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Uplink coverage probability and spectral efficiency for downlink uplink decoupled dense heterogeneous cellular network using multi-slope path loss model

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Abstract

Low powered node densification leading to dense and ultra dense heterogeneous networks is a feature of 5th generation cellular networks. With this densification, the nature of the link between the transmitter and receiver in the network requires even more accurate and reliable models. For performance analysis of such networks where the rates of signal loss over distance becomes a significant parameter, more accurate path loss models should be used when analyzing user equipment (UE) association probability, coverage probability and average spectral efficiency. In this paper, we have considered a two-tier dense heterogeneous cellular network incorporating downlink uplink decoupled technique and have derived generalized expressions for UE association probability, decoupled uplink coverage probability and decoupled uplink average spectral efficiency using multi-slope path loss model. This path loss model broadly incorporates the effects of physical environment on the distance-dependent path loss. For simulation purpose, we have compared network performance while considering single-slope and dual-slope path loss models. The derived analytical expressions have been validated through network simulations and found in good agreement. Through comparison, it has been found that the decoupled UE association probability and uplink coverage probability is higher when incorporating multi-slope path loss model as compared to single-slope path loss model while the decoupled uplink spectral efficiency is observed to be lower when incorporating dual slope path loss model.

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Authors and Affiliations

  1. Department of Electronic Engineering, NED University of Engineering and Technology, Karachi, Pakistan

    Sundus Ali, Muhammad Imran Aslam & Irfan Ahmed

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Appendix A

Appendix A

From (19) in Sect. 4.1, \({I_{{x_S}}}\) is defined as the total uplink interference experienced by the tagged SBS. In order to find \({f_{{x_S}}}({x_S})\), we first consider the CCDF of \({f_{{x_S}}}({x_S})\) defined in (38). Incorporating distance distributions in (38) results in (39). Solving (39), we get (40) which is the PDF of distance distribution obtained by derivating the \(CDF=1-CCDF\) of \({f_{{x_S}}}({x_S})\) with respect to x.

$$\begin{aligned}&F_{{X_S}}^C(x) = \frac{1}{{{P_{decoupled}}}} \nonumber \\&\quad \Pr \left( {X_S} > x|\prod \limits _{i = 1}^{\left| {{n_S} - {n_M}} \right| } R_{\left| {{n_S} - {n_M}} \right| + i - 1}^{\frac{{c}}{{{\alpha _{{n_M}}}}}({\alpha _{\left| {{n_S} -{n_M}} \right| + i}} - {\alpha _{\left| {{n_S} - {n_M}} \right| + i - 1}}} X_S^{{\tilde{\alpha }} }\right. \nonumber \\&\qquad \left. {\le }\, {X_M}<{{{\tilde{P}}}^{{{1}/{{{\alpha _{{n_M}}}}}}}}\prod \limits _{i = 1}^{\left| {{n_S} - {n_M}} \right| } {R_{\left| {{n_S} - {n_M}} \right| + i - 1}^{\frac{c}{{{\alpha _{{n_M}}}}}({\alpha _{\left| {{n_S} - {n_M}} \right| + i}} - {\alpha _{\left| {{n_S} - {n_M}} \right| + i - 1}}}} X_S^{{\tilde{\alpha }} } \right) \end{aligned}$$
(38)
$$\begin{aligned}&F_{{X_S}}^C(x) = \frac{1}{{{P_{decoupled}}}} \nonumber \\&\quad \int \limits _x^\infty \left( e^{ - \pi {\lambda _M}\prod \limits _{i = 1}^{\left| {{n_S} - {n_M}} \right| } R_{\left| {{n_S} - {n_M}} \right| + i - 1}^{\frac{{2c}}{{{\alpha _{{n_M}}}}}({\alpha _{\left| {{n_S} - {n_M}} \right| + i}} - {\alpha _{\left| {{n_S} - {n_M}} \right| + i - 1}}} x_S^{2{\tilde{\alpha }} }} \right. \nonumber \\&\quad \left. - {e^{ - \pi {\lambda _M}{{\tilde{P}}}^{2}/ {\alpha _{{n_M}}}}\prod \limits _{i = 1}^{\left| {{n_S} - {n_M}} \right| } {R_{\left| {{n_S} - {n_M}} \right| + i - 1}^{\frac{{2c}}{{{\alpha _{{n_M}}}}}({\alpha _{\left| {{n_S} - {n_M}} \right| + i}} - {\alpha _{\left| {{n_S} - {n_M}} \right| + i - 1}}} x_S^{2{\tilde{\alpha }} }}} \right) \nonumber \\&\quad {f_{{x_S}}}(x)dx \end{aligned}$$
(39)
$$\begin{aligned}&{f_{{X_S}}}(x) = \frac{1}{{{P_{decoupled}}}}{f_{{x_S}}}(x) \nonumber \\&\quad \left( e^ - \pi {\lambda _M}\prod \limits _{i = 1}^{\left| {{n_S} - {n_M}} \right| } R_{\left| {{n_S} - {n_M}} \right| + i - 1}^{\frac{{2c}}{\alpha _{n_M}}(\alpha _{\left| {{n_S} - {n_M}} \right| + i} - {\alpha _{\left| {{n_S} - {n_M}} \right| + i - 1}}} x_S^2{\tilde{\alpha }}\right. \nonumber \\&\quad \left. - {e^{ - \pi {\lambda _M}{{{\tilde{P}}}^{2}/{\alpha _{{n_M}}}}\prod \limits _{i = 1}^{\left| {{n_S} - {n_M}} \right| } {R_{\left| {{n_S} - {n_M}} \right| + i - 1}^{\frac{{2c}}{{{\alpha _{{n_M}}}}}({\alpha _{\left| {{n_S} - {n_M}} \right| + i}} - {\alpha _{\left| {{n_S} - {n_M}} \right| + i - 1}}}} x_S^{2{\tilde{\alpha }} }}} \right) \nonumber \\ \end{aligned}$$
(40)

For (19), as \({h_{{x_S}}} \sim \exp (1)\), averaging over \({h_{{x_S}}}\) will give us:

$$\begin{aligned} \begin{aligned} \begin{array}{l} \Pr \left[ {{h_{{x_S}}} > \tau {I_{{x_S}}}P_d^{ - 1}\prod \limits _{j = 1}^{{n_S}} {R_j^{{\alpha _{j - 1}} - {\alpha _j}}} {{\left\| {{x_S}} \right\| }^{{\alpha _{{n_S}}}}}} \right] \\ \quad ={E_{{I_{{x_S}}}}}\left[ {{e^{ - \tau P_d^{ - 1}\prod \limits _{j = 1}^{{n_S}} {R_j^{{\alpha _{j - 1}} - {\alpha _j}}} {{\left\| {{x_S}} \right\| }^{{\alpha _{{n_S}}}}}{I_{{x_S}}}}}} \right] = {{{\mathcal {L}}}_{{I_{{x_S}}}}}(s) \end{array} \end{aligned} \end{aligned}$$
(41)

where \({{{\mathcal {L}}}_{{I_{{x_S}}}}}(s)\) is the Laplacian and \(s = \tau P_d^{ - 1}\prod \nolimits _{j = 1}^{{n_S}} {R_j^{{\alpha _{j - 1}} - {\alpha _j}}} {\left\| {{x_S}} \right\| ^{{\alpha _{{n_S}}}}}\). The uplink interference in (41) caused by the Interferer UEs to the serving SBS (uplink) is defined as:

$$\begin{aligned} \begin{aligned} {I_{{x_S}}} = \sum \limits _{{Z_i} \in {\phi _I}_{_d}} {{P_d}{h_{{Z_i}}}\prod \limits _{j = 1}^{{n_S}} {R_j^{{\alpha _j} - {\alpha _{j - 1}}}} {{\left\| {{Z_i}} \right\| }^{ - {\alpha _{{n_S}}}}}} \end{aligned} \end{aligned}$$
(42)

where \({Z_i}\) represent distance between the tagged small BS (serving the test UE) and the Interferer UE. The Laplace functional in (41) can be written as:

$$\begin{aligned} \begin{aligned} \begin{array}{l} {{{\mathcal {L}}}_{{I_{{x_S}}}}}(s) = {E_{{I_{{x_S}}}}}\left[ {{e^{ - s\sum \limits _{{Z_i} \in {\phi _I}_{_d}} {{P_d}{h_{{Z_i}}}\prod \limits _{j = 1}^{{n_S}} {R_j^{{\alpha _j} - {\alpha _{j - 1}}}} {{\left\| {{Z_i}} \right\| }^{ - {\alpha _{{n_S}}}}}} }}} \right] \\ \quad = {E_{{I_{{x_S}}}}}\left[ {\prod \limits _{{Z_i} \in {\phi _I}_{_d}} {{e^{ - s{P_d}{h_{{Z_i}}}\prod \limits _{j = 1}^{{n_S}} {R_j^{{\alpha _j} - {\alpha _{j - 1}}}} {{\left\| {{Z_i}} \right\| }^{ - {\alpha _{{n_S}}}}}}}} } \right] \end{array} \end{aligned} \end{aligned}$$
(43)

As \({h_i} \sim \exp (1)\), (43) can be written as:

$$\begin{aligned} \begin{aligned} {{{{\mathcal {L}}}_{{I_{{x_S}}}}}(s) = {E_{{\phi _{{I_d}}}}}\left[ {\prod \limits _{{Z_i} \in {\phi _I}_{_d}} {\frac{1}{{1 + s{P_d}{h_{{Z_i}}}\prod \nolimits _{j = 1}^{{n_S}} {R_j^{{\alpha _j} - {\alpha _{j - 1}}}} {{\left\| {{Z_i}} \right\| }^{ - {\alpha _{{n_S}}}}}}}} } \right] } \end{aligned}\nonumber \\ \end{aligned}$$
(44)

Since \(E\left[ {\prod {f(X)} } \right] = {e^{\left( { - \lambda \int \limits _{{R^d}} {(1 - f(X))dX} } \right) }}\) , (44) can be reduced to:

$$\begin{aligned} \begin{aligned} {{{{\mathcal {L}}}_{{I_{{x_S}}}}}(s) = {e^{\left( { - {\lambda _{{I_d}}}\int \limits _{{R^d}} {\left( {\frac{1}{{1 + {{\left( {s{P_d}} \right) }^{ - 1}}\prod \nolimits _{j = 1}^{{n_S}} {R_j^{{\alpha _{j - 1}} - {\alpha _j}}} {{\left\| z \right\| }^{{\alpha _{{n_S}}}}}}}} \right) d} {R^d}} \right) }}} \end{aligned}\nonumber \\ \end{aligned}$$
(45)

where \({\lambda _{{I_d}}} = \left( {{\lambda _M} + {\lambda _S}} \right) (1 - {e^{ - \pi \left( {{\lambda _M} + {\lambda _S}} \right) {z^2}}})\) is the effective interference field (an approximate model) as observed from the Serving BS, which is a non-homogeneous PPP \(\phi _{{I_d}}\). Putting \({\lambda _{{I_d}}}\) in (45), we get:

$$\begin{aligned} \begin{aligned} {{{{\mathcal {L}}}_{{I_{{x_S}}}}}(s) = {e^{\left( { - 2\pi ({\lambda _M} + {\lambda _S})\int \limits _0^\infty {\left( {\frac{{z(1 - {e^{ - \pi ({\lambda _M} + {\lambda _S}){z^2}}})}}{{1 + {{\left( {s{P_d}} \right) }^{ - 1}}\prod \nolimits _{j = 1}^{{n_S}} {R_j^{{\alpha _{j - 1}} - {\alpha _j}}} {z^{{\alpha _{{n_S}}}}}}}} \right) } dz} \right) }}} \end{aligned}\nonumber \\ \end{aligned}$$
(46)

Solving (46), we get (47)

$$\begin{aligned} \begin{aligned}&{{{\mathcal {L}}}_{{I_{{x_S}}}}}(s) = \exp \left( - 2\pi ({\lambda _M} + {\lambda _S})\left( \frac{\pi }{{{\alpha _{{n_S}}}\sin \left( {{{{2\pi }} \big / {{ {{\alpha _{{n_S}}}}}} }} \right) }}\right. \right. \\&\quad \left. \left. - \int \limits _0^\infty \left( \frac{{z({e^{ - \pi ({\lambda _M} + {\lambda _S}){z^2}}})}}{{1 + {{\left( {s{P_d}} \right) }^{ - 1}}\prod \nolimits _{j = 1}^{{n_S}} {R_j^{{\alpha _{j - 1}} - {\alpha _j}}} {z^{{\alpha _{{n_S}}}}}}} \right) dz \right) \right) \end{aligned}\nonumber \\ \end{aligned}$$
(47)

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Ali, S., Aslam, M.I. & Ahmed, I. Uplink coverage probability and spectral efficiency for downlink uplink decoupled dense heterogeneous cellular network using multi-slope path loss model. Telecommun Syst 72, 505–516 (2019). https://doi.org/10.1007/s11235-019-00587-3

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