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Downlink Cellular Network Analysis with Multi-slope Path Loss Models

Xinchen Zhang, Jeffrey G. Andrews

arXiv:1408.0549v4cs.IT

TL;DR

Standard path loss models assume a single distance exponent even though real environments often exhibit distance-dependent exponents, motivating more accurate alternatives. The paper analyzes multi-slope cellular networks, especially dual-slope models, and derives coverage and throughput behavior under densification. It finds that SIR decreases while SNR increases with density, and that ultra-densification has a phase transition governed by the near-field exponent α0.

  • Problem

    Standard path loss models inadequately capture distance-dependent exponents and can produce unrealistic received and interference powers in cellular networks.

  • Method

    The paper models downlink base stations with a Poisson point process, uses multi-slope path loss with a dual-slope focus, and derives SIR, SNR, SINR, and throughput expressions.

  • Results

    SIR decreases and SNR increases with network density, so SINR coverage is maximized at finite density; under ultra-densification, throughput grows unboundedly for α0 > 1 but tends to zero for α0 < 1.

  • Takeaways & Limitations

    Multi-slope path loss models revise conclusions about densification and indicate that practical cases with α0 > 1 can remain scalable without intelligent scheduling.

Abstract

from arXiv · show

Existing cellular network analyses, and even simulations, typically use the standard path loss model where received power decays like $\|x\|^{-α}$ over a distance $\|x\|$. This standard path loss model is quite idealized, and in most scenarios the path loss exponent $α$ is itself a function of $\|x\|$, typically an increasing one. Enforcing a single path loss exponent can lead to orders of magnitude differences in average received and interference powers versus the true values. In this paper we study \emph{multi-slope} path loss models, where different distance ranges are subject to different path loss exponents. We focus on the dual-slope path loss function, which is a piece-wise power law and continuous and accurately approximates many practical scenarios. We derive the distributions of SIR, SNR, and finally SINR before finding the potential throughput scaling, which provides insight on the observed cell-splitting rate gain. The exact mathematical results show that the SIR monotonically decreases with network density, while the converse is true for SNR, and thus the network coverage probability in terms of SINR is maximized at some finite density. With ultra-densification (network density goes to infinity), there exists a \emph{phase transition} in the near-field path loss exponent $α_0$: if $α_0 >1$ unbounded potential throughput can be achieved asymptotically; if $α_0 <1$, ultra-densification leads in the extreme case to zero throughput.

I. INTRODUCTION

The paper argues that standard single-slope path loss models can be unrealistic because path loss exponents vary with distance, and develops tractable multi-slope analyses for cellular coverage and throughput. For the dual-slope case, density reduces SIR but increases SNR, producing a finite coverage-maximizing density and a near-field-exponent phase transition in throughput scaling.

  • A. The Case for Multi-Slope Path Loss Models: Distance-dependent path loss exponents make standard single-slope models inaccurate across important propagation environments.The paper notes large errors from replacing distinct regimes with one exponent, including two-ray, dense-network, and millimeter-wave examples.
  • A. The Case for Multi-Slope Path Loss Models: Two-ray propagation changes from α = 2 below Rc ≈ 267 m to α = 4 above it, so using α ≈ 3 creates large errors in both regimes.The threshold uses ht = 10 m, hr = 2 m, and fc = 1 GHz.
  • A. The Case for Multi-Slope Path Loss Models: Millimeter-wave systems are modeled with approximately α0 ≈ 2 for LOS links and α1 ≈ 3.5 for NLOS links, separated by an environment-dependent critical distance.For urban New York City and Chicago, the mean LOS-distance approximation gives Rc ≈ 70 m.
  • B. Contributions: The paper derives integral-form coverage expressions and tight closed-form estimates for multi-slope models with Poisson-distributed base stations.The network model uses a homogeneous PPP for base-station locations and focuses on Rayleigh fading.
  • B. Contributions: For dual-slope models, SIR decreases with density while SNR increases, yielding a finite density that maximizes SINR coverage.This contrasts with SINR invariance under standard path loss models.
  • B. Contributions: Potential throughput scales linearly for α0 > 2, sublinearly for 1 < α0 < 2, and undergoes a phase transition at α0 = 1 under ultra-densification.For α0 < 1, ultra-densification ultimately drives throughput to zero; the conclusions generalize to arbitrary increasing multi-slope exponents.

B. SINR-based Coverage

The paper defines SINR coverage for a typical downlink user and uses coverage density and potential throughput to study network scaling. Under simultaneous densification, potential throughput captures the cell-splitting gain and scales like area spectral efficiency.

  • B. SINR-based Coverage: Coverage probability is the probability that a typical user's received SIR or SINR exceeds target T.The user is associated with the nearest base station, which has the least path loss and highest average received power.
  • B. SINR-based Coverage: The SINR formulation allows any path loss function l and includes receiver-side noise normalized by transmit power and propagation constants.The serving base station is selected as x* ≜ arg maxx l(x).
  • B. SINR-based Coverage: The SINR coverage probability is the complementary cumulative distribution function of SINR at the typical user.The notation may suppress λ and T when they are clear from context.
  • B. SINR-based Coverage: Coverage density and potential throughput are primary metrics for area spectral efficiency under network scaling.Coverage density has units of BSs/Area, while potential throughput has units of bps/Hz/m2.
  • B. SINR-based Coverage: Under simultaneous densification of infrastructure and users, potential throughput and area spectral efficiency have the same scaling.Potential throughput also equals the scaling of coverage density.
  • B. SINR-based Coverage: With standard path loss in the interference-limited case, density does not change the SINR distribution, so potential throughput grows linearly with density.This standard-model result motivates examining how multi-slope path loss changes network scaling.

III. THE GENERAL COVERAGE PROBABILITY EXPRESSIONS

This section develops coverage-probability expressions for general and dual-slope path loss models, then derives consequences for SIR coverage and density dependence.

  • General coverage probability: The stochastic-geometry framework yields an integral-form coverage probability under arbitrary fading and path loss functions.Rayleigh fading is emphasized for tractability, while the general expression supports numerical computation.
  • Dual-slope expressions: The paper simplifies the general coverage expression for dual-slope power-law path loss using Gauss hypergeometric functions.The resulting theorem provides an explicit dual-slope coverage formula and defines Cβ(x), δ0, and δ1.
  • Dual-slope expressions: The dual-slope formula separates coverage contributions according to whether the serving-base-station distance is below or above the critical distance.The two integral intervals arise from a change of variables.
  • SIR coverage: SIR coverage is governed jointly by network density and the critical distance rather than remaining density-invariant as under the standard path loss model.For equal exponents, matching the mean number of near-field base stations makes density and critical-distance effects equivalent; constant coverage requires critical distance to scale with 1/√λ.
  • Interference conditions: Finite interference requires only α1 > 2 under the dual-slope model, because the near-field interfering region is finite and far-field interference must remain bounded.This differs from the usual standard-model requirement α > 2.
  • Special cases: For selected exponent pairs, including [α0 α1] = [2 4], the coverage expressions avoid special functions.The [2 4] case coincides with the well-known two-ray model.

IV. THE INTERFERENCE-LIMITED CASE

Under the dual-slope path loss model, SIR coverage decreases with network density, contrasting with standard-model invariance and creating a critical near-field exponent at α0 = 2.

  • The SIR bounds and ordering hold for arbitrary network topologies and fading realizations, rather than only Poisson networks.The result is attributed purely to the nature of the compared path-loss functions.
  • SIR coverage is ordered below the corresponding standard path-loss coverage under nearest-BS association for arbitrary point processes and fading.The ordering follows from a path-loss comparison and does not depend on statistical assumptions.
  • SIR coverage decreases monotonically as network density increases for arbitrary fading when 0 ≤ α0 ≤ α1.This is formalized by the SIR monotonicity lemma for the dual-slope model.
  • α0 = 2 is the critical exponent separating positive limiting SIR coverage from coverage that decays to zero under densification.For α0 = 3, positive coverage is expected as density grows; for α0 = 2, coverage keeps decreasing to zero.
  • The dual-slope model makes noise important because maximizing SIR by reducing density also drives received signal power toward zero.Thus, the interference-limited approximation is less adequate than under the standard path-loss model.

A. The Tension between SIR and SNR

Under dual-slope path loss, densification creates opposing SIR and SNR trends: interference worsens while received signal strength improves, so SINR coverage peaks at finite density.

  • SNR coverage is analyzed because dual-slope path loss removes the standard-model SINR monotonicity and matters especially in noise-limited systems.The analysis uses the nearest-base-station distance distribution under the Poisson model.
  • For general exponents, SNR coverage lacks a closed form, while α0 = 2 and α1 = 4 admits a simplified expression.The paper also derives a closed-form lower bound for the corresponding SINR coverage.
  • SIR coverage decreases with density, whereas SNR coverage increases, making SINR coverage non-monotonic and maximized at a finite density.SIR and SNR provide upper bounds on SINR coverage.
  • SINR initially increases in the noise-limited regime and then decreases in the interference-limited regime.This explains the finite-density optimum shown for the dual-slope cases.
  • The SINR lower bound uses the Q-function and exponential integral instead of a numerical integral.Simulation comparisons verify its asymptotic tightness as T → 0 and/or λ → 0.

B. Throughput Scaling

Potential throughput can continue growing under ultra-densification even when coverage declines, with scaling determined by the near-field exponent α0 and a phase transition at α0 = 1.

  • α0 > 2 yields potential throughput that grows linearly with BS density as λ → ∞.This scaling holds under full load and extends to noisy networks when interference dominates asymptotically.
  • 1 < α0 < 2 produces sublinear potential-throughput scaling at rate λ^(2−2/α0).Thus, throughput can increase despite vanishing coverage in part of this exponent range.
  • α0 < 1 causes potential throughput to decay to zero under ultra-densification.Theorem 3 identifies α0 = 1 as the phase transition between unbounded and vanishing asymptotic throughput.
  • Theorem 3 excludes α0 = 1 and α0 = 2 because those boundary cases require different proof techniques.By continuity, the authors conjecture linear scaling at α0 = 2 and a finite limit at α0 = 1.
  • Coverage decay for α0 ≤ 2 does not imply decreasing potential throughput, because cell-splitting gain can still scale throughput when α0 > 1.The paper states this range is practically relevant in many cases of interest.

VI. MULTI-SLOPE PATH LOSS MODEL

The multi-slope model extends coverage and throughput analysis beyond dual-slope path loss, with exact coverage expressions and a near-field-exponent phase transition in ultra-dense networks.

  • Coverage probability: Theorem 4 gives an explicit coverage-probability expression for N-slope path loss models, including arbitrarily ordered exponents.For ordered exponents, the dual-slope conclusions extend to the multi-slope case.
  • Throughput scaling: Theorem 6 shows that ultra-dense throughput scaling depends on the near-field exponent α0, while farther-field exponents affect only non-asymptotic SINR.As λ →∞, infinitely many base stations lie in the nearest field, making the asymptotic scaling independent of αn for n ≥1.
  • Throughput scaling: For 1 < α0 < 2, potential throughput scales sublinearly at rate λ^(2−2/α0), while α0 > 2 gives linear scaling with λ.The multi-slope result retains the same critical near-field values as the dual-slope model.
  • Implications: The dual-slope analysis finds that SIR decreases with density, SNR increases, and coverage therefore has a density-dependent optimum.These findings contrast with standard path loss conclusions and motivate scrutiny of that idealized model.
  • Throughput scaling: α0 = 1 is the observed phase transition: α0 < 1 yields zero asymptotic potential throughput, whereas α0 > 1 yields unbounded throughput.At α0 = 1, numerical results suggest convergence to a positive finite value.
  • Implications: The authors argue that multi-slope models can alter predictions for coordination, SIC, HetNet access and load balancing, and D2D efficiency.These implications arise because near-field and far-field links experience different path loss behavior.

APPENDIX A PROOF OF LEMMA 2

The appendix proves Lemma 2 by comparing dual-slope and standard path loss SIR events across serving-link distance regimes and showing the high-density limit through dominated convergence.

  • Case analysis: The proof separates the serving base station into cases inside and outside the critical distance Rc.Each regime permits a direct comparison between the relevant path loss functions and SIR events.
  • Case analysis: For a serving base station within Rc, the dual-slope path loss equals the inner standard model at the serving link and is no larger for interferers.This implication preserves the SIR threshold event under the comparison used in the lemma.
  • Case analysis: For a serving base station beyond Rc, the proof uses the ordering of serving and interfering distances to establish the required SIR comparison.The exponent difference Δα = α1 − α0 enters this case comparison.
  • Asymptotic limit: The high-density limit follows by showing the auxiliary term A(λ, Rc, α0, T) tends to zero via pointwise convergence and the dominated convergence theorem.The integrable dominating function applies for 0 ≤ α0 < 2.

APPENDIX C PROOF OF LEMMA 4

The appendix proves Lemma 4 using radial scaling of a marked homogeneous Poisson point process and monotonicity of the dual-slope path loss under that mapping.

  • Point-process mapping: A radial map f(x) = ax with a > 1 transforms a homogeneous PPP of intensity λ into one of intensity λ/a while preserving iid marks.The mapping is applied to the ground process of the marked point process.
  • Threshold comparison: The proof establishes an indicator inequality comparing threshold events before and after radial scaling.The comparison is checked separately according to whether the serving link remains inside or crosses Rc.
  • Threshold comparison: When the serving distance exceeds Rc, the outer exponent α1 directly determines the scaled SIR comparison.For an inner serving link, the proof accounts for both possible positions relative to Rc after scaling.

APPENDIX D PROOF OF PROP. 2

The appendix derives Proposition 2 by decomposing SINR coverage according to serving distance, lower-bounding the near-serving contribution, and identifying when the bound is asymptotically tight.

  • Coverage decomposition: SINR coverage is decomposed into A and B, the probabilities of coverage from base stations closer to or farther than Rc.The proof lower-bounds A while retaining B exactly in the resulting bound.
  • Lower-bound derivation: The near-serving term is bounded using Jensen’s inequality after expressing the relevant random variable through an exponential-integral representation.The convexity of the exponential function supplies the Jensen step.
  • Asymptotic tightness: As λ → 0, B dominates A, making the lower bound tight in the sparse-network limit.This follows from the equivalence between λ →0 and Rc →0 under the stated fixed quantities.
  • Asymptotic tightness: The bound is also asymptotically tight as T → 0 because both inequalities used in its derivation become tight.The relevant steps are the inequality in (20) and Jensen’s inequality in (22).

APPENDIX E PROOF OF THEOREM 3

The proof analyzes ultra-dense asymptotics by bounding coverage density and establishing convergence properties in the interference-limited regime. It derives a lower-bound scaling for potential throughput and identifies integrability when α0 < 1.

  • As network density tends to infinity, the network becomes interference limited, so the proof sets W = 0; the result also holds with noise.
  • The coverage-density bound combines λA(λ, Rc, α0, T) with a term λ exp(−λπR2c), whose second component vanishes as λ → ∞.
  • The first bound component converges to zero by dominated convergence, using an almost-everywhere vanishing sequence λnfn(·).
  • α0 < 1 ensures the dominating function is integrable on (0, 1), supporting the asymptotic convergence argument.
  • l2(λ, T) = Ω(λ1−δ0) and τl2(λ, T) = Ω(λ2−δ0) = µl2(λ, T) for all T > 0.
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