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Load-Aware Modeling and Analysis of Heterogeneous Cellular Networks
Harpreet S. Dhillon, Radha Krishna Ganti, Jeffrey G. Andrews
TL;DR
Existing random-spatial HCN models typically neglect traffic load by assuming every base station always transmits, which can misrepresent interference from lightly loaded small cells. This paper conditionally thins the interference field according to tier-specific activity factors and derives coverage probabilities for strongest-signal association. The analysis shows that adding lightly loaded small cells increases coverage, while fully loaded models are pessimistic.
Problem
Random-spatial HCN models typically assume all base stations transmit continuously, neglecting traffic load and potentially overstating small-cell interference.
Method
The paper models tier-specific base-station activity by conditionally thinning interfering base stations after a typical mobile connects to its strongest-signal base station.
Results
The analysis derives exact open- and closed-access coverage expressions with arbitrarily tight finite-term bounds and shows that lightly loaded small-cell tiers increase coverage.
Takeaways & Limitations
Fully loaded models are extremely pessimistic for coverage, whereas conditional thinning captures different load levels across heterogeneous base-station tiers.
Abstract
from arXiv · showhide
Random spatial models are attractive for modeling heterogeneous cellular networks (HCNs) due to their realism, tractability, and scalability. A major limitation of such models to date in the context of HCNs is the neglect of network traffic and load: all base stations (BSs) have typically been assumed to always be transmitting. Small cells in particular will have a lighter load than macrocells, and so their contribution to the network interference may be significantly overstated in a fully loaded model. This paper incorporates a flexible notion of BS load by introducing a new idea of conditionally thinning the interference field. For a K-tier HCN where BSs across tiers differ in terms of transmit power, supported data rate, deployment density, and now load, we derive the coverage probability for a typical mobile, which connects to the strongest BS signal. Conditioned on this connection, the interfering BSs of the $i^{th}$ tier are assumed to transmit independently with probability $p_i$, which models the load. Assuming - reasonably - that smaller cells are more lightly loaded than macrocells, the analysis shows that adding such access points to the network always increases the coverage probability. We also observe that fully loaded models are quite pessimistic in terms of coverage.
I. INTRODUCTION
The paper extends tractable random-spatial HCN models to account for heterogeneous base-station load by conditionally thinning interference. It derives coverage results and shows that lightly loaded small cells improve coverage, while fully loaded models are pessimistic.
- Motivation: Fully loaded HCN models assume all base stations transmit concurrently, producing pessimistic coverage and average-rate estimates, especially for smaller cells.Smaller cells generally serve fewer users because of their smaller coverage areas.
- Load-aware model: Conditional thinning models each tier’s interfering base stations as transmitting independently with probability p_i after conditioning on the typical mobile’s serving connection.The activity factors can vary across tiers to represent their different coverage areas and loads.
- Coverage analysis: The paper derives exact open- and closed-access coverage expressions and finite-term upper and lower bounds that can be made arbitrarily tight.The bounds also indicate how many terms are needed to meet a prescribed approximation error.
- Design insights: Adding small cells increases overall open-access coverage in the interference-limited regime when their load is lower than the macrocell load.The analysis also interprets conditional thinning as both interference reduction and bias toward the serving base station through expanded coverage regions.
- Design insights: With equal loads and target SINRs across tiers, interference-limited open-access coverage is invariant to transmit power and deployment density and matches a single-tier network with the same activity factor.
- System model: The model represents a K-tier HCN using PPP-distributed base stations that differ in transmit power, target SINR, deployment density, and activity factor.A typical mobile connects to the strongest received signal, with Rayleigh fading, distance-based path loss, and thermal noise ignored for tractability.
Temporal and spatial correlation in BS activity factors:
The paper models BS activity through conditional thinning but acknowledges that real activity factors exhibit temporal and spatial correlation. For tractability, the analysis assumes independent activity across BSs and derives open- and closed-access coverage results under β_i > 1.
- Temporal and spatial correlation in BS activity factors:: BS activity factors can be temporally and spatially correlated through user mobility, interference, and traffic/load patterns.The paper gives neighboring BSs as an example of positive correlation caused by interference-induced reductions in data rate.
- B. Proposed Load Model and Mathematical Preliminaries: The model conditions on strongest-received-power association and lets each tier-i interferer transmit independently with probability p_i.The probability p_i represents tier-specific load, while 1 − p_i represents idleness.
- B. Proposed Load Model and Mathematical Preliminaries: The derivation assumes target SIR thresholds β_i > 1, ensuring that at most one active BS can satisfy a typical mobile’s target SIR requirements.The paper reports that the resulting analysis remains applicable down to approximately −2 dB in numerical results.
- C. Coverage Regions: Conditional thinning removes inactive interferers from coverage-region illustrations and equivalently expands active-BS coverage regions, biasing mobiles toward their serving BS.These are presented as direct and indirect views of the same load-model effect.
- III. COVERAGE PROBABILITY: The analysis derives exact coverage probabilities for typical mobiles in open- and closed-access K-tier HCNs.The closed-access expression changes the product from all tiers K to the permitted subset B.
B. Special Cases of Interest
The special cases characterize how load affects scale invariance, fully loaded coverage, and the impact of adding tiers. Coverage is unchanged under density scaling only when activity factors are equal, while adding a tier improves coverage when it lowers effective load under the stated condition.
- B. Special Cases of Interest: Fully loaded networks set p_i = 1 for every tier and recover the corresponding fully loaded open-access coverage expression.In this case, A = 0 and g(m) = g_c(m) = 0 for all m.
- B. Special Cases of Interest: In a single-tier open-access network, coverage is independent of BS density λ and transmit power P for any activity factor p.The paper calls this property scale invariance.
- B. Special Cases of Interest: For K-tier networks with equal target SIRs, scale invariance holds when all tiers have the same activity factor, and coverage then matches a single-tier network.With unequal activity factors, coverage is not generally scale invariant.
- B. Special Cases of Interest: If p_1 < p_2, increasing tier-1 density is expected to increase coverage; if p_1 > p_2, it is expected to decrease coverage.The effects cancel when the two tiers have equal activity factors.
- B. Special Cases of Interest: Adding a new tier increases, decreases, or leaves coverage unchanged according to whether its load satisfies, reverses, or meets the stated threshold condition.The condition is expressed through the activity and density–power terms in equation (28).
- B. Special Cases of Interest: Coverage decreases with effective load, so adding a tier increases coverage when it lowers the network’s effective load.This connects the new-tier condition to the effective-load definition.
C. Bounds on the Coverage Probability
The paper derives coverage-probability bounds by truncating the exact infinite series over g(m). Increasing the truncation order makes the bounds arbitrarily tight and determines the terms needed for a prescribed error.
- The exact coverage-probability expression requires an infinite summation over g(m).
- The upper and lower bounds truncate the infinite sum at odd and even numbers of terms, respectively.
- Increasing m makes the coverage-probability bounds arbitrarily tight.
- M_ϵ gives the number of terms needed to keep approximation error below a predefined ϵ.
- For α = 4 and m = 2, some terms of g(m) yield closed-form bounds depending only on the first two terms.
IV. NUMERICAL RESULTS
The numerical-results section presents selected analyses intended to validate key modeling assumptions and visualize important trends.
- The numerical results focus on validating key modeling assumptions and visualizing important trends.
A. Convergence of Infinite Sum
The paper examines convergence of the infinite sum in the coverage-probability expression. Fewer terms are needed at higher BS activity, while extremely small transmission probabilities can require more terms.
- The truncated series P_m is studied as a function of m to assess convergence in a single-tier network.
- Lower BS activity factors require more terms for the series to converge.
- For ϵ = 10−8, the number of required terms is evaluated against the lowest tier's transmission probability.
- When p = .25 and A/η > 1, the series first increases, then decreases, and finally converges to its limiting value.
- The required number of terms is reasonably small unless a tier's transmission probability is extremely small.
1) Comparison with the fully loaded system:
The numerical results compare load-aware coverage with fully loaded and alternative spatial models, including grid, actual deployment, and detailed simulation settings. They show that full-load coverage estimates can be pessimistic and that the analytical model agrees reasonably closely with simulation.
- Comparison with the fully loaded system:: The PPP model is compared with grid and actual macrocell-location models while the second tier remains a PPP.
- Comparison with the fully loaded system:: Fully loaded coverage estimates are quite pessimistic even for reasonably high activity factors such as p = .7 −.8.
- Comparison with the fully loaded system:: The analytical results are compared with detailed simulation using activity factors derived from actual coverage regions.
- Comparison with the fully loaded system:: Coverage increases with λ2 when second-tier BSs are less active than first-tier BSs.
- Comparison with the fully loaded system:: The analytical and detailed-simulation results are reasonably close, especially relative to previously known fully loaded results.
C. Scale Invariance and Effect of Adding Small Cells
Coverage is invariant to density when tier activity factors match, but adding less-active small-cell tiers increases coverage. Closed-access effects also diminish as more small cells become open access, especially under light load.
- Scale invariance: Coverage probability is invariant to second-tier density when p1 = p2.This follows from the two-tier analysis with a common target SIR.
- Adding small cells: Coverage probability increases with λ2 when second-tier BSs are less active than first-tier BSs.The result applies to the regime considered for lightly loaded small cells.
- Open vs Closed Access: The coverage gap between open and closed access decreases as the fraction f of second-tier BSs in open access increases.The closed-access scenario divides the second tier into independent open- and closed-access PPPs.
- Open vs Closed Access: The open-versus-closed-access gap is smaller when second-tier BSs are lightly loaded, making interference from closed-access small cells negligible when enough are open access.This conclusion is reported for the two-tier scenario studied in Fig. 8.
- Adding small cells: Adding lightly loaded access points such as pico or femtocells to a macrocell network always increases coverage probability.The paper attributes this conclusion to conditionally thinning the interference field according to BS load.
- Extensions: The framework identifies temporal and spatial correlation in BS activity, interacting spatial models, queue-based load, and uplink analysis as extensions.The current work focuses on downlink coverage evaluation rather than flow-level performance.
APPENDIX A PROOF OF LEMMA 2:
The appendix evaluates terms in the proof by transforming integrals, applying standard integration results, and combining the resulting expressions to obtain the lemma.
- Integral evaluation: Fubini’s theorem exchanges expectation and integration, while substitutions and integration by parts simplify the resulting integrals.The derivation also uses the gamma function and generalized hypergeometric function before combining the terms.
- Proof setup: Under βi > 1 for all tiers, at most one BS in the network can establish a downlink connection with a typical mobile.This condition is used to simplify the connection event in the proof.
- Interference terms: The proof expresses the relevant expectation using SIR(x), total interference, and the Laplace transform of effective interference.The effective interference is defined as I′ = I − Pihx∥x∥−α, and its Laplace transform matches that of I.
- Conclusion: The appendix combines the simplified terms from the preceding equations to obtain the stated result.The final representation uses the generalized hypergeometric function 2F1.