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Modeling and Analysis of K-Tier Downlink Heterogeneous Cellular Networks
Harpreet S. Dhillon, Radha Krishna Ganti, Francois Baccelli, Jeffrey G. Andrews
TL;DR
Heterogeneous cellular networks make tractable analysis difficult as deployments combine multiple BS classes with different characteristics. This paper develops a tractable K-tier model and derives coverage, outage, rate, and load results, including an interference-limited open-access invariance of coverage with respect to tier count and BS density.
Problem
Increasingly heterogeneous cellular networks are difficult to analyze because multiple BS classes have different traits, while existing approaches rely on simplified models or complex simulations.
Method
The paper models each BS class as a Poisson point-process tier characterized by transmit power, SINR target, and density, then analyzes network metrics under stated strongest-BS, fading, and access assumptions.
Results
The analysis yields coverage and outage expressions with a simple high-SINR closed form, accuracy down to -4 dB under weaker assumptions, and coverage invariance to tier count and BS density for equal-target-SINR open access in dense networks.
Takeaways & Limitations
In interference-limited open-access networks with equal SINR targets, adding tiers or BSs need not change typical SINR while aggregate throughput increases linearly with BS count.
Takeaways & Limitations
The random spatial model could be improved with point processes capturing BS repulsion, minimum separation, or clustering, and requires validation across additional deployment scenarios.
Abstract
from arXiv · showhide
Cellular networks are in a major transition from a carefully planned set of large tower-mounted base-stations (BSs) to an irregular deployment of heterogeneous infrastructure elements that often additionally includes micro, pico, and femtocells, as well as distributed antennas. In this paper, we develop a tractable, flexible, and accurate model for a downlink heterogeneous cellular network (HCN) consisting of K tiers of randomly located BSs, where each tier may differ in terms of average transmit power, supported data rate and BS density. Assuming a mobile user connects to the strongest candidate BS, the resulting Signal-to-Interference-plus-Noise-Ratio (SINR) is greater than 1 when in coverage, Rayleigh fading, we derive an expression for the probability of coverage (equivalently outage) over the entire network under both open and closed access, which assumes a strikingly simple closed-form in the high SINR regime and is accurate down to -4 dB even under weaker assumptions. For external validation, we compare against an actual LTE network (for tier 1) with the other K-1 tiers being modeled as independent Poisson Point Processes. In this case as well, our model is accurate to within 1-2 dB. We also derive the average rate achieved by a randomly located mobile and the average load on each tier of BSs. One interesting observation for interference-limited open access networks is that at a given SINR, adding more tiers and/or BSs neither increases nor decreases the probability of coverage or outage when all the tiers have the same target-SINR.
I. INTRODUCTION
The paper addresses the difficulty of analyzing increasingly heterogeneous cellular networks by proposing a tractable K-tier stochastic model and deriving key performance metrics. It evaluates coverage, outage, rate, and tier load under explicit access, fading, and SINR assumptions.
- Metrics: The analysis derives tractable expressions for SINR statistics, coverage or outage probability, average rate, and average load across tiers.Coverage is analyzed under both open and closed access, while average rate is conditioned on being in coverage.
- Model: K-tier HCNs are modeled as independent PPP-distributed BS tiers differing in transmit power, supported data rate, and density.Each tier is characterized by {P_i, β_i, λ_i}.
- Validation: Coverage analysis is closed-form in the high-SINR regime and remains accurate down to -4 dB under weaker assumptions.The paper also compares the model with an actual 4G macro-cell deployment combined with randomly placed lower tiers.
- Findings: In interference-limited open access with common SINR targets, coverage is independent of tier count and BS density, while aggregate throughput increases linearly with BS count.The corresponding coverage probability generally decreases with tier count and density under closed access.
- Model: The model assumes strongest-BS association, Rayleigh fading, and coverage when some candidate BS exceeds its SINR target β_i.The simplifying assumption β_i > 1 ensures that at most one BS can exceed the required threshold.
B. Coverage Regions
The paper visualizes coverage regions for two- and three-tier heterogeneous networks using independent PPP deployments, including comparisons with actual macro-cell locations. The regions illustrate how transmit power and tier placement shape coverage, while smaller cells help where macrocell coverage is poor.
- Two-tier networks: Two-tier visualizations compare independently PPP-distributed macro and femto BSs with a deployment using actual 4G macro locations and PPP femtocells.The actual-deployment and randomly placed cases have qualitatively similar coverage regions.
- Coverage-region construction: Coverage regions are generated by randomly placing each BS tier as an independent PPP, then tessellating the plane using maximum-SINR connectivity without fading.The maximum-SINR model is equivalent to maximum-SIR and maximum-power connectivity in the absence of fading.
- Three-tier networks: Three-tier visualizations add pico cells between macro and femto tiers, using independent PPPs or actual macro locations with the remaining parameters unchanged.The modeled tiers are macro, pico, and femto cells.
- Transmit-power effects: Typical LTE transmit powers are approximately 50 W for macro, 2 W for pico, and 0.2 W for femtocells.These power differences make femtocell regions much smaller, especially near higher-power BSs.
- Transmit-power effects: Pico-cell coverage footprints increase farther from macro BSs, highlighting the role of smaller cells where macrocell coverage is poor.Femtocell regions are usually much smaller than macro- and pico-cell regions, particularly near higher-power BSs.
- Model scope: The model assumes continuous constant-power transmission in a single frequency band, while random inactivity, orthogonal resources, scheduling, antennas, and random sectoring can extend or improve the framework.If a fraction f of slots is unused at random, the interfering-BS density becomes (1−f)λ.
III. COVERAGE PROBABILITY AND AVERAGE LOAD PER TIER
The paper defines coverage through a randomly located user's ability to connect above an SINR threshold and derives coverage probability for open and closed access. In interference-limited open-access networks with equal tier thresholds, coverage is independent of tier count, density, and transmit-power differences.
- Coverage definition: A user is in coverage when it can connect to at least one BS whose SINR exceeds its threshold; with a common threshold β > 1, coverage is the effective-SINR CCDF.Outage is the complementary CDF, 1−CCDF.
- Open access: Under open access, users may connect to any tier, which reduces to selecting the BS delivering the maximum received SINR.The derivation assumes the strongest candidate BS and SINR thresholds β_i > 1.
- Coverage derivation: Theorem 1 gives a simple general expression for coverage probability when every tier has β_i > 1, using the fact that a mobile can connect to at most one BS.Coverage is expressed as the sum of the probabilities that each BS connects to the user.
- Interference-limited result: In interference-limited open access with equal tier thresholds, coverage is unaffected by the number of tiers, their relative densities, or their transmit powers.The resulting coverage probability is exactly the same as in the single-tier case.
- Interference-limited result: In an interference-limited single-tier network, coverage is independent of BS density and depends only on the target SIR.Scaling BS density changes received and interference powers by the same factor, so the effects cancel.
2) Average Load per Tier:
The paper defines average per-tier load as the fraction of covered users served by that tier and derives results for open and closed access. In interference-limited networks, load increases with tier density and transmit power, while closed access restricts eligible serving tiers.
- Average tier load is the fraction of users in coverage served by that tier, equivalently the fraction of time mobiles connect to it.
- Open access: In interference-limited networks, each tier’s load is directly proportional to λ_jP_j^(2/α).
- Open access: Higher BS density or transmit power increases a tier’s served-user fraction, while a higher SIR threshold decreases it.
- Closed access: Closed access allows connection only to selected tiers, while nonselected tiers remain interferers and restricted strongest-BS associations produce outage.The paper states that this connectivity constraint reduces coverage probability.
- Closed access: With equal thresholds and transmit powers across tiers, closed access has lower coverage than open access by a factor involving the allowed-tier power-density sum.
2) Average Load per Tier:
The paper derives average rate for covered users and closed-access load results. With equal SINR thresholds, covered-user rate is independent of tier BS densities, while tier load follows density and transmit-power effects.
- Average rate: Average rate is computed conditionally on coverage, so it differs from the classic ergodic rate E[R].The metric reflects the average service rate provided to users who are in coverage.
- Open access: For open access, the average covered-user rate is given by Theorem 2 and can be evaluated through a single integral.The theorem is presented for σ^2 = 0, with extension to noise described as straightforward.
- Open access: With the same SINR threshold β for every tier, open-access average rate is independent of the BS density of each tier.The paper attributes this to the density-independent distribution of maximum SIR in this case.
B. Closed Access
Closed-access rate is derived for mobiles restricted to a subset of tiers. With a common threshold, covered-user average rate is unaffected by access control, even though closed access has lower coverage and therefore lower ergodic rate.
- The closed-access average rate is derived for a mobile allowed to connect to only a subset B of the K tiers.
- For a common threshold β across tiers, the paper gives a specialized closed-access expression for average rate among covered users.
- Equal thresholds make covered-user average rate invariant to access control, but lower closed-access coverage leads to lower ergodic rate than open access.
V. NUMERICAL RESULTS
Numerical results examine thermal-noise effects, PPP-model validity, and the β > 1 assumption, while outlining extensions for richer network and traffic models.
- A. Effect of Thermal Noise: Thermal noise has a very limited effect on coverage probability because the typical HCNs are interference limited.The analysis therefore ignores noise in the remainder of the numerical-results section.
- B. Validity of PPP Model: The PPP assumption is nearly as accurate as the grid model for macro-cells, with PPP giving a lower bound and the grid model an upper bound to actual coverage.
- B. Validity of β > 1 Assumption: Theoretical coverage results match simulations reasonably well for β_i > 1 and remain a tight upper bound down to about β_1 = −4 dB.The analysis therefore covers typical cell-edge users.
- C. Average Rate: The average-rate results are also accurate down to about −4 dB target-SIR.
- VI. Conclusions: The model’s extensions include multiple antennas, power control, interference cancellation, scheduling, resource allocation, frequency reuse, and BS cooperation.
- VI. Conclusions: Further modeling should address repulsive or clustered BS locations, additional deployment scenarios, non-homogeneous user distributions, and realistic traffic models.
APPENDIX
The appendix proves that no more than m signal-to-interference ratios can exceed 1/m, using a contradiction argument that generalizes from m = 1.
- At most m of the b_i values can be greater than 1/m for any positive integer m.
- The proof first handles m = 1 by contradiction, showing that two b_i values greater than one violate the relevant inequality.
- For general m, assuming m + 1 values exceed 1/m leads to the analogous contradiction.
B. Proof of Theorem 1
The proof derives K-tier coverage probability under maximum-SINR connectivity by combining a lemma, Campbell–Mecke, Rayleigh fading, and Laplace-transform calculations.
- The coverage probability is derived under maximum-SINR connectivity assuming β_i > 1 for every tier.
- Campbell–Mecke and Rayleigh fading transform the coverage expression into one involving the Laplace transform of cumulative interference from all tiers.Stationarity makes this interference transform independent of the serving BS location, so it is denoted L_Ii.
- Independence of fading and the PPP probability generating functional produce the interference Laplace transform, which is then simplified using Gamma-function properties.
- Substituting the interference transform into the preceding expression yields the K-tier coverage probability P_c({λ_i}, {β_i}, {P_i}).
C. Proof of Proposition 1
The proof expresses the average fraction of users served by tier j using stationarity, ergodicity, Palm calculus, Slivnyak’s theorem, Bayes’ rule, and the single-tier coverage probability.
- The average fraction of users served by tier j is formulated over an increasing sequence of convex regions in R^2.
- Stationarity and ergodicity of the PPP, together with the reduced Palm distribution, Slivnyak’s theorem, and Bayes’ rule, justify the probabilistic reformulation.
- The result follows from Theorem 1 by identifying the relevant single-tier coverage probability for tier j.
D. Proof of Theorem 2
The section derives the average rate for a randomly chosen mobile user under coverage by first obtaining the conditional CCDF of the strongest candidate BS’s SIR and then integrating it.
- The average rate achievable by a randomly chosen mobile user under coverage is expressed using the coverage event and the candidate-BS SIR thresholds.
- The conditional CCDF of the maximum candidate-BS SIR is derived as the starting point for the average-rate calculation.
- The proof applies Bayes’ theorem, Lemma 1 under βi > 1 for all tiers, and Theorem 1, with βmin defined as the minimum tier threshold.
- The derivation denotes the maximum candidate-BS SIR by X and evaluates the average rate through an integral involving X.
- Changing the order of integration and substituting the conditional CCDF yields the final average-rate expression.