Analysis of the decoupled uplink downlink technique for varying path loss exponent in multi-tier HetNet

Abstract

In this paper, the impact of varying path loss exponent (PLE) on user association probability, decoupled uplink coverage probability as well as decoupled uplink average spectral efficiency in downlink uplink decoupled (DUDe) multi-tier heterogeneous networks, is investigated. We investigate the effect of the difference in path loss exponents in both macro and small cell environments over uplink network performance. It is assumed that the mobile user connected to the macro base station experience different path loss exponent as compared to when connected to small base station. It is observed that the difference of path loss exponents in both cases has significant effect on the user association probability, decoupled uplink coverage probability as well as decoupled uplink average spectral efficiency. Moreover, in order to further support key findings and make sound comparison between coupled and DUDe performance in varying PLE environment, generalized analytical expressions for coupled association probabilities, along with coupled uplink coverage probability and coupled uplink average spectral efficiency have been derived. The analytical results evaluated in this paper are compared with the computer simulation and found in good agreement. Our analysis shows that decoupling technique performs suboptimal for cases where the environments around macro and small base stations are different with respect to each other. The work explained in this paper highlights the limitation of applying DUDe technique in realistic conditions where the PLEs of cellular tiers are not exactly equal to one another.

Appendices

A Appendix A: Coupled association probabilities

In coupled association, the UE connects with the BS from which it receives strongest downlink received power (DRP). Hence there are two association cases possible in coupled association: (1) MBS tier association, denoted by $P_{MBS_C}$ (2) SBS tier association, denoted by $P_{SBS_C}$. We have derived UE association probability for MBS tier, which is defined as the following:

$$\begin{aligned} \begin{aligned} {P_{{MBS_C}}} = \Pr ({P_M}{L_M} > {P_S}{L_S}) \end{aligned} \end{aligned}$$

(41)

Equation (41) can be written as:

$$\begin{aligned} \begin{aligned} P_{MB{S_C}} = \Pr ({y_M} < {\tilde{P}}^{\nicefrac {1}{\alpha M}}y_S^{1/\tilde{\alpha }}) \end{aligned} \end{aligned}$$

(42)

Solving (42) using technique discussed in Sect. 3, $P_{MBS_C}$ is found to be:

$$\begin{aligned} \begin{aligned} {P_{MB{S_C}}} = 1 - \int \limits _0^\infty {2\pi {\lambda _M}\tilde{\lambda }{v_S}{e^{ - \pi {\lambda _M}({\tilde{P}}^{\nicefrac {2}{\alpha M}}v_S^{2\tilde{\alpha }} + \tilde{\lambda }v_S^2)}}d{v_S}} \end{aligned} \end{aligned}$$

(43)

Consequently, the SBS tier association probability can be found by the following expression:

$$\begin{aligned} \begin{aligned} P_{SBS_C}=1-P_{MBS_C} \end{aligned} \end{aligned}$$

(44)

B Appendix B: Coupled uplink coverage probability and average spectral efficiency

The uplink coverage probability provided that the UE is positioned in the decoupled region (shown in Fig. 9) and associates with MBS in coupled manner is defined as the following expression:

Fig. 9

DUDe system model showing decoupled region where UE may associate itself with the MBS in the downlink and SBS in the uplink

$$\begin{aligned} \begin{aligned} p_{Coupled,2}^{UL}&= \int \limits _0^\infty {\Pr [SINR_{Coupled,2}^{UL}> \tau ]{f_{{V_{M,2}}}}({v_M})d{v_M}}\\&= \int \limits _0^\infty {\Pr \left\{ {\frac{{{P_d}{h_{{v_M}}}y_M^{ {\alpha _M}(\epsilon -1)}}}{{{I_{{v_M}}} + \sigma _{{v_M}}^2}} > \tau } \right\} {f_{{v_{M,2}}}}({v_M})d{v_M}} \end{aligned} \end{aligned}$$

(45)

where $SINR_{Coupled,2}^{UL}$ is the uplink SINR measured from the MBS serving the test UE provided that the UE is located in the decoupled region, $v_M$ is defined as the distance between tagged MBS and test UE, $I_{v_M }$ is the total uplink interference experienced by the tagged MBS, $\sigma _{v_M }^2$ represent noise power and $f_{V_M ,2}^{} (v_M )$ is the distance distribution conditioned that the test UE is present in the decoupled region and found to be:

$$\begin{aligned} \begin{aligned} {f_{{V_{M,2}}}}(v_M) = \frac{1}{{{P_2}}}({{e^{ - \pi {\lambda _S}{{({1/{\tilde{P}}})}^{\nicefrac {2\bar{\alpha }}{{\alpha _M}}}} }}v_M^{2\bar{\alpha }}} - {e^{ - \pi {\lambda _S}v_M^{2\bar{\alpha }}}}) \end{aligned} \end{aligned}$$

(46)

The total uplink interference, $I_{v_M }$, experienced by the tagged MBS is defined as the following expression:

$$\begin{aligned} \begin{aligned} {I_{{v_M}}}&= \sum \limits _{{Z_{}} \in {\varPhi _{_{{I_{d,M}}}}}} {{P_d}{{\left\| {{Z_{i,M}}} \right\| }^{ - {\alpha _M}}}R_{i,M}^{{\alpha _M}\varepsilon }{h_{{Z_{i,M}}}}}\\&\quad + \sum \limits _{{Z_{i,S}} \in {\varPhi _{_{{I_{d,S}}}}}} {{P_d}{{\left\| {{Z_{i,S}}} \right\| }^{ - {\alpha _M}}}R_{i,S}^{{\alpha _S}\varepsilon }{h_{{Z_{i,S}}}}} \end{aligned} \end{aligned}$$

(47)

Using (47), we solve for Laplacian for interference $I_{v_M }$, which is found to be:

$$\begin{aligned}&{\mathcal {L}_{{I_{{v_M}}}}}\nonumber \\&\quad = \exp \left( { - 2\pi {\lambda _M}\int \limits _0^\infty {\int \limits _0^{z_M^2} \left( {\frac{{\pi {\lambda _M}{e^{ - \pi {\lambda _M}{u_M}}}{z_M}d{u_M}d{z_M}}}{{1 + {\tau ^{ - 1}}{{\left( {{{{{z_M}}}/ {{{v_M}}}}} \right) }^{{\alpha _M}}}u_M^{{{{ - {\alpha _M}\varepsilon }}/ {2}}}}}} \right) } } \right) \nonumber \\&\qquad \exp \left( { - 2\pi {\lambda _S}\int \limits _0^\infty {\int \limits _0^{z_S^2} {\left( {\frac{{\pi {\lambda _S}{e^{ - \pi {\lambda _S}{u_S}}}{z_S}d{u_S}d{z_S}}}{{1 + {\tau ^{ - 1}}{{\left( {{{{{z_S}}}/ {{{v_M}}}}} \right) }^{{\alpha _M}}}u_S^{{{{ - {\alpha _M}\varepsilon }}/ {{2\tilde{\alpha }}}}}}}} \right) } } } \right) \nonumber \\ \end{aligned}$$

(48)

Using (48), and assuming $\epsilon =0$ and $\alpha _M=4$, the coupled uplink coverage probability $p_{Coupled,2}^{UL}$ is found and presented in (49). Using (49), the coupled uplink average spectral efficiency is calculated and shown in (50).

$$\begin{aligned}&p_{Coupled,2}^{UL} \nonumber \\&\quad = \int \limits _0^\infty \exp \left( { - 2\pi {\lambda _M}\int \limits _0^\infty {\left( {\frac{{\left( {1 - {e^{ - \pi {\lambda _M}z_M^2}}} \right) {z_M}d{z_M}}}{{1 + {\tau ^{ - 1}}{{\left( {{{{{z_M}}}/ {{{v_M}}}}} \right) }^4}}}} \right) } } \right) \nonumber \\&\qquad \exp \left( { - 2\pi {\lambda _S}\int \limits _0^\infty {\left( {\frac{{\left( {1 - {e^{ - \pi {\lambda _S}z_S^2}}} \right) {z_S}d{z_S}}}{{1 + {\tau ^{ - 1}}{{\left( {{{{{z_S}}} / {{{v_M}}}}} \right) }^4}}}} \right) } } \right) e^{ - \tau \sigma _{v_M }^2 P_d^{ - 1} v_M^{ -4}}\nonumber \\&\qquad \frac{1}{{{P_2}}}\left( {{e^{ - \pi {\lambda _S}{{({{1} / {{\tilde{P}}}})}^{{{{2\bar{\alpha }}}/ {{{\alpha _M}}}}}}v_M^{2\bar{\alpha }}}} - {e^{ - \pi {\lambda _S}v_M^{2\bar{\alpha }}}}} \right) {f_{{v_M}}}({v_M})d{v_M} \end{aligned}$$

(49)

$$\begin{aligned}&C_{Coupled,2}^{UL} \nonumber \\&\quad = \frac{{{\lambda _M}}}{{({\lambda _M} + {\lambda _S}){P_1}}}\left( {{{\log }_2}(1 + \tau )} \right) \int \limits _0^\infty \nonumber \\&\qquad \exp \left( { - 2\pi {\lambda _M}\int \limits _0^\infty {\left( {\frac{{\left( {1 - {e^{ - \pi {\lambda _M}z_M^2}}} \right) {z_M}d{z_M}}}{{1 + {\tau ^{ - 1}}{{\left( {{{{{z_M}}}/ {{{v_M}}}}} \right) }^4}}}} \right) } } \right) \nonumber \\&\qquad \exp \left( { - 2\pi {\lambda _S}\int \limits _0^\infty {\left( {\frac{{\left( {1 - {e^{ - \pi {\lambda _S}z_S^2}}} \right) {z_S}d{z_S}}}{{1 + {\tau ^{ - 1}}{{\left( {{{{{z_S}}} / {{{v_M}}}}} \right) }^4}}}} \right) } } \right) e^{ - \tau \sigma _{v_M }^2 P_d^{ - 1} v_M^{ -4}}\nonumber \\&\qquad \frac{1}{{{P_2}}}\left( {{e^{ - \pi {\lambda _S}{{({{1} /{{\tilde{P}}}})}^{{{{2\bar{\alpha }}}/ {{{\alpha _M}}}}}}v_M^{2\bar{\alpha }}}} - {e^{ - \pi {\lambda _S}v_M^{2\bar{\alpha }}}}} \right) {f_{{v_M}}}({v_M})d{v_M} \end{aligned}$$

(50)