{k_{JT} }} \\ & = \int\limits_{{0 < l_{1} < \cdots < l_{{k_{JT} }} < \infty }} {\int\limits_{{r_{n} > 0}} {\int\limits_{t > 0} {\exp \left[ { - \frac{{\pi \lambda_{a} \left( {e^{t} - 1} \right)^{{\frac{2}{\alpha }}} }}{{\left( {\sum\nolimits_{i = 1}^{{k_{JT} }} {l_{i}^{ - \alpha } } } \right)^{{\frac{2}{\alpha }}} }}\int_{{\zeta_{bc}^{JT} }}^{\infty } {\frac{1}{{1 + \mu^{{\frac{\alpha }{2}}} }}d\mu } } \right.} } } \\ & \quad \left. { - \frac{{\left( {e^{t} - 1} \right)\sigma^{2} }}{{P_{t} \sum\nolimits_{i = 1}^{{k_{JT} }} {l_{i}^{ - \alpha } } }}} \right]dt \times f_{R} \left( {r_{n} } \right)f_{L} \left( {l_{1} , \ldots ,l_{{k_{JT} }} } \right)dr_{n} dl_{1} \ldots dl_{{k_{JT} }} \\ \end{aligned}$$

(31)

where $\zeta_{bc}^{JT} = \left( {e^{t} - 1} \right)^{{ - \frac{2}{\alpha }}} \left( {\sum\nolimits_{i = 1}^{{k_{JT} }} {l_{i}^{ - \alpha } } } \right)^{{\frac{2}{\alpha }}} r_{n}^{2}$, $f_{R} \left( {r_{n} } \right) = \frac{{2\left( {\pi \lambda_{a} } \right)^{n} }}{{\left( {n - 1} \right)!}}r_{n}^{2n - 1} e^{{ - \pi \lambda_{a} r_{n}^{2} }}$ and

$$f_{L} \left( {l_{1} , \ldots ,l_{{k_{JT} }} } \right) = e^{{ - \pi \lambda_{u} l_{{k_{JT} }}^{2} }} \left( {2\pi \lambda_{u} } \right)^{{k_{JT} }} l_{1} \ldots l_{{k_{JT} }} .$$

Using the expectation (5) to perform approximation, the result in Theorem 2 can be obtained.

Appendix 3

As SSP is defined by the probability that BC doesn’t exceed the given threshold $\gamma^{BC}$, the SSP $p_{s}^{JT}$ in UDN downlink systems with JT can be expressed by

$$p_{s}^{JT} \left( {\gamma^{BC} ,\lambda_{a} ,\alpha } \right) = {\mathbb{P}}\left[ {BC_{n}^{JT} \left( {\lambda_{a} ,\alpha } \right) \le \gamma^{BC} } \right] = 1 - {\mathbb{P}}\left[ {BC_{n}^{JT} \left( {\lambda_{a} ,\alpha } \right) > \gamma^{BC} } \right]$$

(32)

Then follow the same steps in Theorem 3, we further get the specific closed-form of $p_{s}^{JT}$ written as

$$\begin{aligned} p_{s}^{JT} \left( {\gamma^{BC} ,\lambda_{a} ,\alpha } \right) & = 1 - {\mathbb{P}}\left[ {\ln \left( {1 + \frac{{\sum\nolimits_{i = 1}^{{k_{JT} }} {P_{t} g_{i} l_{i}^{ - \alpha } } }}{{\sum\nolimits_{m > n} {P_{t} h_{m} r_{m}^{ - \alpha } } + \sigma^{2} }}} \right) > \gamma^{BC} } \right] \\ & = 1 - {\mathbb{P}}\left[ {\sum\limits_{i = 1}^{{k_{JT} }} {g_{i} l_{i}^{ - \alpha } } > \left( {e^{{\gamma^{BC} }} - 1} \right)\left( {I + \frac{{\sigma^{2} }}{{P_{t} }}} \right)} \right] \\ & = 1 - {\mathbb{E}}_{l} \left[ {\exp \left( { - \frac{{\left( {e^{{\gamma^{BC} }} - 1} \right)\frac{{\sigma^{2} }}{{P_{t} }}}}{{\sum\nolimits_{i = 1}^{{k_{JT} }} {l_{i}^{ - \alpha } } }}} \right){\mathcal{L}}_{I} \left[ {\frac{{\left( {e^{{\gamma^{BC} }} - 1} \right)}}{{\sum\nolimits_{i = 1}^{{k_{JT} }} {l_{i}^{ - \alpha } } }}} \right]} \right] \\ & = 1 - \int\limits_{{0 < l_{1} < \cdots < l_{{k_{JT} }} < \infty }} {\int\limits_{{r_{n} > 0}} {\exp \left[ {\frac{{ - \pi \lambda_{a} \left( {e^{{\gamma^{BC} }} - 1} \right)^{{\frac{2}{\alpha }}} \int_{{\zeta_{bc}^{JT} }}^{\infty } {\frac{1}{{1 + \mu^{{\frac{\alpha }{2}}} }}d\mu } }}{{\left( {\sum\nolimits_{i = 1}^{{k_{JT} }} {l_{i}^{ - \alpha } } } \right)^{{\frac{2}{\alpha }}} }}} \right.} } \\ & \quad \left. { - \frac{{\left( {e^{{\gamma^{BC} }} - 1} \right)\sigma^{2} }}{{P_{t} \sum\nolimits_{i = 1}^{{k_{JT} }} {l_{i}^{ - \alpha } } }}} \right] \times f_{R} \left( {r_{n} } \right)f_{L} \left( {l_{1} , \ldots ,l_{{k_{JT} }} } \right)dr_{n} dl_{1} \ldots dl_{{k_{JT} }} \\ \end{aligned}$$

(33)

Likewise, the desired result in Theorem 3 can be gotten by expectation approximation.