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Modeling and Performance Analysis of Clustered Device-to-Device Networks

Mehrnaz Afshang, Harpreet S. Dhillon, Peter Han Joo Chong

arXiv:1508.02668v2cs.ITcs.NI

TL;DR

The paper addresses the need for a realistic spatial model of D2D networks with clustered, proximate devices that may serve one another. It uses a Poisson cluster process to analyze coverage and ASE under uniform and k-closest content availability, finding an optimal activation level and improved outcomes when content is closer to receivers.

  • Problem

    Existing D2D spatial models do not comprehensively capture clustered devices with multiple proximate candidates that can serve a typical receiver.

  • Method

    The paper models device locations with a Poisson cluster process and derives distance distributions, coverage probability, and ASE for uniform and k-closest content availability.

  • Results

    The analysis reveals an optimal number of simultaneously active transmitters per cluster, with the optimum and resulting ASE increasing as content becomes available closer to receivers.

  • Takeaways & Limitations

    Tuning transmitter activation and content placement provides best- and worst-case coverage and ASE characterizations for clustered D2D networks.

Abstract

from arXiv · show

Device-to-device (D2D) communication enables direct communication between proximate devices thereby improving the overall spectrum utilization and offloading traffic from cellular networks. This paper develops a new spatial model for D2D networks in which the device locations are modeled as a Poisson cluster process. Using this model, we study the performance of a typical D2D receiver in terms of coverage probability under two realistic content availability setups: (i) content of interest for a typical device is available at a device chosen uniformly at random from the same cluster, which we term uniform content availability, and (ii) content of interest is available at the $k^{th}$ closest device from the typical device inside the same cluster, which we term $k$-closest content availability. Using these coverage probability results, we also characterize the area spectral efficiency (ASE) of the whole network for the two setups. A key intermediate step in this analysis is the derivation of the distributions of distances from a typical device to both the intra- and inter-cluster devices. Our analysis reveals that an optimum number of D2D transmitters must be simultaneously activated per cluster in order to maximize ASE. This can be interpreted as the classical tradeoff between more aggressive frequency reuse and higher interference power. The optimum number of simultaneously transmitting devices and the resulting ASE increase as the content is made available closer to the receivers. Our analysis also quantifies the best and worst case performance of clustered D2D networks both in terms of coverage and ASE.

I. INTRODUCTION

The paper develops a stochastic-geometry framework for clustered D2D networks, addressing limitations of models that do not capture multiple proximate serving devices. It analyzes coverage and ASE under uniform and k-closest content availability, revealing an optimal activation level and content-location effects.

  • Motivation: Existing stochastic-geometry models often use PPP or fixed-distance receiver models and do not capture multiple proximate devices that can serve a receiver.The paper identifies this as a limitation of prior D2D network modeling approaches.
  • Contributions: The paper introduces a more realistic clustered spatial model in which device locations follow a Poisson cluster process.The model captures devices grouped around cluster centers and allows multiple nearby devices to potentially serve a given device.
  • Analysis: Coverage probability and ASE are characterized for uniform content availability and k-closest content availability.The analysis derives distance distributions for serving and interfering intra- and inter-cluster devices, along with exact expressions, bounds, and approximations.
  • System insights: An optimal number of simultaneously active links per cluster balances aggressive frequency reuse against higher interference power.Activating the optimal number can provide significant gains over the strictly orthogonal strategy with one active link per cluster.
  • System insights: The optimal activation level and resulting ASE increase when content is available closer to receivers.Tuning k in the k-closest strategy characterizes best and worst cases, corresponding to content at the closest and farthest devices.

II. SYSTEM MODEL

The system model represents clustered devices with a Thomas-based Poisson cluster process and analyzes a typical receiver within a representative cluster. It specifies content placement, activation, interference, and communication assumptions for the two availability strategies.

  • Communication assumptions: Each device has content that can be requested by other devices in the same cluster, while cross-cluster communication is outside the paper’s scope.The scope boundary is motivated by larger inter-cluster distances and potentially different content interests across clusters.
  • Spatial model: Device locations follow a Poisson cluster process whose parent process is a PPP and whose cluster members are i.i.d. normally distributed around each center.The model uses a Thomas cluster process and assumes symmetric normal displacement with variance σ^2.
  • Device activity: The total number of devices per cluster is fixed at N, while active-device counts can vary across clusters.For notational convenience, the number of transmitting devices per cluster is limited to M = N/2, though this can be relaxed.
  • Typical-device analysis: A typical device is randomly selected from a randomly selected cluster and treated as a receiving device at the origin.The representative cluster therefore contains a serving device and intra-cluster interferers selected from the remaining transmitting devices.
  • Content availability: Uniform availability selects a serving transmitter uniformly, whereas k-closest availability selects the kth closest transmitter within the same cluster.Small k places content closer to the receiver and large k places it farther away; remaining transmitters generate interference.

B. Channel Model

The channel model represents clustered D2D devices around cluster centers, with Rayleigh fading and interference split into intra- and inter-cluster components. It derives conditional distance distributions for serving and interfering devices under uniform content availability and both content strategies.

  • Cluster and signal model: Devices are modeled around cluster centers, with the representative cluster centered at x0 and simultaneously active transmitters forming the interference field.The typical receiver’s serving transmitter lies inside the representative cluster.
  • Cluster and signal model: The received signal uses transmit power Pd, exponential fading h0, and path-loss exponent α over serving distance r.The analysis assumes Rayleigh fading and later operates in the interference-limited regime.
  • Interference model: Total interference is the sum of independent intra-cluster and inter-cluster interference from simultaneously active D2D transmitters.The intra-cluster term comes from transmitters inside the representative cluster; the inter-cluster term comes from transmitters outside it.
  • Distance distributions: Conditioned on a cluster-center distance ν, inter-cluster interferer distances are i.i.d. with PDF Ricepdf(u, ν; σ).This distribution applies to both uniform and k-closest content availability because inter-cluster devices are selected uniformly.

B. Distance Distributions in the Ordered Case

The ordered case models content at the kth closest device by ordering the intra-cluster distances and using order statistics. It separately characterizes distances to closer and farther intra-cluster interferers relative to the serving device.

  • Ordered serving distance: The unordered intra-cluster distances are ordered increasingly as S(1) ≤ ... ≤ S(k) ≤ ... ≤ S(M), with serving distance R equal to S(k).This ordering makes the kth closest device the predetermined serving transmitter.
  • Ordered serving distance: The conditional serving-distance PDF for k-closest content availability is the kth order-statistic density of the i.i.d. distance sequence.It is expressed using FS(r|ν0), fS(r|ν0), M, and k.
  • Ordered interference distances: Because the kth closest device is fixed as serving, it cannot also act as an interferer, making the ordered interference analysis more involved.The simultaneously active devices are divided according to whether they are closer or farther than the serving device.
  • Ordered interference distances: Conditioned on serving distance r and cluster-center distance ν0, distances to both closer and farther intra-cluster devices are conditionally i.i.d.Their distributions are characterized separately for the two device subsets.

IV. COVERAGE PROBABILITY AND ASE PERFORMANCE

The paper derives coverage probability and ASE for uniform and k-closest content availability using distance distributions and Laplace transforms of intra- and inter-cluster interference. It develops exact results alongside simpler approximations and bounds for design insight.

  • Analysis framework: Coverage probability and ASE are derived for both uniform and k-closest content availability strategies.The analysis uses previously derived intra- and inter-cluster distance distributions.
  • Interference analysis: Laplace transforms of intra- and inter-cluster interference provide intermediate results for the coverage and ASE analysis.Exact expressions are developed before introducing approximations and lower bounds.
  • Approximations and bounds: The analysis introduces an uncorrelated intra-cluster-distance assumption that treats serving and interferer distances as i.i.d. Rayleigh distributed.This simplifies deconditioning by allowing separate treatment of serving and intra-cluster interferer distances.
  • Approximations and bounds: Jensen’s inequality yields a closed-form lower bound on the Laplace transform of intra-cluster interference.The bound is combined with an inter-cluster interference lower bound to obtain closed-form coverage and ASE approximations.
  • Operating regime: The simplified inter-cluster expression assumes M ≫ m̄, corresponding to relatively few simultaneously active devices per cluster.This regime is also identified as one in which ASE is usually optimized, especially for uniform content availability.

2) Coverage Probability:

Coverage probability is defined by whether the typical receiver’s SIR exceeds threshold β. The paper obtains it from the serving-distance distribution and the Laplace transforms of independent intra- and inter-cluster interference.

  • Definition: Coverage probability is the probability that the typical device’s SIR exceeds threshold β for successful demodulation and decoding.The definition is based on the receiver’s SIR exceeding a predetermined threshold.
  • Exact analysis: The exact coverage expression uses the Laplace transforms of intra-cluster and inter-cluster interference.These transforms are supplied by Lemmas 5 and 6.
  • Exact analysis: The derivation separates the independent interference terms and then deconditions over serving distance R and cluster-center distance ν0.The serving-link distribution comes from Corollary 1, while ν0 is Rayleigh distributed because the typical device is Gaussian-distributed around the cluster center.

3) Area Spectral Efficiency:

The paper defines ASE from active-transmitter density, spectral efficiency, and coverage, then optimizes the number of simultaneously active links per cluster. This optimization exposes a tradeoff between increased reuse and increased interference.

  • ASE is the average number of transmitted bits per unit time, bandwidth, and area.
  • ASE equals λ log2(1+β)Pc, where λ is active-transmitter density and Pc is typical-device coverage probability.
  • The clustered-network ASE uses average active-transmitter density ¯mλc and the corresponding coverage probability Pc.
  • Uniform content availability: The number of simultaneously active links has an optimum because more active links increase potential ASE while also increasing interference.The optimum is obtained by numerically solving the ASE maximization problem.
  • Bounds and approximations: Closed-form coverage and ASE approximations are derived from tight bounds and simplified distance-interference analyses.The coverage approximation uses independent de-conditioning over serving and intra-cluster interfering distances, while the reported numerical results show tight agreement.
  • Design observations: ASE and coverage increase for denser clusters, whereas increasing cluster-center density decreases coverage but increases ASE.The latter tradeoff supports increasing cluster-center density while coverage remains acceptable.
  • k-closest content availability: For k-closest availability, k = 1 and k = M characterize the best and worst links, respectively.The corresponding interference transforms are given for the first-closest and Mth-closest serving devices under ¯m ≪ M.

2) Coverage Probability Analysis:

The coverage analysis derives exact expressions for uniform and k-closest content availability using serving-distance distributions and interference Laplace transforms. It also develops simpler approximations for the more involved k-closest case.

  • Exact coverage: The k-closest coverage expression combines the serving-distance distribution with inter- and intra-cluster interference Laplace transforms.
  • Approximation motivation: The exact general-k interference expression is more complicated because serving-device selection creates dependence and involves two summations.
  • ASE: ASE for k-closest availability is parameterized by k, with maximum and minimum ASE achieved at k = 1 and k = M, respectively.
  • Approximations: Two coverage approximations simplify the k-closest analysis by ignoring serving-device exclusion or intra-cluster distance correlation.The first treats intra-cluster interferers as uniformly selected; the second uses a simpler interference transform under the additional independence assumption.
  • Approximation validation: The numerical study reports that the k-closest approximations are remarkably tight and can serve as proxies for the exact result.

V. RESULTS AND DISCUSSION

Simulation results validate the analytical coverage expressions and show that intra-cluster interference strongly governs coverage. They also reveal stable ASE optima and systematic effects of cluster spread and center density.

  • Coverage interference: Coverage is strongly dictated by intra-cluster interference; without inter-cluster interference, it is independent of scattering variance.The independence follows from counter-balancing effects on desired-link quality and intra-cluster interference.
  • Coverage interference: Smaller scattering variance increases coverage under both inter-cluster-only and total-interference conditions.
  • Cluster density: With σ = 50, reducing cluster-center density improves coverage, while increased cluster variance reduces it and has the dominant effect.
  • ASE trends: Smaller scattering variances produce higher ASE, highlighting the importance of short-range D2D communication.
  • ASE trends: ASE increases with cluster-center density because the active-transmitter-density gain outweighs the associated inter-cluster-interference penalty.
  • Optimal activation: Figures 4 and 6 show the same optimal average number of simultaneously active transmitters per cluster across scattering variances.The stability is attributed to the dominant role of intra-cluster interference in the ASE tradeoff.

1) Best and worst link analyses:

The paper analyzes clustered D2D networks under best- and worst-link content availability, showing how serving distance, active transmitters, and approximations shape coverage and ASE. It develops a Poisson-cluster framework with distance distributions and exact or approximate performance expressions.

  • Validation: Analytical and simulation results match perfectly for the k-closest content availability case.
  • Best and worst links: Larger cluster size M decreases the minimum serving distance for the best link but increases the maximum serving distance for the worst link.This order-statistics effect gives the two cases conflicting responses to M.
  • Content availability: Both coverage probability and ASE increase significantly as the serving device becomes closer, corresponding to smaller k.The optimum number of simultaneously active D2D transmitters also increases for smaller k.
  • Framework: The proposed framework models clustered D2D devices with a Poisson cluster process and derives distance distributions for the Thomas cluster process.These distributions support analysis of more general wireless networks with clustered nodes.
  • Framework: The paper derives exact coverage-probability and ASE expressions, together with closed-form bounds and approximations, for uniform and k-closest content availability.It also characterizes distance distributions from a typical device to intra- and inter-cluster devices.
  • ASE optimization: The optimal number of simultaneously active transmitters maximizes ASE through a tradeoff between higher interference power and more aggressive frequency reuse.

APPENDIX

The appendix derives distance distributions for intra-cluster devices and uses them to characterize interference through conditional and unconditional Laplace transforms.

  • Distance distributions: A uniformly selected device is represented as z = x0 + y, with distance S = ∥z∥ from the typical device.Conditioning on the cluster-center distance ν0 enables derivation of fS(s|ν0).
  • Distance distributions: Conditioned on x0, the Cartesian coordinates of z are Gaussian with means x1 and x2 and variance σ2.The transformation z1 = s cos θ and z2 = s sin θ converts the joint distribution into a distance distribution.
  • Order statistics: For k-closest availability, ordered intra-cluster distances use order-statistics distributions, with nearer interfering devices characterized relative to the serving distance.The derivation uses symmetry and the serving-link distribution, while uniform selection removes the permutation factor.
  • Interference transforms: The intra-cluster interference Laplace transform averages Rayleigh fading, device locations, and the conditioned number of interfering devices.Under m̄ ≪ M, it reduces using the distance distribution and a change to polar coordinates.
  • Interference transforms: The unconditional intra-cluster interference transform is obtained by de-conditioning over the cluster-center location and applying Jensen’s, convolution, and Young’s inequalities.The distance-variable change and Corollary 2 provide the distributional basis for this step.

E. Proof of Lemma 6

This proof derives the inter-cluster interference transform and a lower bound, then handles disjoint intra-cluster interferer sets using conditioned device counts and independent distances.

  • Inter-cluster interference: The inter-cluster interference Laplace transform averages fading and interfering-device counts using the PPP probability generating functional.The derivation then converts the resulting expression to polar coordinates.
  • Inter-cluster interference: Under m̄ ≪ M, the inter-cluster interference transform is simplified using the stated sparsity assumption.The approximation is explicitly conditioned on the mean cluster size being much smaller than M.
  • Lower bound: A lower bound follows from the exponential Taylor expansion, the inequality 1 − exp(−ax) ≤ a, and a Rician distribution property.The distance density fU(u|ν) is supplied by Lemma 2.
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