Source-linked AI summary
Analytical Modeling of Uplink Cellular Networks
Thomas D. Novlan, Harpreet S. Dhillon, Jeffrey G. Andrews
TL;DR
The paper addresses the difficulty of obtaining insightful, tractable cellular-uplink analysis beyond simplified Wyner models and simulation-heavy approaches. It uses stochastic geometry with coupled mobile and base-station placements, one scheduled mobile per base station, and per-mobile power control to derive uplink coverage. The resulting framework supports power-control analysis, indicating partial channel inversion at low SINR and full power transmission at higher SINR.
Problem
Existing uplink analysis relies either on simplified Wyner-type interference models or complex simulations that provide limited analytical insight.
Method
The paper models mobile locations with a PPP, associates users with base stations through Voronoi cells, accounts for dependence and one active uplink user per base station, and includes fractional power control.
Results
The paper derives tractable uplink coverage probability and average-rate expressions and evaluates coverage and transmit-power utilization across power-control parameters.
Takeaways & Limitations
Partial channel inversion benefits cell-edge users at low and moderate SINR, while more aggressive power control can reduce overall mobile power utilization; full power transmission is optimal at higher SINR.
Abstract
from arXiv · showhide
Cellular uplink analysis has typically been undertaken by either a simple approach that lumps all interference into a single deterministic or random parameter in a Wyner-type model, or via complex system level simulations that often do not provide insight into why various trends are observed. This paper proposes a novel middle way using point processes that is both accurate and also results in easy-to-evaluate integral expressions based on the Laplace transform of the interference. We assume mobiles and base stations are randomly placed in the network with each mobile pairing up to its closest base station. Compared to related recent work on downlink analysis, the proposed uplink model differs in two key features. First, dependence is considered between user and base station point processes to make sure each base station serves a single mobile in the given resource block. Second, per-mobile power control is included, which further couples the transmission of mobiles due to location-dependent channel inversion. Nevertheless, we succeed in deriving the coverage (equivalently outage) probability of a typical link in the network. This model can be used to address a wide variety of system design questions in the future. In this paper we focus on the implications for power control and see that partial channel inversion should be used at low signal-to-interference-plus-noise ratio (SINR), while full power transmission is optimal at higher SINR.
I. INTRODUCTION
Uplink analysis is harder than downlink analysis because mobile-generated interference and location-dependent power control make interference statistics highly variable. The paper develops a tractable stochastic-geometry framework to obtain uplink SINR distributions and related performance measures.
- Uplink interference comes from mobile devices distributed throughout the network rather than from fixed transmitters.
- Location-dependent power control makes mobile transmit power highly variable, significantly changing uplink interference statistics.
- Wyner-type models simplify interference analytically, whereas grid deployments generally require approximations and exhaustive Monte Carlo simulations.
- Random spatial models can provide tractable expressions, broader performance characterizations, and useful intuition for cellular networks.
- Existing uplink analytical work also used asymptotic assumptions, multiple users per base station, and averaged interference transmit power.
- Power control addresses the near-far problem by reducing the received-power disparity between cell-edge and cell-interior users.
C. Contributions
The paper derives tractable uplink coverage results using a coupled mobile–base-station model with fractional power control, while approximating weak dependence to preserve analytical tractability.
- The main contribution is uplink coverage probability for a randomly chosen mobile using fractional power control.The framework is intended to incorporate virtually all modern cellular systems.
- The analysis assumes these distances are i.i.d., because their dependence is weak and can be ignored with minimal accuracy impact.
- Mobile users form a PPP, each associated base station is modeled within the mobile’s Voronoi cell, and each base station schedules one active uplink user.
- For tractability, a randomly chosen base station is approximated by a point uniformly chosen in R2, yielding a Rayleigh distribution for its nearest-mobile distance.
- The distance variables of interfering mobiles are identically distributed but generally dependent because of the Poisson–Voronoi structure and one-base-station-per-cell restriction.
B. Channel Model
The channel model combines distance-based path loss, Rayleigh fading, baseline transmit power, and fractional power control governed by ϵ.
- Path loss is inversely proportional to distance with exponent α, and the desired link includes small-scale Rayleigh fading.
- Fractional power control uses distance-proportional transmit power R^αϵ, where ϵ ∈[0, 1].Closer users require less transmit power to maintain the same received signal power, benefiting battery-powered devices.
- The associated SINR is defined at a base station located at the origin and includes the desired signal and interference from the set Z of interfering mobiles.
- ϵ = 1 fully inverts path loss in the desired-link numerator, while ϵ = 0 applies no channel inversion and gives all mobiles equal transmit power.
C. Summary of Special Cases
The paper validates its tractable uplink model through several special cases, contrasting analytically tractable PPP variants with simulation-based grid and fitted-interference approaches.
- The unrestricted PPP case retains the mobile PPP and uniformly drops one base station in each mobile’s Voronoi cell, but lacks direct tractable analysis.
- PPP-Rayleigh imposes i.i.d. Rayleigh serving distances and corresponds to the main result for irregular base-station deployment.
- PPP-Uniform instead models each serving base station uniformly within a circle around its mobile and corresponds to the regular-deployment result.
- The Grid case places base stations at hexagonal-cell centers and evaluates performance using Monte Carlo simulations because it lacks tractable expressions.
- The Log-normal approach replaces inter-cell interference with a fitted log-normal random variable based on grid-model simulations.
III. COVERAGE PROBABILITY
The section defines uplink coverage as the SINR ccdf and derives a general coverage expression for i.i.d. mobile-to-serving-base-station distances using the interference Laplace transform. It also gives a simplified full-power, interference-limited result.
- Coverage probability is the probability that uplink SINR exceeds target T, equivalently the average fraction of users in coverage.
- The analysis models interference from transmitting mobiles beyond the typical link distance, with transmit power depending on each mobile's serving distance and power-control factor ϵ.
- Theorem 1 gives uplink coverage for i.i.d. Rz through an expression involving the Laplace transform of interference.
- The coverage derivation integrates over the typical-link distance and uses Rayleigh fading, the Laplace-transform definition, independence, and the PGFL of a PPP.
- For full power control, ϵ = 1, with negligible noise, the coverage expression simplifies to a corollary of Theorem 1.
A. Distribution of Rz and Comments on Independence Assumption
The paper approximates the serving-distance distribution Rz with a tractable marginal model and tests the i.i.d. assumption through neighboring Voronoi-cell distances. The measured dependence is weak, and the resulting analytical coverage closely matches simulations.
- Under the independence assumption, Rz is approximated by the distance from a randomly chosen point to its closest base station and modeled with a Rayleigh distribution.
- The joint distribution of distances Rz1 and Rz2 is studied for mobiles in neighboring Voronoi cells as a worst-case dependence test.
- The actual and independent joint densities are surprisingly similar, although independence produces a slightly more dispersed density.
- The simulated correlation coefficient between neighboring serving distances is 0.07.
- Using the approximated Rz density, the paper derives the PPP interference Laplace transform and finds that analytical coverage closely approximates true-power-control simulations.
- The proposed model captures the SINR-distribution shape more accurately than a log-normal interference approximation and remains parameterized by key network features.
B. Comments on Regular (Grid) Model
For regular deployments, the paper replaces hexagonal cells with an analytically tractable random spatial model whose serving-distance distribution approximates the grid case. Its coverage predictions closely match true-power-control grid simulations.
- Hexagonal grid models are useful for numerical studies but do not provide analytical tractability.
- The grid approximation randomizes user locations and selects an Rz distribution that generates analytical coverage expressions.
- Hexagons are approximated by equal-area circles, with each base station uniformly located within a circle around its corresponding mobile.
- The regular-model Rz density closely approximates the grid-model distribution and supports computation of the interference Laplace transform.
- For α = 4, ϵ = 1 and α = 3.25, ϵ = .75 with λ = .25 and σ2 = 0, analytical coverage closely approximates true-power-control hexagonal-grid simulations.
IV. SYSTEM DESIGN APPLICATIONS
The model is applied to realistic network parameters to study average rate and compare uplink and downlink coverage under fractional power control.
- The analysis focuses on a scenario where each base station is uniformly located in its corresponding mobile user's Voronoi cell, with Rayleigh-distributed nearest-base-station distance.This setting is intended to capture non-uniform topology in modern deployments.
- Average user rate is evaluated as a function of fractional power control for pathloss exponents α = 2.5, 3.25, and 4, with µ−1 = 200 mW.The figure includes both no-noise and σ2 = −104dBm cases.
- Downlink and uplink coverage are compared using 40W downlink transmission and uplink fractional power control with ϵ = .6, .8, and 1, capped at 200 mW.
A. Average rate
The paper extends its uplink coverage framework to average rate by integrating Shannon-capacity rates over the SINR and fading distributions.
- Average data rate is obtained from the SINR distribution, enabling analytical rate expressions as functions of λ, α, and ϵ.The paper identifies this as an application previously unavailable for deterministic network-topology models.
- Assuming adaptive modulation and coding, the rate is defined using the Shannon expression E[ln(1 + SINR)] and reported in nats/Hz.One bit equals loge(2) nats.
- The average-rate derivation integrates the SINR exceedance probability over t and the nearest-base-station distance r.The distance density is fR(r) = 2πλre−πλr2, and the interference transform depends on the network model for Rz.
- Average rate increases with α across all ϵ values in the evaluated cases, while increasing ϵ decreases the rate for a randomly selected user.The result combines the effects of power control on high-, medium-, and low-SINR users.
B. Downlink vs. Uplink Coverage
The paper compares downlink and uplink coverage under two downlink spatial models and examines how fractional power control changes uplink SINR behavior.
- The comparison uses a downlink model based on the proposed user-induced base-station locations and another with PPP-distributed base stations.Both assume equal transmit power across downlink base stations.
- The two downlink models produce sufficiently similar SINR distributions to support direct comparison with the paper's uplink results.
- Fractional power control makes uplink interference depend on ϵ and the distribution of serving distances because mobile transmit powers are variable rather than constant.
- Uplink coverage is lower in the comparison partly because uplink and downlink transmit powers differ, while larger ϵ further reduces high-SINR coverage.Users near their serving base stations reduce transmit power relative to cell-edge users.
- Different uplink and downlink coverage regions can affect handoff decisions and motivate jointly balancing coverage and capacity across network devices.
C. Fractional Power Control
Fractional power control produces a coverage tradeoff: moderate compensation benefits lower-SINR users, whereas full power is favored at high SINR, while stronger control reduces mobile transmit power.
- C. Fractional Power Control: ϵ = 0.25 gives the highest coverage for users in the lower 50 percentile, followed by ϵ = 0.5, with gains over fixed power before crossing below ϵ = 0.The crossover occurs at 5 dB for ϵ = 0.25 and 0 dB for ϵ = 0.5.
- C. Fractional Power Control: For SINR thresholds below −10 dB, coverage differences among ϵ = 0, 0.25, and 0.5 are negligible.
- C. Fractional Power Control: Coverage decreases as ϵ increases, with ϵ = .75 performing much worse than fixed power above 5 dB and ϵ = 1 reducing coverage across all thresholds.
- C. Fractional Power Control: Coverage maximization has two regimes: ϵ = .25 to .3 is best at low SINR, while ϵ = 0 is best at high SINR.The transition between regimes spans approximately 5 dB and depends somewhat on α.
- C. Fractional Power Control: The tradeoff arises because cell-interior users are more noise-limited, whereas cell-edge users are more interference-limited and benefit from pathloss-proportional power control.
- C. Fractional Power Control: At high ϵ, 10–15% of users transmit below 0 dBm, a 23 dB reduction from the 23 dBm maximum transmit power.The paper therefore frames power-control selection as a balance between coverage and battery utilization.
V. CONCLUSION
The paper derives tractable uplink coverage and average-rate expressions for uniform and irregular topologies, exposing power-control tradeoffs while leaving heterogeneous networks for future study.
- The model provides tractable coverage-probability and average-rate expressions for both uniform and irregular cellular network topologies.These expressions depend on network topology and system-design parameters, including SINR targets, base-station density, and fractional power-control parameters.
- The results clarify the tradeoff between improving cell-edge performance through fractional power control and reducing overall mobile power utilization.
- Heterogeneous network topologies remain a major area for future work because multiple overlapping coverage tiers may substantially alter interference distributions and system-design tradeoffs.