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Performance Analysis of mmWave Ad Hoc Networks

Andrew Thornburg, Tianyang Bai, Robert W. Heath

arXiv:1412.0765v2cs.IT

TL;DR

Ad hoc networks face interference from nearby transmissions, while the outdoor performance of mmWave systems with blockage and directional antennas remains to be established. The paper uses stochastic geometry to analyze one-way and two-way mmWave ad hoc networks and their SINR and INR behavior. It finds higher transmission capacity and area spectral efficiency than UHF under certain conditions, with optimized two-way allocation retaining substantial one-way performance.

  • Problem

    Outdoor mmWave ad hoc network performance incorporating directional antennas and building blockage is not yet understood, despite ad hoc networks’ interference limitations and mmWave bandwidth advantages.

  • Method

    The paper uses stochastic geometry to model one-way and two-way mmWave ad hoc networks with blockage, antenna alignment, interferer and user positions, and slotted ALOHA access.

  • Results

    MmWave networks can improve on UHF performance and efficiency for LOS and NLOS communication; optimized two-way allocation achieves 75% of one-way capacity and twice equal-allocation efficiency.

  • Takeaways & Limitations

    LOS-aware protocols and LOS interference mitigation are motivated because NLOS interference is negligible while LOS interference can remain limiting.

Abstract

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Ad hoc networks provide an on-demand, infrastructure-free means to communicate between soldiers in war zones, aid workers in disaster areas, or consumers in device-to-device (D2D) applications. Unfortunately, ad hoc networks are limited by interference due to nearby transmissions. Millimeter-wave (mmWave) devices offer several potential advantages for ad hoc networks including reduced interference due to directional antennas and building blockages, not to mention huge bandwidth channels for large data rates.. This paper uses a stochastic geometry approach to characterize the one-way and two-way signal-to-interference ratio distribution of a mmWave ad hoc network with directional antennas, random blockages, and ALOHA channel access. The interference-to-noise ratio shows that a fundamental limitation of an ad hoc network, interference, may still be an issue. The performance of mmWave ad hoc networks is bounded by the transmission capacity and area spectral efficiency. The results show that mmWave networks can support much higher densities and larger spectral efficiencies, even in the presence of blockage, compared with lower frequency communication for certain link distances. Due to the increased bandwidth, the rate coverage of mmWave can be much greater than lower frequency devices.

I. INTRODUCTION

The paper addresses interference and limited understanding in outdoor mmWave ad hoc networks by modeling blockage, directionality, and user randomness, then comparing performance with lower-frequency UHF networks.

  • Motivation: Outdoor mmWave ad hoc network performance with directional antennas and building blockage remains insufficiently understood despite potential interference and bandwidth advantages.Ad hoc networks require high rates and broad coverage but face uncoordinated interference from nearby transmitters.
  • Approach: The paper formulates mmWave ad hoc network performance using stochastic geometry with random blockage, antenna alignment, interferer position, and user position.The framework also compares mmWave performance against a lower-frequency UHF ad hoc network.
  • Contributions: The analysis derives a bound for the SINR CCDF, strengthens prior LOS results, extends the bound to NLOS communication, and uses a Taylor approximation for transmission capacity and area spectral efficiency.It also evaluates random receiver location and motivates LOS-aware protocols based on the performance increase from LOS communication.
  • Contributions: The paper computes the INR distribution and studies its behavior when the network operates at transmission capacity.This targets the transition between noise-limited and interference-limited operation in mmWave ad hoc networks.
  • Contributions: Optimal bandwidth allocation is shown to produce large gains in transmission capacity and area spectral efficiency for two-way communication.The paper characterizes how two-way communication changes both performance metrics.
  • Network Model: The network model uses a homogeneous PPP for transmitters, fixed-length dipoles, constant transmit power, and synchronized slotted ALOHA access.The analysis focuses on a typical dipole pair whose performance characterizes the network through Slivnyak’s theorem.

C. Blockage Model

The model represents blockage as independent random buildings, producing LOS or NLOS links with different path-loss exponents. It then derives a stochastic-geometry SINR distribution bound by decomposing interference across blockage and antenna-gain classes.

  • Outdoor links are modeled as either LOS or NLOS, with each condition assigned a different path-loss exponent.
  • Directional beamforming is approximated with a sectored antenna model, while Nakagami fading provides a gamma-distributed received-power model.
  • Random buildings form an independent Poisson point process with random widths, lengths, and orientations, yielding LOS probability P[LOS] = e−βd.
  • The analysis assumes outdoor transmitters and independently determines blockage on each link, while noting that ignoring blockage correlations may affect SINR tails but has small practical-range impact.
  • D. SINR: The SINR combines aligned desired-signal gain, fading, path loss, aggregate interference, and noise for a fixed dipole link.
  • SINR Distribution: Interference is decomposed into independent sub-processes indexed by antenna gain and LOS/NLOS status, whose Laplace transforms are multiplied.
  • SINR Distribution: Theorem 1 tightly upper-bounds the SINR distribution by combining conditional LOS and NLOS terms with noise and LOS/NLOS interference components.

B. Validation of the Model

Monte Carlo simulations validate the analytical SINR model across sparse and dense mmWave ad hoc networks. The bound matches simulations well, while denser and longer-link cases exhibit low-SINR plateaus.

  • The analytical model uses Monte Carlo simulation over a 4km2 Poisson point process, 500MHz thermal noise of −117dB, and Nakagami parameter Nh = 3.
  • At λ = 5 × 10−5m−2, corresponding to 50 users/km2, Theorem 1 matches the empirical SINR distribution extremely well.
  • At λ = 5 × 10−4m−2, corresponding to 500 users/km2, the analytical result again matches simulation and shows bimodal CCDF behavior for larger link distances.
  • For larger link distances in the dense network, CCDF plateaus appear around −10dB.

C. LOS Protocol-Gain

Restricting communication to LOS links substantially improves SINR, motivating LOS-aware protocols. The improvement increases with link distance and can be especially large in denser networks.

  • Interference shifts most NLOS-link SINR probability toward very low values in dense networks, preventing communication except at thresholds below −20dB.
  • Assuming LOS communication sets P[LOS] = 1 and motivates protocols such as LOS relaying around buildings.
  • 20dB improvement is observed for a 25m LOS link in the denser network, motivating the term LOS protocol-gain.

D. Distributions of r

The fixed-length dipole assumption is extended to random receiver distances using location-density functions. Uniform and Rayleigh receiver geometries are compared with the fixed 25 m link setting to assess SINR effects.

  • D. Distributions of r: The dipole model fixes the communication-link length, which is analytically tractable but unrealistic when users move.A receiver location density can integrate the fixed-distance result to represent varying receiver distances.
  • D. Distributions of r: The varying-distance SINR distribution uses a receiver support S and density f_R while retaining Theorem 1 as the conditional result.This formulation directly replaces the fixed receiver distance with a distribution over receiver locations.
  • D. Distributions of r: Fig. 5 compares uniform and Rayleigh receiver geometries with a fixed-link baseline, using links averaging 25 m.The figure reports both overall LOS/NLOS and LOS-only SINR distributions.
  • D. Distributions of r: Random receiver distances improve larger-threshold SINR because shorter links sometimes occur and make LOS communication more likely.The same random locations reduce performance at lower SINR thresholds.

E. LOS Interference Limited Networks

The paper characterizes when mmWave ad hoc networks transition from noise-limited to interference-limited operation. It derives a bounded INR distribution while noting dependence on design parameters including transmission power.

  • E. LOS Interference Limited Networks: The INR CDF is used to characterize the transition from noise-limited to interference-limited operation as user density, building density, antenna pattern, and distance vary.Interference remains a central design limitation for ad hoc networks.
  • E. LOS Interference Limited Networks: The noise-limited threshold T is left to system designers, with 1 (0 dB) and 10 (10 dB) suggested as natural choices.The threshold defines the operational classification rather than being fixed by the analysis.
  • E. LOS Interference Limited Networks: The INR analysis replaces a constant term with a low-variance gamma random variable and bounds the resulting distribution analytically.The derivation uses total probability, a gamma CDF approximation, the Binomial Theorem, and PPP independence.
  • E. LOS Interference Limited Networks: Theorem 2 states that the INR distribution of a mmWave ad hoc network can be tightly bounded.The bound summarizes the preceding Laplace-transform construction.
  • E. LOS Interference Limited Networks: The INR scales with transmit power, motivating future transmission-power control based on the proximity of the nearest interferer.The paper focuses on node density and directional-beamforming beamwidth, while identifying power as an additional dependency.

F. One-Way Performance Analysis

The one-way analysis converts the SINR distribution into transmission capacity and area spectral efficiency. Taylor-style bounds produce a closed-form transmission-capacity solution and an ASE in bits/sec/Hz/m2.

  • F. One-Way Performance Analysis: Transmission capacity λ_ϵ is the largest network density supported at SINR threshold T and outage ϵ.It also represents successful transmissions per unit area and therefore connects to supported users.
  • F. One-Way Performance Analysis: Theorem 1’s exponential terms are approximated with a tight small-x bound to analyze optimal density when coverage probability is near one.The approximation is applied to both the Laplace-functional and NLOS terms.
  • F. One-Way Performance Analysis: The bounded coverage probability becomes a quadratic equation in λ, allowing transmission capacity to be solved in closed form.The exact solution depends on N_h.
  • F. One-Way Performance Analysis: Area spectral efficiency characterizes network-wide performance rather than only a single link’s SINR.The resulting ASE is expressed in bits/sec/Hz/m2.
  • F. One-Way Performance Analysis: The efficiency factor is log2(1 + T), which enters the area spectral-efficiency expression.

IV. TWO-WAY AD HOC COMMUNICATION

The one-way derivations are extended to concurrent forward and reverse communication under an outage constraint. The two-way model uses FDD to split bandwidth between links with separate rate requirements.

  • IV. TWO-WAY AD HOC COMMUNICATION: The one-way analysis omits the reverse link, although practical successful transmission usually relies on two-way communication.The two-way transmission capacity constrains outages on both forward and reverse links.
  • IV. TWO-WAY AD HOC COMMUNICATION: The forward link runs from transmitter to receiver, while the reverse link carries receiver-to-transmitter control information.Both links operate concurrently under frequency division duplexing.
  • IV. TWO-WAY AD HOC COMMUNICATION: FDD divides total bandwidth B_total into B_F for the forward link and B_R = B_total − B_F for the reverse link.The two links have differing rate requirements and separately defined SINR thresholds.

A. Two-way SINR Analysis

The two-way SINR analysis bounds the probability that both forward and reverse links exceed their thresholds, then uses that bound to characterize two-way network performance and bandwidth allocation.

  • Two-way SINR probability: The two-way SINR probability is the probability that both forward and reverse links exceed their respective SINR thresholds.The forward and reverse thresholds may differ because the reverse control link is generally lower-rate than the forward link.
  • Two-way SINR probability: Adding an interferer decreases SINR and makes the successful-transmission event {SINR > T} decreasing.This monotonicity enables application of the FKG inequality to the two-way probability.
  • Two-way SINR probability: The FKG inequality provides a lower bound on the two-way SINR probability by relating the joint forward-and-reverse event to the product of their probabilities.The bound is tight when the forward and reverse channels are independent, and dependence may be low in ad hoc networks.
  • Two-way performance: Because both transmitter and receiver must succeed, the two-way transmission capacity is no greater than the one-way transmission capacity.A Taylor expansion within the transmission-capacity framework yields an analytic expression for the two-way case.
  • Two-way performance: The two-way area spectral efficiency is optimized by allocating bandwidth between forward and reverse links according to their rate requirements.The paper explores the trade-off for given forward and reverse rate requirements, RF and RR.

V. PERFORMANCE RESULTS

The performance results compare mmWave and UHF ad hoc networks across transmission capacity, area spectral efficiency, optimal density, and rate coverage. MmWave advantages are strongest for suitable link conditions, especially LOS communication, but longer blocked links can become noise limited.

  • Transmission Capacity: A linear increase in SINR in dB produces an exponential decrease in the density of users that satisfies a 10% outage constraint.The shortest dipole length supports the highest density, while higher SINR requirements reduce allowable density.
  • Transmission Capacity: For 25m dipole links with LOS and NLOS communication, mmWave supports larger densities, whereas at 50m or 75m lower-frequency networks can support higher densities at higher thresholds.At longer mmWave links, blockage and NLOS path loss can make the network noise limited and reduce transmission capacity to zero.
  • Transmission Capacity: Restricting communication to LOS links enables longer mmWave links to maintain positive transmission capacity at higher SINR thresholds.The LOS-only comparison uses a different y-axis scale from the mixed LOS/NLOS case.
  • Area Spectral Efficiency: 10× area-spectral-efficiency gains over UHF occur when mmWave transmission capacity is non-zero.The reported gain is attributed to directional-antenna interference reduction and the larger path-loss exponent for NLOS links.
  • Area Spectral Efficiency: The ASE-maximizing density decreases exponentially with link distance and corresponds to an average neighbor distance one-half the link distance in the LOS-only case.The optimal density is obtained numerically from the density producing the largest ASE.
  • Rate Analysis: With 500MHz mmWave bandwidth versus 50MHz at lower frequency, mmWave rate coverage increases by orders of magnitude and all link lengths exceed 1Gbps a majority of the time.The rate-coverage analysis considers networks allowing both LOS and NLOS communication.

D. INR Distribution

The INR results show that mmWave ad hoc networks can remain interference limited, especially in dense deployments, with LOS interference dominating under key conditions. Two-way performance improves substantially through optimized bandwidth allocation, while LOS-only operation can yield major capacity and efficiency gains.

  • INR Distribution: LOS interference largely determines INR distributions in dense networks, whereas NLOS interference is negligible at those densities.The dense-network CDFs with only LOS interference are nearly identical to the full-interference results.
  • INR Distribution: P[INR < 0dB] = 0.4 for the sparsest 30° network, while denser networks become interference limited.At 90°, P[INR < 0dB] = 0.05 for the sparsest network, indicating stronger interference dominance across networks.
  • INR Distribution: At transmission capacity, INR is nearly always > 0dB for LOS-conditioned deployments, while interference remains non-negligible without LOS-only transmission.For a 25m link, INR is > 0dB 70% of the time in the reported non-LOS-only setting.
  • Two-Way Communication Results: 75% of one-way area spectral efficiency is achieved for 10% outage, representing a 100% increase over equal bandwidth allocation.The two-way results use bandwidth allocation to split resources between directions.
  • Two-Way Communication Results: A nearly 2x transmission-capacity improvement results from increasing forward bandwidth allocation from 50% to the optimal 90%.A rate-based 96% allocation performs nearly as poorly as the naive 50% split, and the optimal allocation is invariant to outage constraint.
  • Two-Way Communication Results: With optimal allocation, the two-way system reaches 75% of one-way area spectral efficiency and supports 2× the equal-split user density.Equal resource splitting reduces supported density by nearly a factor of 3.
  • Conclusions: LOS-only communication can produce 10-100× improvements in transmission capacity and area spectral efficiency compared with communication considering both LOS and NLOS links.The conclusion motivates LOS-aware protocols and LOS-interference mitigation strategies.

APPENDIX

The appendix develops supporting mathematical results for the paper’s stochastic-geometry analysis, including a normalized gamma-variable transformation and parameter definitions.

  • Proof of Lemma 1: Lemma 1 is proved using a result from prior work involving β = [Γ(1 + 1/p)]−p with p ∈ (0, 1).The cited theorem supplies the starting mathematical relation for the lemma.
  • Proof of Lemma 1: A normalized gamma random variable y ∼ Γ(k, θ) has inverse shape and scale parameters, giving E[y] = 1 and θ = 1/k.The appendix sets k = 1/p and x^p = kz for the transformation.
  • Proof of Lemma 1: The transformed parameter is a = k [Γ(1 + k)]−1/k = k(k!)−1/k.This expression follows after substituting the normalized gamma parameterization.
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