Uplink performance analysis in D2D-enabled cellular networks with clustered users

Abstract

This paper provides an analytical framework for the coexistence of uplink cellular network and D2D network in the same frequency band. D2D devices are assumed to be distributed according to a Poisson Cluster Process (PCP), while the locations of the cellular users and macro BSs are modeled as an independent Poisson Point Process (PPP) respectively. We study the performance of the uplink in cellular network with power control and the performance of this D2D network for two content availability cases: (1) Uniform content availability, which means content of interest to a typical device is available on devices selected randomly and uniformly from the same cluster, and (2) Closest content availability, which means content of interest is available on the closest devices to a typical device in the same cluster. Using this model, the distribution of the Signal to Interference Ratios (SIR) of a typical receiving node (tagged BS or D2D Rx) can be obtained. We derive the coverage probability, area spectrum efficiency (ASE) and average rate for both cellular and D2D links to analyze the performance of the whole network, and our analysis shows that an optimal number of D2D links must be simultaneously activated per cluster in order to maximize ASE.

Appendices

Appendix A: Proof of Eq. (15)

Laplace transform of the interference at the typical D2D Rx from intra-cluster interfering D2D devices $\text{L}_{{\text{I}_{{{\text{d2d}}}}^{{{\text{intra}}}} }} \left( {s/v_{0} } \right)$ is

$$ \begin{aligned} \text{L}_{{\text{I}_{{{\text{d2d}}}}^{{{\text{intra}}}} }} \left( {s/v_{0} } \right){ = } & {\text{ E}}_{{\text{I}_{{{\text{d2d}}}}^{{{\text{intra}}}} }} \left[ {\exp \left( { - s\text{I}_{{{\text{d2d}}}}^{{{\text{intra}}}} } \right)} \right] \\ = & {\text{E}}\left[ {\exp \left( { - s\sum\limits_{{y \in \text{A}^{{x_{0} }} \backslash y_{0} }} {P_{d} h\left\| {x_{0} + y} \right\|^{ - \alpha } } } \right)} \right] \\ = & {\text{E}}_{{\text{A}^{{x_{0} }} }} \left[ {\prod\limits_{{y \in \text{A}^{{x_{0} }} \backslash y_{0} }} {\left[ {{\text{E}}_{h} \left[ {\exp \left( { - sP_{d} h\left\| {x_{0} + y} \right\|^{ - \alpha } } \right)} \right]} \right]} } \right] \\ \mathop = \limits^{\left( a \right)} & {\text{E}}_{{\text{A}^{{x_{0} }} }} \left[ {\prod\limits_{{y \in \text{A}^{{x_{0} }} \backslash y_{0} }} {\int_{0}^{\infty } {\exp \left( { - sP_{d} h\left\| {x_{0} + y} \right\|^{ - \alpha } } \right)\mu e^{ - \mu h} } } dh} \right] \\ = & {\text{E}}_{{\text{A}^{{x_{0} }} }} \left[ {\prod\limits_{{y \in \text{A}^{{x_{0} }} \backslash y_{0} }} {{\text{E}}_{Y} \left[ {\frac{\mu }{{\mu + sp_{d} \left\| {x_{0} + y} \right\|^{ - \alpha } }}} \right]} } \right] \\ \mathop = \limits^{\left( b \right)} & \sum\limits_{k = 0}^{M - 1} {\left( {\int\limits_{{R^{2} }} {\frac{\mu }{{\mu + sp_{d} \left\| {x_{0} + y} \right\|^{ - \alpha } }}f_{Y} \left( y \right)dy} } \right)}^{k} {\text{P}}\left( {{\text{K}} = k{\text{|K}} < M{ - 1}} \right) \\ = & \sum\limits_{k = 0}^{M - 1} {\left( {\int\limits_{{R^{2} }} {\frac{\mu }{{\mu + sp_{d} \left\| {x_{0} + y} \right\|^{ - \alpha } }}f_{Y} \left( y \right)dy} } \right)}^{k} \frac{{\left( {m - 1} \right)^{k} }}{k!}\frac{{e^{{ - \left( {m - 1} \right)}} }}{\xi },\xi {\text{ is }}\sum\limits_{j = 0}^{M - 1} {\frac{{\left( {m - 1} \right)^{j} e^{{ - \left( {m - 1} \right)}} }}{j!}} \\ \mathop = \limits^{\left( c \right)} & \exp \left( {\left( {m - 1} \right)\int\limits_{{R^{2} }} {\frac{\mu }{{\mu + sp_{d} \left\| {x_{0} + y} \right\|^{ - \alpha } }}f_{Y} \left( y \right)dy} + \left( { - \left( {m - 1} \right)} \right)} \right) \\ = & \exp \left( { - \left( {m - 1} \right)\int\limits_{{R^{2} }} {\frac{{sp_{d} \left\| {x_{0} + y} \right\|^{ - \alpha } }}{{\mu + sp_{d} \left\| {x_{0} + y} \right\|^{ - \alpha } }}f_{Y} \left( y \right)dy} } \right) \\ \mathop = \limits^{\left( d \right)} & \exp \left( { - \left( {m - 1} \right)\int_{0}^{\infty } {\frac{{sp_{d} r_{d1}^{ - \alpha } }}{{\mu + sp_{d} r_{d1}^{ - \alpha } }}f_{{R_{d1} }} \left( {r_{d1} /v_{0} } \right)dr_{d1} } } \right) \\ \end{aligned} $$

where (a) represents the expectation of h∼exp(μ) and (b) represents the expectation of the number of interfering devices in each cluster, (c) under the assumption $M > > m$, the formula is simplified based on the Taylor formula, and (d) follows from the change of variable $\left\| {x_{0} + y} \right\| = r_{d1}$.

Appendix B: Proof of Eq. (16)

Laplace transform of the interference at the typical D2D Rx from inter-cluster interfering D2D devices $\text{L}_{{\text{I}_{{{\text{d2d}}}}^{{{\text{inter}}}} }} \left( s \right)$ is

$$ \begin{aligned} \text{L}_{{\text{I}_{{{\text{d2d}}}}^{{{\text{inter}}}} }} \left( s \right) = & {\text{E}}_{{\text{I}_{{{\text{d2d}}}}^{{{\text{inter}}}} }} \left[ {\exp \left( { - s\text{I}_{{{\text{d2d}}}}^{{{\text{inter}}}} } \right)} \right] \\ = & {\text{E}}\left[ {\exp \left( { - s\sum\limits_{{x \in \Phi_{C} }} {\sum\limits_{{y \in \text{A}^{\text{X}} }} {P_{d} h\left\| {x + y} \right\|^{ - \alpha } } } } \right)} \right] \\ = & {\text{E}}_{{\Phi_{C} }} \left[ {\prod\limits_{{x \in \Phi_{C} }} {{\text{E}}_{{\text{A}^{x} }} \left[ {\prod\limits_{{y \in \text{A}^{\text{X}} }} {\left[ {{\text{E}}_{h} \left[ {\exp \left( { - sP_{d} h\left\| {x + y} \right\|^{ - \alpha } } \right)} \right]} \right]} } \right]} } \right] \\ \mathop = \limits^{\left( a \right)} & {\text{E}}_{{\Phi_{C} }} \left[ {\prod\limits_{{x \in \Phi_{C} }} {{\text{E}}_{{\text{A}^{x} }} \left[ {\prod\limits_{{y \in \text{A}^{\text{X}} }} {{\text{E}}_{Y} \left[ {\frac{\mu }{{\mu + sp_{d} \left\| {x + y} \right\|^{ - \alpha } }}} \right]} } \right]} } \right] \\ = & {\text{E}}_{{\Phi_{C} }} \left[ {\prod\limits_{{x \in \Phi_{C} }} {{\text{E}}_{{\text{A}^{x} }} \left[ {\prod\limits_{{y \in \text{A}^{\text{X}} }} {\int\limits_{{R^{2} }} {\frac{\mu }{{\mu + sp_{d} \left\| {x + y} \right\|^{ - \alpha } }}f_{Y} \left( y \right)dy} } } \right]} } \right] \\ \mathop = \limits^{\left( b \right)} & {\text{E}}_{{\Phi_{C} }} \left[ {\prod\limits_{{x \in \Phi_{C} }} {\sum\limits_{k = 0}^{M} {\left( {\int\limits_{{R^{2} }} {\frac{\mu }{{\mu + sp_{d} \left\| {x + y} \right\|^{ - \alpha } }}f_{Y} \left( y \right)dy} } \right)}^{k} {\text{P}}\left( {{\text{K}} = k{\text{|K}} < M} \right)} } \right] \\ \mathop = \limits^{\left( c \right)} & \exp \left( { - \lambda_{c} \int\limits_{{R^{2} }} {\left( {1 - \sum\limits_{k = 0}^{M} {\left( {\int\limits_{{R^{2} }} {\frac{\mu }{{\mu + sp_{d} \left\| {x + y} \right\|^{ - \alpha } }}f_{Y} \left( y \right)dy} } \right)}^{k} \frac{{m^{k} }}{k!}\frac{{e^{ - m} }}{\eta }} \right) \cdot 2\pi vdv} } \right),\eta {\text{ is }}\sum\limits_{j = 0}^{M} {\frac{{m^{j} e^{ - m} }}{j!}} \\ \mathop = \limits^{\left( d \right)} & \exp \left( { - 2\pi \lambda_{c} \int_{0}^{\infty } {1 - \exp \left( { - m\left( {1 - \int\limits_{{R^{2} }} {\frac{\mu }{{\mu + sp_{d} \left\| {x + y} \right\|^{ - \alpha } }}f_{Y} \left( y \right)dy} } \right)} \right)vdv} } \right) \\ \mathop = \limits^{\left( e \right)} & \exp \left( { - 2\pi \lambda_{c} \int_{0}^{\infty } {1 - \exp \left( { - m\left( {1 - \int_{0}^{\infty } {\frac{\mu }{{\mu + sp_{d} r_{d2}^{ - \alpha } }}f_{{R_{d2} }} \left( {r_{d2} /v} \right)dr_{d2} } } \right)} \right)vdv} } \right) \\ & \quad \mathop \ge \limits^{\left( f \right)} \exp \left( { - 2\pi \lambda_{c} \int_{0}^{\infty } {m\int_{0}^{\infty } {\frac{{sp_{d} r_{d2}^{ - \alpha } }}{{\mu + sp_{d} r_{d2}^{ - \alpha } }}f_{{R_{d2} }} \left( {r_{d2} /v} \right)dr_{d2} } vdv} } \right) \\ \mathop = \limits^{\left( g \right)} & \exp \left( { - 2\pi \lambda_{c} m\int_{0}^{\infty } {\frac{{sp_{d} r_{d2}^{ - \alpha } }}{{\mu + sp_{d} r_{d2}^{ - \alpha } }}r_{d2} dr_{d2} } } \right) \\ = & \exp \left( { - \pi \lambda_{c} m\left( {sp_{d} } \right)^{{\frac{2}{\alpha }}} \frac{2\pi /\alpha }{{\sin \left( {2\pi /\alpha } \right)}}} \right) \\ \end{aligned} $$

where (a) represents the expectation of h∼exp(μ) and (b) represents the expectation of the number of interfering devices in each cluster, (c) follows from the probability generating functional (PGFL) of PPP [20], (d) under the assumption $M > > m$, the formula is simplified based on the Taylor formula, (e) follows from the change of variable $\left\| {x + y} \right\| = r_{d2}$, (f) according to Taylor, the approximate $1{\text{ - exp}}\left( { - ax} \right) \le a$ simplification formula is developed, and (g) follows by converting from Cartesian to polar coordinates where $\int_{0}^{\infty } {\frac{1}{{1 + su^{ - \alpha } }}} f_{U} \left( {u/v} \right)du$.

Appendix C: Proof of Eq. (18)

Laplace transform of the interference at the typical D2D Rx from cellular users $\text{L}_{{\text{I}_{{{\text{c2b}}}} }} \left( s \right)$ is

$$ \begin{aligned} \text{L}_{{\text{I}_{{{\text{c2b}}}} }} \left( s \right) = & {\text{E}}_{{\text{I}_{{{\text{c2b}}}} }} \left[ {\exp \left( { - s\text{I}_{{{\text{c2b}}}} } \right)} \right] \\ = & {\text{E}}_{{R_{z} ,h,D_{Z} }} \left[ {\exp \left( { - s\sum\limits_{{z \in \Phi_{CU} }} {P_{cu} hR_{z}^{\alpha \varepsilon } D_{z}^{ - \alpha } } } \right)} \right] \\ = & {\text{E}}_{{R_{z} ,D_{Z} }} \left[ {\prod\limits_{{z \in \Phi_{CU} }} {{\text{E}}_{h} \left[ {\exp \left( { - sP_{cu} hR_{z}^{\alpha \varepsilon } D_{z}^{ - \alpha } } \right)} \right]} } \right] \\ \mathop = \limits^{\left( a \right)} & {\text{E}}_{{R_{z} ,D_{Z} }} \left[ {\prod\limits_{{z \in \Phi_{CU} }} {\int_{0}^{\infty } {\mu \exp \left( { - \left( {sP_{cu} R_{z}^{\alpha \varepsilon } D_{z}^{ - \alpha } + \mu } \right)x} \right)dx} } } \right] \\ = & {\text{E}}_{{R_{z} ,D_{Z} }} \left[ {\prod\limits_{{z \in \Phi_{CU} }} {\frac{\mu }{{sP_{cu} R_{z}^{\alpha \varepsilon } D_{z}^{ - \alpha } + \mu }}} } \right] \\ \mathop = \limits^{\left( b \right)} & \exp \left( { - 2\pi \lambda \int_{0}^{\infty } {1 - {\text{E}}_{{R_{z} }} \left[ {\frac{\mu }{{sP_{cu} R_{z}^{\alpha \varepsilon } x_{{}}^{ - \alpha } + \mu }}} \right]xdx} } \right) \\ = & \exp \left( { - 2\pi \lambda \int_{0}^{\infty } {{\text{E}}_{{R_{z} }} \left[ {\frac{{sP_{cu} R_{z}^{\alpha \varepsilon } x_{{}}^{ - \alpha } }}{{sP_{cu} R_{z}^{\alpha \varepsilon } x_{{}}^{ - \alpha } + \mu }}} \right]xdx} } \right) \\ \mathop = \limits^{\left( c \right)} & \exp \left( { - 2\pi \lambda \int_{0}^{\infty } {\int_{{0}}^{\infty } {\frac{{sP_{cu} R_{z}^{\alpha \varepsilon } x_{{}}^{ - \alpha } }}{{sP_{cu} R_{z}^{\alpha \varepsilon } x_{{}}^{ - \alpha } + \mu }}} f_{{R_{z} }} \left( {r_{z} } \right)xdr_{z} dx} } \right) \\ \end{aligned} $$

where (a) represents the expectation of h∼exp(μ) and (b) follows from the probability generating functional (PGFL) of PPP [24], (c) represents the distance distribution of an interfering user to its closest BS.

Appendix D: Proof of Eq. (21)

Coverage probability of a tagged BS $P_{c}^{\left( c \right)}$ is

$$ \begin{aligned} P_{c}^{\left( c \right)} = & {\text{E}}_{{R_{c} }} \left[ {{\text{P}}\left( {{\text{SIR}}_{C} > \beta |R_{c} } \right)} \right] \\ = & \int_{0}^{\infty } {{\text{P}}\left( {{\text{SIR}}_{C} > \beta } \right)} f_{{R_{c} }} \left( {r_{c} } \right)dr_{c} \\ = & \int_{0}^{\infty } {{\text{P}}\left( {\frac{{P_{cu} hr_{c}^{\alpha \varepsilon } r_{c}^{ - \alpha } }}{{\text{I}_{{{\text{c2b}}}} + \text{I}_{{{\text{d2b}}}} }} > \beta } \right)} f_{{R_{c} }} \left( {r_{c} } \right)dr_{c} \\ = & \int_{0}^{\infty } {{\text{P}}\left( {h > \frac{{\beta \left( {\text{I}_{{{\text{c2b}}}} + \text{I}_{{{\text{d2b}}}} } \right)}}{{P_{cu} r_{c}^{\alpha \varepsilon } r_{c}^{ - \alpha } }}} \right)} f_{{R_{c} }} \left( {r_{c} } \right)dr_{c} \\ \mathop = \limits^{\left( a \right)} & \int_{0}^{\infty } {{\text{E}}_{\text{I}} \left[ {\exp \left( { - \mu \beta \left( {\text{I}_{{{\text{c2b}}}} + \text{I}_{{{\text{d2b}}}} } \right)P_{cu}^{ - 1} r_{c}^{{\alpha \left( {1 - \varepsilon } \right)}} } \right)} \right]} f_{{R_{c} }} \left( {r_{c} } \right)dr_{c} \\ \mathop = \limits^{\left( b \right)} & \int_{0}^{\infty } {\text{L}_{{\text{I}_{{{\text{c2b}}}} }} \left( s \right)\text{L}_{{\text{I}_{{{\text{d2b}}}} }} \left( s \right)f_{{R_{c} }} \left( {r_{c} } \right)dr_{c} } \\ \end{aligned} $$

where (a) represents the expectation of h∼exp(μ) and (b) follows from the change of variable $s = \mu \beta \, r_{c}^{{\alpha \left( {1 - \varepsilon } \right)}} P_{cu}^{ - 1}$.

Appendix E: Proof of Eq. (26)

Average rate of a tagged BS is $R_{{}}^{\left( c \right)}$ is

$$ \begin{aligned} R_{{}}^{\left( c \right)} = & {\text{E}}\left[ {{\text{ln}}\left( {1 + {\text{SIR}}} \right)} \right] \\ = & \int_{0}^{\infty } {\text{P}} \left[ {{\text{ln}}\left( {1 + {\text{SIR}}} \right) > t} \right]dt \\ = & \int_{0}^{\infty } {\int_{0}^{\infty } {\text{P}} \left[ {{\text{ln}}\left( {1 + \frac{{P_{cu} hr_{c}^{\alpha \varepsilon } r_{c}^{ - \alpha } }}{{\text{I}_{{{\text{c2b}}}} + \text{I}_{{{\text{d2b}}}} }}} \right) > t} \right]f_{{R_{c} }} \left( {r_{c} } \right)dr_{c} dt} \\ = & \int_{0}^{\infty } {\int_{0}^{\infty } {\text{P}} \left[ {h > \frac{{\left( {e^{t} - 1} \right)\left( {\text{I}_{{{\text{c2b}}}} + \text{I}_{{{\text{d2b}}}} } \right)}}{{P_{cu} r_{c}^{\alpha \varepsilon } r_{c}^{ - \alpha } }}} \right]f_{{R_{c} }} \left( {r_{c} } \right)dr_{c} dt} \\ \mathop = \limits^{\left( a \right)} & \int_{0}^{\infty } {\int_{0}^{\infty } {{\text{E}}_{\text{I}} \left[ {\exp \left( { - \mu \left( {e^{t} - 1} \right)P_{cu}^{ - 1} r_{c}^{{\alpha \left( {1 - \varepsilon } \right)}} \left( {\text{I}_{{{\text{c2b}}}} + \text{I}_{{{\text{d2b}}}} } \right)} \right)} \right]f_{{R_{c} }} \left( {r_{c} } \right)dr_{c} dt} } \\ \mathop = \limits^{\left( b \right)} & \int_{0}^{\infty } {\int_{0}^{\infty } {\text{L}_{{\text{I}_{{{\text{c2b}}}} }} \left( s \right)\text{L}_{{\text{I}_{{{\text{d2b}}}} }} \left( s \right)f_{{R_{c} }} \left( {r_{c} } \right)dr_{c} dt} } \\ \end{aligned} $$

where (a) represents the expectation of h∼exp(μ) and (b) follows from the change of variable $s = \mu \left( {e^{t} - 1} \right) \, r_{c}^{{\alpha \left( {1{ - }\varepsilon } \right)}} P_{cu}^{ - 1}$.