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Tractable Model for Rate in Self-Backhauled Millimeter Wave Cellular Networks
Sarabjot Singh, Mandar N. Kulkarni, Amitava Ghosh, Jeffrey G. Andrews
TL;DR
MmWave cellular networks need dense deployments and directional antennas, but their high rates make scalable backhaul and rate analysis important. This paper develops and validates a tractable self-backhauled mmWave model, finding that densification strongly improves cell-edge rates while bandwidth has limited cell-edge impact, and that different wired-backhaul fractions can yield the same median rate.
Problem
The paper addresses the need for a general tractable model that integrates self-backhauling, shared access/backhaul resources, and rate analysis in mmWave cellular networks.
Method
The paper models PPP-distributed mmWave BSs, directional antennas, LOS/NLOS propagation, two-hop minimum-path-loss association, and shared access/backhaul resources, then validates the analysis with building-based simulations.
Results
Cell-edge rates improve drastically with BS density, bandwidth has little cell-edge effect, and equivalent median rates can be achieved with different wired-backhaul fractions and BS densities.
Takeaways & Limitations
Self-backhauling supports dense mmWave deployments, while rate outcomes depend jointly on BS density and wired-backhaul provisioning.
Takeaways & Limitations
The framework leaves indoor-user offloading and multihop backhaul in sparser A-BS deployments for future investigation.
Abstract
from arXiv · showhide
Millimeter wave (mmW) cellular systems will require high gain directional antennas and dense base station (BS) deployments to overcome high near field path loss and poor diffraction. As a desirable side effect, high gain antennas provide interference isolation, providing an opportunity to incorporate self-backhauling--BSs backhauling among themselves in a mesh architecture without significant loss in throughput--to enable the requisite large BS densities. The use of directional antennas and resource sharing between access and backhaul links leads to coverage and rate trends that differ significantly from conventional microwave ($μ$W) cellular systems. In this paper, we propose a general and tractable mmW cellular model capturing these key trends and characterize the associated rate distribution. The developed model and analysis is validated using actual building locations from dense urban settings and empirically-derived path loss models. The analysis shows that in sharp contrast to the interference limited nature of $μ$W cellular networks, the spectral efficiency of mmW networks (besides total rate) also increases with BS density particularly at the cell edge. Increasing the system bandwidth, although boosting median and peak rates, does not significantly influence the cell edge rate. With self-backhauling, different combinations of the wired backhaul fraction (i.e. the faction of BSs with a wired connection) and BS density are shown to guarantee the same median rate (QoS).
I. INTRODUCTION
mmWave cellular networks motivate dense BS deployments and directional antennas, while self-backhauling offers a scalable way to address backhaul demands. The paper develops a tractable model linking propagation, association, resource sharing, and rate outcomes.
- 20−100 GHz bands offer large amounts of underused spectrum for meeting projected terrestrial traffic demands.
- Dense BS deployments are needed for acceptable mmWave coverage and rate, but GHz-scale bandwidth creates substantial backhaul requirements.
- Self-backhauling lets wired-backhaul BSs serve multiple unwired BSs over mmWave links, while access and backhaul share each BS’s radio resources.
- The model captures LOS/NLOS access and backhaul propagation, adaptive blockage, PPP-distributed BSs, and separate user and BS association areas.
- Cell edge rates improve drastically with BS density, whereas bandwidth has little effect on power- and noise-limited cell-edge users.
- Rate saturation occurs when BS density increases while wired-backhaul BS density remains fixed, but equal median rates can arise from different density–wired-backhaul combinations.
C. Blockage model
The blockage model approximates access-link LOS probability using a geography-dependent LOS fraction within a finite-radius ball. Users and BSs associate through minimum path loss in the resulting two-hop network.
- An access link is LOS with probability C for distance d ≤ D and non-LOS beyond D.C represents the average LOS-area fraction within a radius-D circular ball.
- The tagged A-BS provides wireless backhaul to associated BSs and access links to associated users.
- The parameters (C, D) depend on geography and deployment, and may differ between access and backhaul links.
- Users select the BS offering minimum access path loss, while unwired BSs select the A-BS offering minimum backhaul path loss.This creates the paper’s two-hop access/backhaul association structure.
- Access and backhaul SINR calculations include thermal noise N ≜ N0B and corresponding interference.
E. Validation methodology
The analysis characterizes typical-link SNR, SINR, and rate distributions under PPP assumptions and a shared-resource model, then validates the framework with Monte Carlo simulations and empirical propagation settings.
- The model is evaluated through Monte Carlo simulations using actual building locations and empirically derived propagation inputs.
- Access and backhaul links share orthogonally divided radio resources, so user rate depends on both user load at BSs and BS load at A-BSs.
- Resource fractions are allocated between backhaul and access, with each A-BS sharing its reserved backhaul resources equally among associated BSs.
- The framework derives path-loss distributions using propagation-process intensity measures for access and backhaul links.
- Under stationary PPP assumptions, typical-link SNR coverage represents the network-wide SNR distribution, with analogous interpretations for SINR and rate coverage.
- Theorem 1 provides SNR distributions for typical downlink, uplink, and backhaul links, whose coverage scales with link density, transmit power, and antenna gain.
B. Interference in mmWave networks
The analysis upper-bounds interference in mmWave networks and shows that directional, large-bandwidth deployments are typically noise-limited. It also characterizes how density, directivity, bandwidth, and frequency affect the INR bound.
- Analytical interference bound: The downlink interference is upper-bounded by total received power, whose shot-noise Laplace functional yields an analytical INR bound.The bound uses the total received power from interfering base stations and an Euler characterization of its complementary distribution.
- Noise-limited behavior: INR exceeds 0 dB in fewer than 20% of cases even at about 200 BS/km^2 in the considered large-bandwidth, narrow-beam network.The analytical upper bound agrees with simulations, and interference does not dominate noise in this setting.
- Parameter equivalence: The INR upper bound is invariant to increasing BS density or beam-width when bandwidth or carrier frequency scales appropriately.This is the density-directivity-bandwidth-frequency equivalence under uniform path-loss exponents and shadowing variance.
- Rate-analysis approximation: Because SNR closely approximates SINR in directional, large-bandwidth, densely blocked settings, rate analysis can deliberately ignore interference while simulations retain it for validation.For interference-limited settings, the analytical rate results can instead be obtained by replacing SNR with SINR.
C. Load characterization
Self-backhauled networks require separate association cells for users and backhauled BSs because access and backhaul loads are shared differently. Exact area distributions are difficult, so the model uses tractable approximations anchored by exact mean areas.
- Association cells: Two association cells are defined: a BS’s user cell determines access load, while an A-BS’s BS cell determines backhaul load.The user cell contains users served by a BS or A-BS; the BS cell contains BSs served by a wired-backhaul A-BS.
- Association-area challenge: The exact distribution of these irregular association areas is highly non-trivial to characterize.The complexity arises from blockage-sensitive, path-loss-based associations rather than simple distance-based cells.
- Mean association areas: Under minimum-path-loss association, the mean user-cell area is 1/λ and the mean A-BS BS-cell area is 1/(λω).Stationary association gives these exact mean areas for typical BS and A-BS cells.
- Load characterization: The tagged BS and A-BS area distributions are area-biased relative to corresponding typical-cell distributions.This distinction affects the load distributions experienced by users associated with tagged network elements.
- Tractable load model: The model approximates association-area distributions using Poisson-Voronoi areas with the same mean area, then derives user and BS load PMFs.The approximation is stated for typical BS and A-BS association areas and is subsequently checked through simulations.
D. Rate coverage
The rate-coverage analysis combines access and backhaul sharing through separate load distributions and link-SNR distributions. It derives self-backhauled and hybrid-network rate expressions, with mean-load simplifications and an explicit minimum-SNR constraint.
- Self-backhauled rate coverage: Lemma 2 gives the rate coverage of a typical user in a self-backhauled mmWave network as a function of rate threshold ρ and bandwidth B.The expression uses ρ̂ = ρ/B, v(x) = 2^x − 1, and the SNR coverage functions from Theorem 1.
- Load and association effects: The rate expression combines the cases of association with an A-BS and with a non-wired BS, accounting for access and backhaul loads.The derivation conditions on the A-BS association event, whose probability is ω, and invokes independence among loads and SNRs.
- Mean-load approximation: Corollary 2 simplifies rate coverage by replacing the different loads with their respective mean values.This approximation uses the load distributions characterized in Proposition 2.
- Impact of wired fraction: Increasing the wired A-BS fraction ω raises A-BS service probability, A-BS access rate, and backhaul rate for users associated with non-wired BSs.The stated mechanisms are reduced user and BS load per A-BS and increased probability of A-BS association.
- Minimum-SNR constraint: A minimum-SNR threshold τ0 can be imposed so that the rate is zero whenever SNR falls below τ0.This models the practical inability to transmit reliably below a specified modulation-and-coding threshold.
- Hybrid-network rate coverage: Lemma 3 extends rate coverage to a hybrid mmWave-UHF network by combining mmWave coverage for offloaded users with UHF coverage under macrocell load.The mmWave term adjusts user density using the offloading threshold, while the UHF term uses the tagged macrocell’s user-load distribution.
E. Validation
The model is fitted to geography through the average LOS fractional area, then validated against simulations in Manhattan and Chicago. Analytical rate distributions closely match simulations, while larger fitting radii generally improve the match.
- Geographic fitting: The model uses C and D as geography-dependent parameters, with C representing average LOS fractional area within radius D.These parameters are obtained from empirical LOS-area measurements for each region.
- Geographic fitting: Monte Carlo simulations estimate LOS fractional area versus radius D for users in the Manhattan and Chicago regions.The resulting empirical C values are used with Table II’s D and C parameters.
- Model scope: The match is generally better for fitting pairs with larger D, particularly D = 200–250 m.The simplification sets LOS fractional area beyond D to zero, while larger D makes that approximation more accurate.
A. Coverage and density
In dense urban mmWave settings, analytical coverage tracks simulation closely, and increasing BS density improves downlink and uplink coverage and spectral efficiency. At very high densities, finite user populations can prevent interference from growing without bound.
- Validation: Analytical SNR tracks simulated SINR coverage closely for both downlink and uplink.An example downlink case at 250 BSs per sq. km and 10 dB shows a gap below 10%.
- Density effects: Increasing BS density improves both downlink and uplink coverage and therefore spectral efficiency.This differs from conventional interference-limited networks, whose SINR is nearly density-invariant.
- High-density regime: At very large densities, SINR coverage saturates quickly in lightly blocked scenarios when finite user population is ignored.The saturation reflects interference becoming dominant over thermal noise.
- High-density regime: Accounting for finite user density can make SINR coverage continue improving with BS density.The tagged-BS path loss improves while interference is implicitly capped by the finite user population.
B. Rate coverage
Rate improves with densification through better coverage and lower load, while bandwidth mainly benefits median and peak users. Self-backhaul and UHF coexistence provide distinct ways to address rate bottlenecks and cell-edge reliability.
- Rate coverage: Increasing infrastructure density improves downlink and uplink rates by reducing cell size and base-station load.For fixed wired-backhaul fraction ω = 0.5, analytical and simulation rate coverage agree well.
- Bandwidth: Larger bandwidth considerably increases median and peak rates but does not increase cell-edge rates.Low edge-user SNR counterbalances the bandwidth gain; with Rate = 0 for SNR < τ0, edge rates can decrease.
- Coexistence: Offloading users with mmWave link SNR below −10 dB to a UHF network significantly improves edge rates under the minimum-SNR constraint.The gain decreases as mmWave density rises because fewer users have poor SNR; without the constraint, mmWave is preferred due to 100x larger bandwidth.
- Self-backhauling: Increasing the wired-backhaul fraction improves rate distribution but yields diminishing returns as the bottleneck shifts to the air interface.Different wired-backhaul fractions and BS densities can produce similar rate distributions.
- Self-backhauling: A median rate of 400 Mbps can be achieved with either (ω = 0.9, λ = 110) or (ω = 0.3, λ = 200).The result supports equivalent median QoS through different combinations of wired-backhaul fraction and BS density.
- Self-backhauling: Rate coverage saturates with density for each fixed A-BS density, and the saturation density increases with A-BS density.The saturation threshold is defined as the point beyond which only marginal rate-coverage gains occur.
V. CONCLUSION AND FUTURE CHALLENGES
The paper develops an analytical framework for mmW rate distributions with self-backhauling and characterizes how bandwidth, BS density, and A-BS density affect rates. It also identifies future extensions involving indoor-user offloading, multihop backhaul, and other mmW network architectures.
- The framework characterizes mmW rate distributions while integrating self-backhauling among BSs and coexistence with a conventional macrocellular network.This integration is presented as a first contribution of the work.
- Bandwidth has minimal impact on power- and noise-limited cell-edge rates, whereas increasing BS density improves those rates drastically.
- With self-backhauling and fixed A-BS density, rate saturates as BS density increases, with saturation density directly proportional to A-BS density.
- Future challenges: The analytical framework provides tools for analyzing architectures including device-to-device and ad hoc mmW networks.
- Future challenges: Future work could investigate indoor-user offloading to 4G or WiFi and multihop backhaul in sparser A-BS deployments.
APPENDIX A
The appendix derives tractable distributions for path loss, coverage, uplink SNR, and SINR in the modeled mmW network. It uses propagation-process intensity measures, LOS/NLOS link probabilities, fading, and finite-user corrections.
- Path loss distribution: The path-loss derivation models N := L(X) as a propagation process and obtains its intensity measure.The process includes shadowing and distance-dependent path loss.
- Path loss distribution: LOS and NLOS links are assigned probabilities that depend on whether link length is below or above the threshold D.The probabilities satisfy Cl,D + Cn,D = 1 and Cl,¯D + Cn,¯D = 1.
- Path loss distribution: Because N is a PPP, the tagged BS path-loss distribution follows from the intensity measure as P(inf_X∈Φ L(X) > t) = exp(−Λ((0, t])).
- SINR distribution: The SINR distribution is characterized using the path-loss intensity measure, conditional CDFs, and a conditional Laplace transform.The conditional CDF is obtained from the Laplace transform using Euler’s characterization.
- SINR distribution: Rayleigh fading on each link improves tractability by avoiding computationally intensive inverse Laplace-transform calculations in some cases.The fading variable is specified as H ∼ exp(1).
- Finite-user correction: Finite user populations are incorporated by replacing λ with λ(1 − K(λu, λ, 0)) to account for BSs without users.