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Analytical Modeling of Mode Selection and Power Control for Underlay D2D Communication in Cellular Networks

Hesham ElSawy, Ekram Hossain

arXiv:1405.2017v2cs.NI

TL;DR

Underlay D2D communication can improve reuse and capacity but creates cross-mode interference, motivating tractable analysis of uplink network performance. The paper combines stochastic-geometry modeling, flexible bias-based mode selection using both link qualities, and truncated channel inversion power control. Results indicate that the proposed selection scheme outperforms distance-only selection, while bias and cutoff thresholds govern important performance tradeoffs.

  • Problem

    Underlay D2D communication creates cross-mode interference, requiring analytical models that quantify uplink performance while accounting for power control, maximum transmit power, and mode selection.

  • Method

    The paper uses stochastic geometry and PPP models to analyze truncated channel inversion power control with a bias-based mode-selection scheme that considers D2D and cellular link quality.

  • Results

    The proposed mode selection scheme outperforms traditional selection based only on D2D link distance, while bias and cutoff thresholds produce tradeoffs in SINR outage, transmit power, and capacity.

  • Takeaways & Limitations

    Bias factor and power-control cutoff threshold are key design parameters for balancing D2D traffic offloading, interference protection, SINR outage, and truncation outage.

Abstract

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Device-to-device (D2D) communication enables the user equipments (UEs) located in close proximity to bypass the cellular base stations (BSs) and directly connect to each other, and thereby, offload traffic from the cellular infrastructure. D2D communication can improve spatial frequency reuse and energy efficiency in cellular networks. This paper presents a comprehensive and tractable analytical framework for D2D-enabled uplink cellular networks with a flexible mode selection scheme along with truncated channel inversion power control. Different from the existing mode selection schemes where the decision on mode selection is made based only on the D2D link distance (i.e., distance between two UEs using D2D mode of communication), the proposed mode selection scheme for a UE accounts for both the D2D link distance and cellular link distance (i.e., distance between the UE and the BS). The developed framework is used to analyze and understand how the underlaying D2D communication affects the cellular network performance. Through comprehensive numerical analysis, we investigate the expected performance gains and provide guidelines for selecting the network parameters.

I. INTRODUCTION

The paper develops a tractable stochastic-geometry framework for underlay D2D uplink networks, combining flexible mode selection with power control to analyze interference and performance. Its mode-selection scheme considers both D2D and cellular link quality, and numerical results examine performance gains and parameter tradeoffs.

  • Motivation: Underlay spectrum sharing improves reuse but introduces cross-mode interference that can degrade SINR and limit network performance.The framework accounts for power control, maximum UE transmit power, and mode selection in an uplink cellular network.
  • Contribution: The proposed mode-selection scheme compares biased D2D link quality with cellular uplink quality, using bias factor T_d to control the decision.Disabling D2D, enforcing D2D, and distance-based selection arise as special cases through extreme bias settings or model parameters.
  • Contribution: The paper develops a tractable analytical framework evaluating SINR outage, average transmit power, average link capacity, and total network capacity.The analysis combines the proposed selection scheme with truncated channel inversion-based power control and uses stochastic geometry with PPP network models.
  • Results: The proposed scheme outperforms traditional D2D-link-distance-only selection, while bias factor and cutoff threshold variation creates tradeoffs among transmit power, SINR outage, and link capacity.The paper uses numerical analysis to study these tradeoffs and provide network-parameter guidance.
  • Modeling scope: The model avoids the circular approximation for BS coverage and includes maximum transmit power constraints in a multi-cell uplink environment.These modeling choices address limitations associated with circular coverage approximations in uplink analysis.

B. Radio Channel Model

The model uses power-law path loss with distinct cellular and D2D exponents, nearest-BS association, biased mode selection, and truncated channel inversion. These choices classify UEs into covered or uncovered and cellular, potential-D2D, or non-potential subsets.

  • Radio channel model: Signal power decays as r^-η with η > 2, using separate exponents ηc and ηd for cellular and D2D links.
  • Association: UEs associate with their nearest BS, while D2D receivers may lie outside the transmitter’s serving cell.
  • Mode selection: A potential D2D UE selects D2D mode when Td r_d^-ηd ≥ r_c^-ηc; Td = 0 disables D2D, whereas Td = ∞ forces potential D2D UEs into D2D mode.
  • Interference protection: The mode-selection criterion correlates D2D-transmitter locations and link distances with BS locations, providing inherent interference protection for cellular uplinks.
  • UE classification: The UE population is partitioned into four non-overlapping subsets based on BS coverage and potential-D2D status, with uncovered non-potential UEs transmitting nothing.

IV. ANALYSIS OF TRANSMIT POWER OF UES

The analysis derives transmit-power distributions and moments for UEs under truncated channel inversion, conditioning on their mode and classification. Potential D2D UEs operating cellularly receive a separate conditional power characterization.

  • Transmit-power cases: Case #1 UEs do not transmit, so their transmit power is P1 = 0.
  • D2D mode: D2D-mode UEs use Pd = ρo r_d^ηd subject to the mode-selection condition r_d^ηd ≤ Td r_c^ηc.
  • D2D mode: The paper gives a pdf and moments for transmit power in the D2D mode under truncated channel inversion and bias factor Td.
  • Cellular mode: Cellular-mode UEs include non-potential covered UEs and potential D2D UEs whose cellular link quality exceeds their biased D2D link quality.
  • Cellular mode: For case #2, cellular transmit power follows P2 = ρo r_c^ηc under the maximum-power constraint and truncated channel inversion.
  • Aggregate cellular power: The generic cellular-mode power combines mutually exclusive case #2 and case #4 contributions, allowing its moments to be obtained from their conditional moments.

V. ANALYSIS OF SINR

The SINR analysis models cellular and D2D interference separately and derives tractable outage and capacity expressions under PPP approximations. The resulting theorems provide outage probabilities for both link types.

  • SINR formulation: For mode χ ∈ {c, d}, SINR is evaluated at the associated receiver using the intended signal and aggregate cellular and D2D interference.
  • Interference model: Interference from cellular-mode and D2D-mode UEs is represented by separate point processes, ˜Φc and ˜Φd.
  • Outage analysis: SINR outage is computed through Laplace transforms of the aggregate cellular and D2D interference.
  • Analytical approximation: The analysis approximates the non-tractable softcore and Poisson hole interferer processes by equi-dense PPPs with independent interfering transmit powers.
  • Outage results: Theorem 1 gives the outage probability for a generic D2D UE, while Theorem 2 gives the corresponding outage probability for a generic cellular UE.
  • Capacity analysis: Average link capacity is derived from the SINR distribution using Shannon’s logarithmic rate expression and interference Laplace transforms.

A. System Parameters and Model Validation

The model is validated against simulation before numerical evaluation. Increasing the power-control cutoff threshold reduces SINR outage while increasing average transmit power.

  • Parameters: The default evaluation uses λ = 5 BSs/km2, U = 50 UE/km2, Pu = 1 W, ηc = ηd = 4, Td = 1, and θ = 1.
  • Simulation setup: Simulations realize PPP cellular networks and UEs, schedule transmissions according to coverage, mode selection, and same-cell channel constraints, and repeat each scenario 10000 times.
  • Model validation: Fig. 4 shows that the analytical model accurately captures SINR outage in simulation.
  • Cutoff-threshold effects: Increasing ρo decreases SINR outage because scheduled D2D intensity falls and the desired received signal becomes stronger relative to interference.
  • Cutoff-threshold effects: Increasing ρo also increases average UE transmit power under channel inversion, while the useful-signal improvement dominates its interference contribution.

B. Proposed Scheme vs. Distance-Based Mode Selection Scheme

The proposed bias-based mode selection outperforms distance-only selection at matched admitted D2D-link intensity, while the bias factor and cutoff threshold create explicit performance trade-offs. Appropriate parameter choices can improve rate, capacity, and transmit power, but excessive D2D activation or cutoff levels worsen outage or truncation.

  • Proposed Scheme vs. Distance-Based Mode Selection Scheme: At equal admitted D2D-link intensity, the proposed scheme yields lower cellular SINR outage probability and lower average D2D transmit power than distance-based selection.The comparison uses the point where both schemes admit the same D2D-link intensity.
  • Proposed Scheme vs. Distance-Based Mode Selection Scheme: The proposed criterion protects cellular BSs by selecting D2D mode only when received power at the nearest BS is below Tdρo.Distance-based selection neglects the cellular link distance and therefore does not impose the same BS-protection condition.
  • Design Parameters: Increasing Td raises cellular SINR outage by increasing D2D interferer intensity and reducing interference protection around BSs.For Td > 1, D2D UEs also require more power to invert channels toward D2D receivers than toward the nearest BS.
  • Design Parameters: An optimal Td maximizes the expected rate of a generic potential D2D UE and the total network capacity, whereas excessive Td causes high D2D interference and poor SINR.At low Td, most potential D2D UEs use cellular mode; increasing Td enables more exclusive D2D channel use before interference dominates.
  • Design Parameters: Td = 1 minimizes UE transmit powers because each potential D2D UE selects the mode requiring less power for channel inversion.Td = 0 disables D2D communication, while Td = ∞ forces all potential D2D UEs into D2D mode.
  • Design Parameters: Increasing ρo improves SINR outage and expected link capacity but increases average transmit power and truncation outage.The useful-signal gain from a higher cutoff dominates the accompanying increase in aggregate interference, while the assumed uniform D2D-distance distribution makes D2D truncation especially sensitive.

D. Discussions

The proposed biasing-based mode selection scheme improves D2D-enabled uplink performance relative to distance-only selection while exposing tradeoffs governed by the bias factor and power-control cutoff threshold. The analysis also identifies pessimistic assumptions and extensions relevant to practical deployment.

  • Design parameters: Increasing the cutoff threshold ρ_o lowers SINR outage probability and raises link capacity, but requires higher transmit power and increases truncation outage probability.A relatively low threshold can limit truncation outage, while interference management can improve SINR performance.
  • Scope and assumptions: The framework evaluates a pessimistic setting using PPP cellular modeling, one-channel sharing, aggressive D2D access, and uniformly distributed D2D receivers.These assumptions increase cross-mode and intra-mode interference and make larger D2D link distances more likely.
  • Interference effects: Enforcing all potential D2D UEs into D2D mode can offset spatial frequency reuse gains through increased interference and degraded SINR performance.For T_d > 1, an individual D2D transmitter contributes interference power T_dρ_0 at a BS.
  • Scope and extensions: Although developed for single-tier networks, the framework can be extended systematically to multi-tier networks with differing cutoff thresholds and path-loss exponents.With common cutoff thresholds and path-loss exponents, the multi-tier analysis reduces to the single-tier case.
  • Design parameters: The bias factor T_d controls both cellular interference protection and the intensity of enabled D2D communication.An optimal value depends on the network objective.

APPENDIX A PROOF OF LEMMA 1

The appendix derives the D2D-mode selection probability and conditional transmit-power distribution from the cellular and D2D link-distance distributions under truncated channel inversion and a maximum-power constraint.

  • Distance distributions: The D2D link distance has pdf f_r_d(r)=2r/R^2 for 0≤r≤R, while the cellular link distance is Rayleigh distributed under the PPP base-station model.These distributions provide the inputs for subsequent transmit-power transformations.
  • Power transformation: Required channel-inversion powers are defined as X_c=ρ_o r_c^η_c and X_d=ρ_o r_d^η_d for cellular and D2D links, respectively.The variables represent unconditional powers needed to invert the corresponding channels.
  • Power distribution: The pdfs of X_c and X_d are obtained by transforming the cellular and D2D link-distance distributions.The D2D transformed power has support bounded by ρ_oR^η_d.
  • Mode selection: The D2D mode is selected when X_d≤T_dX_c, subject to the maximum transmit-power constraint P_d≤P_u.The conditional D2D transmit-power pdf is then obtained for users operating in D2D mode.

APPENDIX C PROOF OF LEMMA 4

This appendix derives the distribution and moments of a truncated cellular transmit-power variable, accounting for its finite support and the bias factor used in mode selection.

  • Variable transformation: The transformed cellular variable X̃_c=ρ_o r̃_c^η_c has a pdf obtained from the cellular link-distance distribution.The transformed variable is used in the mode-selection probability derivation.
  • Mode selection: Because X̃_c and X_d have finite support [0,P_u], the bias factor T_d affects the mode-selection probability.The finite support follows from the maximum transmit-power constraint.
  • Case analysis: For T_d>1, the appendix derives a corresponding case expression and combines it with the complementary case to obtain the target result.The combined expression is identified as equation (6).
  • Moments: The moments of P̃_4 are computed by integrating x^α f_P̃_4(x) over its support.The moment expression is stated in the appendix after the distribution derivation.

APPENDIX D PROOF OF THEOREM 1

The appendix derives Laplace transforms for aggregate interference at a D2D receiver by approximating interfering transmitter locations with Poisson point processes and applying PPP functional tools.

  • D2D interference: The D2D-interference Laplace transform is derived by modeling interfering D2D UEs as a PPP rather than a Poisson hole process.This approximation supplies a tractable expression for aggregate interference.
  • Derivation: Independence among the interfering-transmitter process, transmit powers, and fading enables expectation factorization in the transform derivation.The PPP moment generating functional is then used to obtain the approximate transform.
  • Cellular interference: The cellular-interference transform models interfering uplink UEs as a PPP with intensity λ and independent transmit powers.The intensity λ results from scheduling one user per base station to avoid intra-cell interference.
  • Model scope: The analysis uses η_d as the path-loss exponent between UEs and invokes Slivnyak’s theorem to generalize the transforms to a generic D2D receiver location.The resulting transforms apply to any D2D receiver in a generic position.

APPENDIX E PROOF OF THEOREM 2

The theorem is proved by deriving approximate Laplace transforms for cellular and D2D interference at a base station. The derivation uses PPP and power-independence approximations, while incorporating interference protection from mode selection.

  • The proof first derives approximate LTs for interference from cellular UEs and aggregate interference from D2D transmitters at a BS at the origin.Orthogonal channel assignment creates correlations among interfering cellular UE locations, complicating the analysis.
  • Nearest-BS association and truncated channel inversion ensure that the average interference from each cellular interfering UE is strictly less than ρo.
  • Because each BS assigns a unique channel to each UE, the interfering UE intensity is λ.
  • The cellular interference approximation assumes interfering UEs form a PPP with independent transmit powers.
  • For D2D interference, the approximation replaces a Poisson hole process with a PPP while bounding each individual interferer's interference by Tdρo.
  • Slivnyak’s theorem establishes that the obtained LTs apply to a BS in any generic position.
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