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Downlink Rate Distribution in Heterogeneous Cellular Networks under Generalized Cell Selection
Harpreet S. Dhillon, Jeffrey G. Andrews
TL;DR
The paper addresses HetNet models that do not realistically distinguish long-term shadowing from small-scale fading during cell selection. It derives downlink rate distributions under generalized selection using an equivalent model, then studies shadowing’s effect on load balancing and rate-optimal bias. The results show that shadowing can naturally balance tier loads and reduce the need for artificial selection bias in certain regimes.
Problem
Prior models either ignore shadowing in cell selection or base selection on instantaneous received power, rather than distinguishing long-term effects from small-scale fading.
Method
The paper derives downlink rate distributions under generalized cell selection and represents shadowing through an equivalent model with appropriately scaled transmit powers.
Results
The analysis shows that shadowing can naturally balance load across tiers, while rate distributions and fifth-percentile-rate behavior are validated against simulations.
Takeaways & Limitations
In certain regimes, shadowing reduces the artificial cell-selection bias needed for rate maximization.
Abstract
from arXiv · showhide
Considering both small-scale fading and long-term shadowing, we characterize the downlink rate distribution at a typical user equipment (UE) in a heterogeneous cellular network (HetNet), where shadowing, following any general distribution, impacts cell selection while fading does not. Prior work either ignores the impact of channel randomness on cell selection or lumps all the sources of randomness into a single variable, with cell selection based on the instantaneous signal strength, which is unrealistic. As an application of the results, we study the impact of shadowing on load balancing in terms of the optimal per-tier selection bias needed for rate maximization.
I. INTRODUCTION
The paper addresses unrealistic cell-selection assumptions in HetNet models by distinguishing long-term path loss and shadowing from small-scale fading. It extends analysis from SINR to downlink rate, which also depends on BS load, and uses an equivalent model to study biasing.
- Motivation: Prior models either ignore shadowing in cell selection or base selection on maximum instantaneous received power.These assumptions do not capture the distinct roles of long-term channel effects and small-scale fading.
- Motivation: Downlink rate analysis must account for BS load in addition to SINR.The paper therefore focuses on rate rather than only signal quality.
- Motivation: Small cells can maximize downlink rate despite poorer SINR when their lower load provides each UE more time-frequency resources.This motivates analyzing cell selection jointly with load.
- Contribution: The proposed equivalence captures shadowing by scaling tier transmit powers and then selecting the closest BS within each tier.This extends service-area and load analysis to generalized cell selection.
II. SYSTEM MODEL
The system model represents a K-tier HetNet with tier-specific powers, densities, biases, fading, and general shadowing. Cell selection uses long-term biased received power, while displacement transforms shadowing into an equivalent PPP model that supports rate and load analysis.
- Network model: The HetNet contains K independent PPP tiers differing in transmit power, deployment density, and cell-selection bias.Resources are orthogonally partitioned so each resource block serves one UE without intra-cell interference.
- Channel model: Shadowing may follow any distribution whose relevant fractional moment is finite.The model includes Rayleigh fading, shadowing X_k, and power-law path loss with exponent α; lognormal shadowing is a common special case.
- Cell selection: Cell selection uses long-term biased received power, excluding rapidly varying fading from the selection decision.A UE first identifies the candidate BS with highest long-term received power in each tier, then selects across tiers using bias B_k.
- Equivalent model: The displacement theorem converts shadowed tier locations into an equivalent homogeneous PPP with transformed density.In this representation, the candidate serving BS in each tier is simply the closest point to the origin.
- Equivalent model: Equivalent tier densities extend closest-BS SIR results to generalized selection and provide the basis for extending analysis to rate distributions involving BS load.The transformation preserves the relevant distributional representation while incorporating shadowing through the equivalent model.
III. DOWNLINK RATE DISTRIBUTION
This section characterizes downlink rate coverage under generalized cell selection by combining selection probability, conditional SIR, and tagged-BS load. It uses an equivalent model to obtain a tractable rate-coverage expression while accounting for service-area bias and shadowing.
- Rate coverage is the probability that a typical UE's downlink rate exceeds a required threshold T, and its CCDF characterizes the rate distribution.
- Tagged-BS load reflects service-area randomness and Poisson UE arrivals, with equal time-frequency resource allocation across served UEs.
- Because tagged-BS service area and SIR are jointly difficult to characterize, the analysis assumes their associated load and SIR variables are independent.The paper states that this assumption does not compromise the accuracy of the analysis.
- The rate analysis combines selection probability, conditional SIR, and the load distribution of the tagged serving BS.The resulting expression is obtained by substituting the corresponding lemmas into the rate-coverage formulation.
- If all tiers share the same relevant shadowing fractional moment, the downlink rate distribution is invariant to the shadowing distributions.
- A no-shadowing HetNet with appropriately scaled transmit powers yields the same rate-coverage expression as the generalized cell-selection model.This equivalence extends results derived under no shadowing to generalized cell selection.
IV. NUMERICAL RESULTS
The numerical results validate the load approximation and show that generalized shadowing-aware cell selection yields tight rate-distribution approximations. Shadowing can naturally balance tier loads, reducing the artificial bias needed to maximize rate.
- The analytic load approximation is fairly accurate, and exact SIR and selection-probability components produce a tight rate-distribution approximation.The rate expression is validated against numerical results in Fig. 2.
- Fig. 1 evaluates the CDF of load Ψk for two-tier and single-tier settings under shadowing standard deviations [4 4] dB and [4 8] dB.The plotted configurations use λu = 20λ1, λ2 = 2λ1 for K = 2, and λ2 = 0 for K = 1.
- Shadowing produces natural load balancing across tiers compared with the no-shadowing baseline.The equivalent model interprets this as shadowing increasing small-cell effective transmit power and expanding their coverage areas.
- The optimal selection bias maximizing fifth-percentile rate is smaller when shadowing provides part of the load-balancing effect.The smaller artificial bias in this case also achieves the highest rate.
V. CONCLUSIONS
The paper derives downlink rate distributions under generalized cell selection while distinguishing long-term shadowing and path loss from small-scale fading. Its equivalent interpretation shows that shadowing can balance tier loads and reduce the need for artificial selection bias in certain regimes.
- The paper derives downlink rate distributions under generalized cell selection while distinguishing long-term shadowing and small-scale fading.
- Shadowing can be equivalently represented by appropriately scaling transmit powers.
- Fig. 2 reports rate coverage and fifth-percentile rate under specified tier densities, user densities, and selection-bias settings.
- In certain regimes, shadowing naturally balances load across tiers and reduces the need for artificial cell-selection bias.