A Tractable Approach to Coverage and Rate in Cellular Networks

Jeffrey G. Andrews; Francois Baccelli; Radha Krishna Ganti

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A Tractable Approach to Coverage and Rate in Cellular Networks

Jeffrey G. Andrews, Francois Baccelli, Radha Krishna Ganti

arXiv:1009.0516v2Sep 2, 2010cs.ITcs.NImath.PR

TL;DR

Cellular-network coverage and rate analysis needs models that are less idealized and more tractable than conventional grid-based approaches. This paper uses stochastic geometry to derive computable SINR and rate expressions, finding that its model lower-bounds real-deployment coverage while matching the grid model’s accuracy.

Problem
Existing cellular-network models are highly idealized or insufficiently tractable for full-network downlink capacity and coverage analysis.
Method
The paper uses stochastic geometry with independently positioned base stations to derive general downlink SINR coverage and mean-rate expressions.
Results
The proposed model lower-bounds real-deployment coverage and is about as accurate as the grid model, which upper-bounds coverage.
Takeaways & Limitations
The framework offers a more tractable cellular-network analysis and may better represent increasingly opportunistic, dense base-station deployments.
Takeaways & Limitations
Common downlink models can be overly simple, and simplified approaches to other-cell interference modeling remain in use.

Abstract

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Cellular networks are usually modeled by placing the base stations on a grid, with mobile users either randomly scattered or placed deterministically. These models have been used extensively but suffer from being both highly idealized and not very tractable, so complex system-level simulations are used to evaluate coverage/outage probability and rate. More tractable models have long been desirable. We develop new general models for the multi-cell signal-to-interference-plus-noise ratio (SINR) using stochastic geometry. Under very general assumptions, the resulting expressions for the downlink SINR CCDF (equivalent to the coverage probability) involve quickly computable integrals, and in some practical special cases can be simplified to common integrals (e.g., the Q-function) or even to simple closed-form expressions. We also derive the mean rate, and then the coverage gain (and mean rate loss) from static frequency reuse. We compare our coverage predictions to the grid model and an actual base station deployment, and observe that the proposed model is pessimistic (a lower bound on coverage) whereas the grid model is optimistic, and that both are about equally accurate. In addition to being more tractable, the proposed model may better capture the increasingly opportunistic and dense placement of base stations in future networks.

I. INTRODUCTION · A. Common Approaches and Their Limitations

The paper addresses the lack of accurate, tractable models for other-cell interference, which limits analysis and improvement of cellular coverage and capacity. It develops tractable downlink models while highlighting the substantial limitations of simplified, grid-based, and simulation-heavy approaches.

I. INTRODUCTION: Accurate and tractable models for other-cell interference remain unavailable, hindering techniques to improve spectral efficiency, especially in dense urban networks.The paper identifies other-cell interference as the most important obstacle to higher spectral efficiency in today’s cellular networks.
I. INTRODUCTION: The paper develops accurate and tractable models for downlink capacity and coverage under full-network interference.This contribution is motivated by the need for models that are both realistic and analytically manageable.
A. Common Approaches and Their Limitations: The Wyner model reduces other-cell interference to fixed gains or constants, making it highly inaccurate except when interference is heavily averaged.The model is typically one-dimensional and assumes unit gain from the serving base station and equal subunit gains from neighboring cells.
A. Common Approaches and Their Limitations: For LTE, WiMAX, and similar orthogonal-access systems, mean-value interference models are particularly inaccurate because SINR varies dramatically across a cell.Such simplified approaches nevertheless remained common in multicell capacity and cooperation analyses.
A. Common Approaches and Their Limitations: Single-interferer analyses capture some user-position dependence but neglect most network interference and remain highly idealized.The paper notes that simplified interference models were still considered state-of-the-art because more realistic tractable approaches were difficult to find.
A. Common Approaches and Their Limitations: Grid-based analyses can evaluate worst-case metrics at cell corners, but they are pessimistic and provide limited guidance about most users.With shadowing or fading, SINR remains random, allowing worst-case average rate and outage probability to be determined.
A. Common Approaches and Their Limitations: For random user locations, general tractable SINR expressions are unavailable, so typical coverage and outage results require complex, time-consuming simulations.Private simulations are onerous to construct and run, and they also raise repeatability and transparency concerns.
A. Common Approaches and Their Limitations: Regular grid models are highly idealized and may become inaccurate for heterogeneous, ad hoc deployments whose cell radii vary substantially.Urban and suburban networks may include inserted picocells and haphazardly scattered femtocells alongside centrally planned infrastructure.

B. Our Approach and Contributions · II. DOWNLINK SYSTEM MODEL

The paper models base-station locations as a homogeneous Poisson point process, enabling tractable stochastic-geometry analysis of cellular coverage and rate. It derives general and simplified coverage results, mean rate, and frequency-reuse tradeoffs, then compares predictions with grid simulations and an actual deployment.

B. Our Approach and Contributions: Base stations are modeled as a homogeneous Poisson point process of density λ, making their positions independent and enabling stochastic-geometry analysis.Mobile users form an independent homogeneous point process and connect to their nearest base station, while all others interfere.
B. Our Approach and Contributions: The model derives a general cellular coverage probability for arbitrary interference fading and shadowing distributions without requiring Monte Carlo methods.Coverage is the CCDF probability that a typical user achieves a threshold SINR.
B. Our Approach and Contributions: Under Rayleigh fading, path loss exponent 4, and interference-limited operation, the coverage expression simplifies to a formula depending only on threshold SINR.These assumptions progressively increase tractability, with the simplest expression obtained when all three apply.
B. Our Approach and Contributions: Compared with grid simulations and an actual deployment, the Poisson model provides a reliable lower bound on reality while the grid model provides an approximately equally loose upper bound.The results remain simple, tractable, and accurate even under the simplifying assumptions.
B. Our Approach and Contributions: The paper derives the mean achievable rate and analyzes the competing objectives of coverage and rate through frequency reuse.Frequency reuse is used to increase coverage or equivalently cell-edge rates.
B. Our Approach and Contributions: The coverage and rate expressions incorporate frequency reuse to determine the reuse needed for a specified coverage probability and its mean-rate cost.Mean rate declines because reuse uses the total bandwidth less efficiently.
II. DOWNLINK SYSTEM MODEL: Interference is the sum of received powers from all non-home base stations and is treated as noise, while same-cell interference is absent and noise power is σ2.The interference distribution g may include fading, shadowing, and other random effects; lognormal interference degrades coverage but does not significantly affect analytical accuracy.

III. COVERAGE · A. Distance to Nearest Base Station

The paper defines downlink coverage as the probability of achieving a target SINR and derives it through increasingly specialized stochastic-geometry models. For nearest-base-station association, the serving distance follows a Poisson-process distribution, while interferers lie outside the corresponding exclusion disc.

III. COVERAGE: Coverage probability is defined as the probability that a randomly chosen user achieves a target SINR T.It is also equivalent to the average fraction of users or network area achieving SINR T.
III. COVERAGE: Coverage is equivalently the SINR CCDF over the entire network, because the CDF gives P[SINR ≤ T].A user is in coverage when its nearest-base-station SINR exceeds threshold T and is dropped below T.
III. COVERAGE: The analysis places the mobile user at the origin without loss of generality and expresses its SINR using distance r from the associated base station.Interference is accumulated from other base stations, excluding the tagged base station.
A. Distance to Nearest Base Station: Nearest-base-station association makes r the key geometric quantity separating a typical user from its tagged base station.Every interfering base station must be farther from the user than r.
A. Distance to Nearest Base Station: The nearest-base-station distance distribution is derived from the null probability of a 2-D Poisson process over the relevant area.This yields the CDF P[r ≤ R] = Fr(R) = 1 − e−λπR2.
A. Distance to Nearest Base Station: Interference is modeled as standard M/M shot noise generated by a Poisson point process of intensity λ outside a radius-r disc.Known results for this exclusion-disc shot-noise process are applied subsequently.

B. General Case and Main Result

Theorem 1 gives a general coverage-probability result for a target SINR T in the cellular-network model. Although not closed-form, its integrals are fairly easy to evaluate and motivate simplified special cases.

B. General Case and Main Result: Theorem 1 states the main and most general result for the coverage probability of a typical randomly located mobile user.All other results in the section follow from this theorem.
B. General Case and Main Result: The theorem gives the probability of achieving a target SINR T in the general cellular network.The result is derived by conditioning on the nearest base station’s distance and averaging coverage over the plane.
B. General Case and Main Result: The general coverage expression is not closed-form, but its integrals are fairly easy to evaluate.The derivation uses the interferer-channel distribution, the Laplace transform of interference, and the PPP probability generating functional.
B. General Case and Main Result: The analysis next considers relevant special cases in which significant simplification is possible.These cases follow directly from the general theorem.

C. Special Cases: Interference Experiences General Fading … 3) General Fading, Small but Non-zero Noise:

The paper develops tractable special cases for general interference fading, including a quasi-closed-form coverage expression when α = 4, density-independent coverage without noise, and a low-noise approximation that improves computability while retaining accuracy for nonzero noise.

C. Special Cases: Interference Experiences General Fading: The special cases vary the path-loss exponent, noise regime, and interference-fading assumptions while retaining a general interference-power distribution in the cases considered here.The subsection focuses on combinations involving α = 4 and negligible noise, while treating interference fading generally.
1) General Fading, Noise, α = 4:: For α = 4, Theorem 1 yields a form whose coverage probability is quasi-closed form once β(T, 4) is numerically calculated.The resulting expression uses the standard Gaussian tail probability Q(x), which is readily evaluated by modern calculators and software.
2) General Fading, No Noise, α > 2:: With negligible thermal noise, coverage is obtained from Theorem 1 under the limit σ2 → 0, corresponding to an interference-limited network.Noise can be negligible both in cell interiors because of high SNR and at cell edges because of high INR.
2) General Fading, No Noise, α > 2:: In the no-noise case, the coverage probability does not depend on base-station density λ, so both very dense and very sparse networks retain a positive coverage probability.Increasing density raises signal and interference power together, leaving coverage unchanged when noise is negligible.
2) General Fading, No Noise, α > 2:: Increasing coverage in interference-limited networks generally requires interference management, such as frequency reuse, rather than merely deploying more base stations.Additional base stations can nevertheless cover more simultaneous users because the model assumes one active user per cell.
3) General Fading, Small but Non-zero Noise:: For small but nonzero noise, the low-noise success-probability approximation is easier to compute than the constant-noise expression and more accurate than the no-noise approximation.The approximation is derived using exp(−x) = 1 − x + o(x) as x → 0 and integration by parts.
3) General Fading, Small but Non-zero Noise:: For α = 4, the finite-noise expression (14) converges to the no-noise expression (15) as σ2 → 0, despite this equivalence not being immediately apparent.A large-x series expansion of Q(x) simplifies the finite-noise form to (15).

D. Special Cases: Interference is Rayleigh Fading … IV. AVERAGE ACHIEVABLE RATE

The paper derives simplified coverage results for exponential interference, including a remarkably simple no-noise expression at α = 4, then formulates mean achievable rate under adaptive modulation and coding. These results assume interference is treated as noise, while extensions to other receiver structures and interference distributions are noted.

D. Special Cases: Interference is Rayleigh Fading: Exponential interference, corresponding to Rayleigh fading with neglected shadowing, enables significant simplification of the coverage analysis.The resulting coverage probability is stated as Theorem 2 for a typical randomly located mobile user.
D. Special Cases: Interference is Rayleigh Fading: The exponential-interference coverage result follows as a special case of Theorem 1 and is proved in Appendix A.The proof is described as substantially simpler than the general case.
1) Exponential Fading, Noise, α = 4:: For exponential fading with noise and α = 4, the coverage expression is practically closed-form and requires only one integral computation.The no-noise result has a similar appearance to the general fading result.
2) Exponential Fading, No Noise, α > 2:: For exponential fading without noise and α > 2, ρ(T, α) is faster and easier to compute than the more general β(T, α).The simplified form retains the same general fading appearance while reducing computational complexity.
2) Exponential Fading, No Noise, α > 2:: At α = 4 without noise, coverage depends only on the SIR threshold T and approaches 1 as T →0 and 0 as T →∞.This is characterized as a remarkably simple coverage-probability expression.
2) Exponential Fading, No Noise, α > 2:: 0.56 is the coverage probability at T = 1, or 0 dB, in the fully loaded network, calculated as 4(4 + π)−1.A threshold of T = 1 would allow a maximum rate of 1 bps/Hz.
IV. AVERAGE ACHIEVABLE RATE: The mean achievable rate is computed in nats/Hz for a typical user using adaptive modulation and coding at the instantaneous Shannon bound ln(1 + SINR).Interference is treated as noise, so true channel capacity requiring a multiuser receiver is not achieved.
IV. AVERAGE ACHIEVABLE RATE: The rate framework can incorporate practical modulation, coding, receiver structures, and general interference distributions through a gap approximation and extensions of Theorem 1.The gap uses ln(1 + SINR/G) with G ≥1, while future work could relax the multiuser-receiver constraint.

A. General Case and Main Result · B. Special Case: α = 4 · C. Special Case: No Noise

The paper states a general theorem for the average ergodic rate of a typical mobile user, with special simplifications for α = 4 and for the no-noise regime. In the interference-limited case, per-user capacity is independent of base-station density, while network throughput increases with density.

A. General Case and Main Result: Theorem 3 gives the average ergodic rate of a typical mobile user and its associated base station.
A. General Case and Main Result: Computing τ in the general case requires three numerical integrations.
B. Special Case: α = 4: For α = 4, expression (28) can be evaluated numerically with one numerical integration, assuming an available Q(x) lookup table.
C. Special Case: No Noise: In the no-noise case, the ergodic capacity per user does not depend on λ, so increasing base-station density does not increase it.
C. Special Case: No Noise: The density independence follows because the nearest-serving distance and average nearest-interferer distance both scale as Θ(λ−1/2), causing the scaling effects to cancel.
C. Special Case: No Noise: Overall sum throughput and area spectral efficiency increase linearly with the number of base stations.
C. Special Case: No Noise: 2.15 bps/Hz is the predicted no-noise mean downlink rate with Rayleigh fading when α = 4.

V. VALIDATION OF THE PROPOSED MODEL · A. The Grid Model and An Actual BS Deployment

The paper validates its tractable stochastic-geometry model against both a regular grid and an actual urban base-station deployment. The grid is analytically difficult and optimistic, while the proposed model is intended to capture less regular deployments more naturally.

V. VALIDATION OF THE PROPOSED MODEL: The validation compares analytical coverage and rate results with the widely accepted grid model and coverage predicted from precise coordinates of an actual provider deployment.The deployment is neither a perfect grid nor Poisson.
V. VALIDATION OF THE PROPOSED MODEL: The grid model is an upper bound on coverage because its perfectly regular geometry is optimal from a coverage point of view.Neglecting outer-tier background interference adds further optimism, although that effect is not very significant.
A. The Grid Model and An Actual BS Deployment: The square grid places a home base station at the origin and interfering stations in square tiers, with each coverage area measuring 2R × 2R.Any user within distance R of a base station is guaranteed coverage.
A. The Grid Model and An Actual BS Deployment: The grid base-station density is 1/4R2 base stations per unit area, and a two-tier example uses N = 24.A hexagonal lattice would change results only by a very small constant.
A. The Grid Model and An Actual BS Deployment: Grid-model coverage retains the same principle as the general SINR formulation but is difficult to evaluate analytically, so numerical integration is used.The difficulty arises from the structured interference term.
A. The Grid Model and An Actual BS Deployment: Unlike the grid, the proposed model allows tagged and interfering base stations to be arbitrarily close to the user, subject only to the handoff ordering constraint.The grid guarantees a base station within R and excludes an interfering station closer than R.
A. The Grid Model and An Actual BS Deployment: The actual deployment spans approximately 100 ×100 km in a relatively flat, uniform urban area, with a 40 ×40 km middle-region zoom shown.Its plotted cell boundaries use a Voronoi tessellation based only on Euclidean distance, unlike potentially more complex practical boundaries.
A. The Grid Model and An Actual BS Deployment: Because this is only a single deployment, further validation is needed, although irregular terrain and concentrated populations may make random spatial models better suited to many cities.The paper suggests such environments may have less regular topologies than the example deployment.

B. Coverage Comparison · VI. FREQUENCY REUSE: COVERAGE VS. RATE

The PPP model provides conservative, analytically tractable coverage predictions that are generally no less accurate than the optimistic grid model, while frequency reuse reduces interference to improve coverage at a rate cost. For α = 4 and SINR = 1, the grid and PPP models predict success probabilities of about 0.7 and 0.53, respectively, motivating reuse-based coverage enhancement.

B. Coverage Comparison: For N = 8 versus N = 24, coverage is only slightly more optimistic for N = 8 when α = 4, with the gap increasing slightly for smaller α.The compared curves exhibit the same basic shape across SINR targets.
B. Coverage Comparison: The grid model predicts higher coverage than the PPP model for every SINR target, while the PPP model forms a lower bound and the grid model an upper bound.Both models have similar accuracy, but the PPP model is preferable for conservative predictions and greater analytical tractability.
B. Coverage Comparison: Less than 1 dB separates the SNR = 10 and SNR →∞ cases, supporting the assumption that dense cellular networks are interference-limited.The analysis therefore neglects noise in subsequent plots.
B. Coverage Comparison: The PPP model becomes more accurate at lower path loss exponents because it captures distant interference that one- or two-tier grid models omit.Far-off interference is more significant for small α, improving the PPP model’s relative accuracy in that regime.
B. Coverage Comparison: Increasing lognormal interference raises the modeled coverage probability, and the model still reasonably tracks a real deployment.The increase occurs because randomness gives cell-edge users with poor mean SINR a greater chance of meeting the target.
VI. FREQUENCY REUSE: COVERAGE VS. RATE: For α = 4 and SINR = 1, the grid model gives a success probability of about 0.7, whereas the PPP model predicts 0.53.Neither probability is sufficient for a commercial network, motivating methods that increase coverage probability.
VI. FREQUENCY REUSE: COVERAGE VS. RATE: Frequency reuse reduces interference by assigning one of δ frequency bands per cell, with larger δ monotonically decreasing interference.In the square grid, δ = 4 permits base-station separation of 4R; in the PPP model, each base station selects a band randomly.
VI. FREQUENCY REUSE: COVERAGE VS. RATE: Random PPP reuse can place adjacent base stations on the same frequency even for large δ, unlike idealized pre-planned grid reuse.The authors nevertheless restrict attention to straightforward per-cell frequency reuse and note that random allocation may resemble dynamic allocation in OFDMA networks.

A. Increasing Coverage via Frequency Reuse

The section derives coverage results under random frequency reuse, including a coverage limit as the number of bands grows and a corollary for the bands required to meet an outage target. Comparisons show central frequency planning outperforms random allocation, with increasing divergence between PPP predictions and actual base-station coverage as reuse increases.

A. Increasing Coverage via Frequency Reuse: Theorem 4 gives the coverage probability when δ frequency bands are randomly allocated to cells and interference power is exponentially distributed.Same-band interferers form a thinned PPP with density λ/δ.
A. Increasing Coverage via Frequency Reuse: As δ →∞, coverage approaches a limit that depends only on the noise power.
A. Increasing Coverage via Frequency Reuse: Corollary 1 specifies the minimum number of frequency bands needed to ensure a prescribed outage probability in the no-noise case.The result is obtained by setting coverage equal to 1 − ǫ.
A. Increasing Coverage via Frequency Reuse: Central frequency planning outperforms random frequency allocation, because random reuse more often assigns the same band to adjacent cells.The comparison uses optimal central planning for the grid model and a centralized greedy allocation benchmark for the actual base-station network.
A. Increasing Coverage via Frequency Reuse: As δ increases, the gap between PPP-predicted coverage and actual-base-station coverage grows, limiting the model’s fidelity for large reuse factors.The paper states that the proposed model in its current form is not a faithful predictor of coverage for large frequency reuse factors.

B. Frequency Reuse’s Effect on Rate · VII. CONCLUSIONS

Frequency reuse improves coverage but reduces each cell’s available bandwidth, so the average rate is maximized without reuse, at δ = 1. The proposed stochastic-geometry framework is more tractable than grid-based models, lower-bounds real-deployment performance, and offers broad extensions for future cellular-network analysis.

B. Frequency Reuse’s Effect on Rate: 1.49 nats/sec/Hz, 1.1 nats/sec/Hz, and 0.87 nats/sec/Hz are the numerical average rates for δ = 1, 2, and 3, respectively, when α = 4 and noise is absent.The rates decrease as the reuse factor increases.
VII. CONCLUSIONS: The paper presents a downlink cellular-network framework that is significantly more tractable than traditional grid-based models.Its analysis uses stochastic geometry to provide tractable coverage and rate results.
VII. CONCLUSIONS: The proposed model lower-bounds a real deployment about as accurately as the traditional grid model, which upper-bounds it.The authors state that a final accuracy verdict requires extensive comparison with additional real base-station deployments.
VII. CONCLUSIONS: Increasingly opportunistic deployments with higher density and variable cell radii may make the proposed model increasingly accurate while preserving its tractability.This conclusion is tied to current deployment trends rather than established accuracy across all future networks.
VII. CONCLUSIONS: Future work could extend the model to uplinks, repulsive base-station placements, and heterogeneous macro, micro, pico, and femto networks.The heterogeneous extension would allow differing transmit powers and coverage areas.
VII. CONCLUSIONS: Further analysis could examine how multiple-antenna techniques, opportunistic scheduling, and base-station cooperation affect network performance.These topics are identified as additional directions enabled by the framework’s focus on neighboring base stations.

APPENDIX A · APPENDIX B

Appendix A simplifies the proof under exponential desired-signal fading, while Appendix B derives the typical-user ergodic rate and reports coverage, reuse, and rate comparisons across spatial models.

APPENDIX A: APPENDIX A: The proof follows Theorem 1 through step (a) of (12), then uses the new assumption gi ∼exp(µ) to obtain a simpler form.The integration limits remain r to ∞, with s = µTr^α substituted afterward.
APPENDIX A: APPENDIX A: Substituting (40) into (11) with v → r2 yields the desired result and completes the proof.
APPENDIX B: APPENDIX B: The typical user’s ergodic rate is defined as τ ≜ E[ln(1 + SINR)], averaging over the spatial PPP and fading distribution.The derivation applies the tail-integral identity for positive random variables and parallels Theorems 1 and 2.
APPENDIX B: APPENDIX B: The Poisson model places base stations and mobiles randomly, associates each mobile with its nearest base station, and produces a Voronoi tessellation.
APPENDIX B: APPENDIX B: With α = 4, the no-noise approximation is quite accurate, and coverage is only slightly lower with 24 interfering base stations than with 8.The comparison uses the square-grid model with N = 8, 24.
APPENDIX B: APPENDIX B: The proposed model is a lower bound and becomes more accurate at lower path-loss exponents in coverage comparisons with the grid model.The plotted cases use α = 2.5 and α = 4, SNR = 10, and exponential interference.
APPENDIX B: APPENDIX B: Lower spatial reuse, represented by higher δ, improves outage performance, while all three reuse curves exhibit similar behavior.The comparison considers frequency-reuse factors δ = 2 and δ = 4.
APPENDIX B: APPENDIX B: Average rate is maximized when every cell uses the same frequency and therefore the complete bandwidth.The rate comparison uses SNR = 10 dB for Poisson-distributed and actual base-station locations.