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Performance Impact of LoS and NLoS Transmissions in Dense Cellular Networks

Ming Ding, Peng Wang, David Lopez-Perez, Guoqiang Mao, Zihuai Lin

arXiv:1503.04251v3cs.IT

TL;DR

Dense small cell networks need models that distinguish LoS and NLoS transmissions because existing analyses often use simplistic path loss models. This paper develops such a model and finds that coverage and ASE vary non-monotonically with BS density, with ASE becoming nearly linear only in ultra-dense regimes.

  • Problem

    Dense SCN deployment requires deeper theoretical understanding, while most prior studies use path loss models that do not differentiate LoS and NLoS transmissions.

  • Method

    The paper analytically evaluates coverage probability and ASE using a path loss model incorporating LoS and NLoS transmissions, including a piece-wise linear approximation for 3GPP Case 2 and an SINR-dependent ASE definition.

  • Results

    Coverage probability first increases and then decreases with BS density; consequently, ASE may slow, decrease, and later grow nearly linearly in ultra-dense networks.

  • Takeaways & Limitations

    LoS/NLoS-aware modeling produces quantitatively and qualitatively different dense-SCN performance results, including an ASE valley at practical 4G/5G densities and near-linear growth beyond λ1.

Abstract

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In this paper, we introduce a sophisticated path loss model incorporating both line-of-sight (LoS) and non-line-of-sight (NLoS) transmissions to study their impact on the performance of dense small cell networks (SCNs). Analytical results are obtained for the coverage probability and the area spectral efficiency (ASE), assuming both a general path loss model and a special case with a linear LoS probability function. The performance impact of LoS and NLoS transmissions in dense SCNs in terms of the coverage probability and the ASE is significant, both quantitatively and qualitatively, compared with the previous work that does not differentiate LoS and NLoS transmissions. Our analysis demonstrates that the network coverage probability first increases with the increase of the base station (BS) density, and then decreases as the SCN becomes denser. This decrease further makes the ASE suffer from a slow growth or even a decrease with network densification. The ASE will grow almost linearly as the BS density goes ultra dense. For practical regime of the BS density, the performance results derived from our analysis are distinctively different from previous results, and thus shed new insights on the design and deployment of future dense SCNs.

I. INTRODUCTION

Dense small cell networks can increase capacity through spatial reuse, but realistic analysis must distinguish LoS and NLoS transmissions. The paper proposes a general model and tractable special-case analysis that reveal densification effects differing qualitatively from simplistic path-loss studies.

  • Motivation: SCNs are pursued to meet rapidly increasing mobile traffic and network-load demands through high spatial spectrum reuse.Network capacity could potentially grow linearly with the number of small cells.
  • Research gap: Most prior SCN studies use simplistic path-loss models that do not differentiate LoS and NLoS transmissions.The paper notes that LoS may occur at short distances, whereas NLoS transmission is common in offices and central business districts.
  • Contributions: The paper proposes a general piece-wise path-loss model with probabilistic LoS and NLoS transmissions and derives analytical results for coverage probability and ASE.The model is intended to apply to several channel models capturing LoS and NLoS transmissions.
  • Contributions: For a linear LoS probability function, the paper derives numerically tractable integral-form expressions for coverage probability and ASE without using a step-function approximation.The resulting analysis is more tractable than the exponential-function treatment described for prior work while representing practical networks more accurately than simplistic models.
  • Main finding: The analysis finds that ASE may grow slowly or decrease with BS densification, a result qualitatively different from studies that do not differentiate LoS and NLoS transmissions.The paper identifies this finding as applying without the assumption of near-field path-loss exponents.

IV. ANALYSIS FOR THE PROPOSED PATH LOSS MODEL

The paper analyzes a typical UE in an HPPP-based SCN using SINR, coverage probability, and SINR-dependent ASE. Its general path-loss model yields analytical coverage results, with LoS, NLoS, and LoS-probability components all affecting performance.

  • Performance measures: The analysis considers a typical UE at the origin and defines coverage probability as the probability that its SINR exceeds threshold γ.The approach uses HPPP properties to study association with the serving BS.
  • System model: SINR combines received power through path loss and Rayleigh fading with aggregate interference and AWGN noise.The serving BS is at distance r, while interfering BSs contribute according to their path loss and fading gains.
  • Performance measures: ASE is defined using the SINR density and a minimum working SINR γ0, producing an SINR-dependent rate rather than a deterministic threshold-based rate.This definition is more realistic but requires one additional numerical integral than the alternative definition in prior work.
  • Analytical results: Theorem 1 derives coverage probability for the proposed path-loss model, whose LoS path loss, NLoS path loss, and LoS probability function all affect the final result.The ASE is then obtained by inserting coverage-related results into its defining expression.
  • Analytical tractability: The general coverage calculation requires three integral folds, and ASE calculation adds another fold.The paper contrasts this general four-fold computation with a more tractable special case.

V. STUDY OF A 3GPP SPECIAL CASE

The paper specializes its general model to 3GPP path-loss and LoS-probability functions, with a linear LoS probability yielding tractable expressions. It also discusses a more complicated 3GPP case and piece-wise linear approximation.

  • Integral evaluation: The computation of the 3GPP Case 1 terms divides LoS and NLoS contributions across distance ranges determined by d1 and related boundaries.Some NLoS calculations require separate cases because one range includes both LoS and NLoS interference while another includes only NLoS interference.
  • 3GPP Case 1: 3GPP Case 1 represents the general model with two path-loss pieces and a linear LoS probability function.The case uses N = 2 and is compatible with dense SCNs because its path-loss and LoS-probability functions apply to small cells.
  • 3GPP Case 2: 3GPP Case 2 uses a more complicated LoS probability function with shape parameters R1 and R2 ensuring continuity.The paper investigates this case numerically using the general theorem.
  • 3GPP Case 1: The linear LoS probability function is used because it provides more tractable results while supporting extensions to complicated practical path-loss models.The paper states that more complicated LoS functions can be approximated by piece-wise linear functions.
  • Integral evaluation: The derived Case 1 expressions use component terms for LoS and NLoS transmission and can be evaluated through the stated lemmas and integral forms.The paper presents separate computations for T_L1, T_NL1, T_L2, and T_NL2.

E. The Results of pcov (λ, γ) and AASE (λ, γ0)

For 3GPP Case 1, the paper combines the derived component terms to compute coverage probability and then ASE. The linear LoS probability function makes these results substantially more tractable than the general model.

  • Coverage probability: Coverage probability for 3GPP Case 1 is assembled from the component terms T_L1, T_NL1, T_L2, and T_NL2.These terms are computed using numerically tractable integral-form expressions.
  • ASE: ASE for 3GPP Case 1 is obtained by inserting the computed coverage probability into the ASE expression.The paper uses the SINR-dependent ASE definition introduced earlier.
  • Tractability: Only one integral fold is required for Case 1 coverage probability, while ASE requires a 2-fold integral computation.The general case requires three folds for coverage and four folds for ASE.

VI. SIMULATION AND DISCUSSION

The numerical study adopts practical 3GPP Case 1 parameters to evaluate dense small cell network performance.

  • The evaluation uses 3GPP Case 1 parameters including d1 = 0.3 km, αL = 2.09, and αNL = 3.75.The parameter set also specifies path loss gains, transmit power, and noise power.

A. Validation of the Analytical Results of pcov (λ, γ) for 3GPP Case 1

For 3GPP Case 1, the proposed LoS/NLoS model matches simulations and predicts a coverage peak followed by density-dependent decline, unlike the single-slope baseline.

  • The proposed analytical coverage results perfectly match simulation results for γ = 0 dB and γ = 3 dB.
  • For λ ≤ 10 BSs/km2, coverage probability increases with BS density in the proposed analysis.This regime is characterized as sparse and noise-limited.
  • For 10 BSs/km2 < λ < 10^3 BSs/km2, coverage probability decreases as density increases because many interference paths transition from NLoS to LoS.Nearby interfering BSs can therefore reach the typical UE through strong LoS paths.
  • For λ ≥ 10^3 BSs/km2, coverage declines more slowly because signal and interference are both LoS-dominated and statistically stable.
  • Coverage peaks at λ0, which can be obtained by setting the partial derivative of pcov(λ, γ) with respect to λ to zero.The reported numerical optima are 19.01 BSs/km2 for γ = 0 dB and 16.52 BSs/km2 for γ = 3 dB.

B. Discussion on the Analytical Results of AASE (λ, γ0) for 3GPP Case 1

The ASE has three density regimes under the proposed LoS/NLoS model: rapid growth when sparse, slowdown or decline at practical densities, and near-linear growth when ultra-dense.

  • The ASE analysis evaluates γ0 = 0, 3, 6 dB using coverage results and compares them with the single-slope analysis from.
  • For λ ≤ λ0, the ASE quickly increases with density because the network is generally noise-limited.In Fig. 2, λ0 is approximately 20 BSs/km2.
  • For λ ≥ λ1, ASE growth accelerates because the declining coverage probability becomes a minor factor relative to increasing density.
  • For λ > 10^3 BSs/km2, ASE follows a nearly linear trajectory because signal and interference are LoS-dominated and statistically stable.
  • The ASE may suffer slow growth or decline above λ0, a conclusion whose generality beyond the investigated parameters requires further study.
  • The practical-density valley complicates evolution from 4G to 5G, while the results provide guidance for cost-efficient network densification.

C. Discussion on Various Values of αL for 3GPP Case 1

ASE degradation in the intermediate-density regime becomes more pronounced as the gap between NLoS and LoS path loss exponents increases.

  • Fig. 3 compares ASE for γ0 = 0 dB across LoS path loss exponents αL and against the single-slope result from.
  • The ASE slowdown or decline over λ ∈ [λ0, λ1] is more pronounced when the difference between αNL and αL is larger.Here λ0 ≈ 20 BSs/km2 and λ1 ≈ 10^2 BSs/km2.
  • With αL = 1.09, ASE decline in the intermediate-density range is significant and hardly recovers after λ1.This observation agrees with prior results for αL < 2.
  • The proposed analysis finds ASE decline even when αL ≥ 2, differing from because the paper uses an SINR-dependent rate definition.The SINR-dependent definition is more realistic but requires one additional numerical-integral fold compared with.

D. Investigation of 3GPP Case 2

For 3GPP Case 2, the paper evaluates ASE using direct numerical integration and a three-piece linear approximation of the LoS probability function. The approximation matches the original model well and preserves the qualitative densification findings.

  • 3GPP Case 2 uses R1 = 0.156 km and R2 = 0.03 km for its LoS probability function.
  • A three-piece linear function approximates the complicated 3GPP Case 2 LoS probability function.The approach extends the tractable linear-LoS analysis to more complicated models.
  • The approximation uses d1 = 0.0184 km and d2 = 0.1171 km, chosen from the original LoS probability curve.
  • Piece-wise linear LoS approximation makes complicated path loss models tractable through the analytical results derived for a linear LoS probability function.
  • The Approximated 3GPP Case 2 results match the original 3GPP Case 2 ASE results well, with only quantitative deviations.
  • Incorporating LoS and NLoS transmissions significantly affects SCN ASE performance, including qualitatively different densification behavior.

APPENDIX A: PROOF OF THEOREM 1

The proof of Theorem 1 derives coverage probability by separating LoS and NLoS association events, obtaining their distance distributions, and conditioning SINR analysis on serving distance and path type.

  • Coverage analysis first derives distance PDFs for UE association with LoS and NLoS serving BSs.
  • LoS and NLoS association events are disjoint, so their coverage contributions are summed directly.
  • The LoS serving-distance PDF is constructed from the nearest LoS-BS event and a conditional event excluding better NLoS competitors.
  • Piecewise distance ranges yield the segment PDFs for LoS and NLoS serving distances.
  • Conditioned SINR calculations use exponential fading and Laplace transforms of interference.
  • The NLoS serving-distance PDF is derived analogously using the nearest NLoS-BS event and exclusion of better LoS competitors.

APPENDIX B: PROOF OF LEMMA 2

The proof of Lemma 2 evaluates the interference Laplace transform by accounting for the relevant LoS and NLoS interferers across distance ranges.

  • For 0 < r ≤ d1, the interference calculation includes both LoS and NLoS paths.
  • The resulting Laplace-transform expression is obtained by substituting the interference terms and applying the definitions of ρ1 and ρ2.
  • The proof concludes after deriving the stated expression for LI_r(s).

APPENDIX C: PROOF OF LEMMA 3

The proof of Lemma 3 derives interference expressions separately for regimes with both LoS and NLoS paths and with NLoS paths only.

  • The analysis considers interference from both LoS and NLoS paths in the applicable distance regime.
  • For r > d1, the interference calculation uses only NLoS paths.
  • The proof is completed by substituting the derived expressions into the relevant lemma equations.
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