Source-linked AI summary
Heterogeneous Cellular Networks with Flexible Cell Association: A Comprehensive Downlink SINR Analysis
Han-Shin Jo, Young Jin Sang, Ping Xia, Jeffrey G. Andrews
TL;DR
The paper addresses tractable SINR analysis for heterogeneous cellular networks with flexible, biased cell association. It models multi-tier networks with randomly located base stations and derives outage, rate, and throughput metrics. Under interference-limited full loading and unbiased association, outage probability and average ergodic rate are unaffected by the number of base stations or tiers, while biasing changes these metrics.
Problem
Existing HCN analysis motivates a more general model that supports flexible cell association and accurate SINR characterization across the relevant range.
Method
The paper models multi-tier HCNs with random BS locations and derives SINR metrics under biased association based on long-term averaged received power.
Results
Under interference-limited full loading with unbiased association, the number of tiers and BS density does not affect outage probability or average ergodic rate under the stated assumptions.
Takeaways & Limitations
Biasing alters outage, rate, and related metrics, so the framework supports evaluating flexible association policies in HCN design.
Abstract
from arXiv · showhide
In this paper we develop a tractable framework for SINR analysis in downlink heterogeneous cellular networks (HCNs) with flexible cell association policies. The HCN is modeled as a multi-tier cellular network where each tier's base stations (BSs) are randomly located and have a particular transmit power, path loss exponent, spatial density, and bias towards admitting mobile users. For example, as compared to macrocells, picocells would usually have lower transmit power, higher path loss exponent (lower antennas), higher spatial density (many picocells per macrocell), and a positive bias so that macrocell users are actively encouraged to use the more lightly loaded picocells. In the present paper we implicitly assume all base stations have full queues; future work should relax this. For this model, we derive the outage probability of a typical user in the whole network or a certain tier, which is equivalently the downlink SINR cumulative distribution function. The results are accurate for all SINRs, and their expressions admit quite simple closed-forms in some plausible special cases. We also derive the \emph{average ergodic rate} of the typical user, and the \emph{minimum average user throughput} -- the smallest value among the average user throughputs supported by one cell in each tier. We observe that neither the number of BSs or tiers changes the outage probability or average ergodic rate in an interference-limited full-loaded HCN with unbiased cell association (no biasing), and observe how biasing alters the various metrics.
I. INTRODUCTION
The paper motivates stochastic modeling for heterogeneous cellular networks and develops an analytical framework that characterizes SINR and related metrics under arbitrary per-tier association biases.
- Network motivation: HCNs combine conventional macrocells with diverse lower-power base stations, including picocells, femtocells, and relay BSs.Tiers can differ in transmit power, path loss exponent, and spatial density.
- Network motivation: Positive biasing can encourage users to associate with lightly loaded lower-tier cells even when the macrocell signal is stronger.The smaller coverage area of lower-tier cells usually results in lighter load.
- Paper objective: The paper proposes a model and analytical framework for SINR, outage or coverage, and data rate in HCNs with arbitrary per-tier association biases.The framework targets flexible cell association as an important factor in overall HCN performance.
- Modeling gap: Hexagonal-grid SINR expressions depend on multiple random variables and are typically estimated by Monte Carlo methods, complicating metric analysis.The paper notes that base-station locations vary across deployments, motivating abstraction through spatial random processes.
- Modeling gap: PPP-based HCN analyses existed but did not jointly provide all-SINR accuracy, flexible biasing, and per-tier coverage probabilities.The cited prior results were exact only for positive SINR in dB, excluded biasing, and omitted per-tier coverage probabilities.
B. Contributions
The paper develops a stochastic HCN model with biased long-term-power association and derives association, outage, rate, and throughput metrics, including notable invariances in special cases.
- Flexible cell association: Users associate with the BS offering maximum long-term averaged received power with biasing, while fading is ignored for association.This differs from connecting to the BS with the highest instantaneous SINR.
- Flexible cell association: The analysis derives per-tier association probabilities, serving-distance distributions, and the average number of users associated with a BS in each tier.The resulting cell-load quantity captures the effect of cell association on tier loading.
- SINR analysis: The framework derives complete outage probabilities over all SINRs for a randomly selected mobile in the network or a specified tier.Outage probability at an SINR threshold is equivalently the SINR CDF; general expressions are easily computable and become closed-form in some cases.
- Performance metrics: In interference-limited HCNs with no biasing, outage probability and average ergodic rate are independent of BS transmit power, BS density, and the number of tiers.The paper also derives minimum average user throughput across tiers and observes that adding infrastructure can primarily increase simultaneous service capacity.
- Performance metrics: Biasing strictly worsens outage and rate in fully loaded networks, while tier loading can significantly affect the relative merits of association policies.The paper presents the framework for evaluating bias choices rather than drawing conclusions about their relative merits.
- System model: The framework models K tiers using transmit power, BS spatial density, path loss exponent, and biasing factors, with BSs and users represented by homogeneous PPPs.It also assumes Rayleigh fading, orthogonal multiple access within cells, additive noise, and common transmit power within each tier.
A. Flexible Cell Association and Cell Load
The paper models open-access association by maximum biased received power and derives per-tier association probabilities and cell loads. Biasing and tier density redistribute users across tiers, with density increasing tier-wide association while reducing users per BS.
- Flexible Cell Association: Maximum biased-received-power association lets users access any tier and connects them to the BS offering the strongest biased received power.With all bias factors equal to one, this becomes conventional unbiased association based on strongest average received power.
- Per-Tier Association: The per-tier association probability depends on each tier’s BS density, transmit power, path loss exponent, and bias factor.The paper states that BS density is more dominant in determining the association probability than transmit power or bias factor.
- Cell Load: Lemma 2 converts per-tier association probabilities into the average number of users associated with a BS, which quantifies each tier’s cell load.The result is obtained from the average numbers of users and BSs in each tier.
- Cell Load: Increasing a tier’s BS density raises the number of users associated with that tier but decreases its users per BS.The decrease follows because the tier’s association area grows more slowly than its BS population under the stated model.
- Load Balancing: Increasing transmit power or bias sends more users to the corresponding tier, increasing its cell load while decreasing load in other tiers.Deploying more BSs instead reduces the cell load of the corresponding tier and other tiers.
B. Statistical Distance to Serving Base Station
For a typical user associated with tier k, the paper treats the serving-BS distance as a random variable and derives its probability density under the association rule. The interference model accounts for tier-specific exclusion distances induced by association.
- Serving Distance: The paper defines X_k as the distance from a typical user at the origin to its serving BS in tier k.The typical user is analyzed after conditioning on its associated tier.
- Serving Distance: The serving-BS distance X_k is random because base stations are modeled as Poisson point processes.Its probability density function is given in Lemma 3.
- Serving Distance: Conditioned on association with tier k, the distribution of X_k is derived from the joint event that the nearest tier-k BS lies beyond a given distance and tier k is selected.The derivation uses P[n = k] = A_k, the per-tier association probability.
- Interference Geometry: Given serving distance X_k = x, interfering BSs in every tier lie outside an association-dependent exclusion radius.This lower distance limit differs from models with closed access that lack the same exclusion constraint.
III. OUTAGE PROBABILITY
The paper defines outage as the probability that a randomly located user’s instantaneous SINR falls below a target and derives a general outage expression for each tier and the network.
- Definition: Outage is the probability that a randomly located user has instantaneous SINR below a target threshold.For a target SINR τ, the metric averages outage over the serving-cell distance distribution.
- Network Outage: The network outage probability is obtained by averaging tier-specific outage probabilities using the per-tier association probabilities.Because each user associates with at most one tier, the result follows from the law of total probability.
- Interpretation: The outage metric is the average fraction of cell area in outage and is also the SINR cumulative distribution function over the entire network.The cell-area averaging uses the serving-distance density derived earlier.
- Main Result: Theorem 1 gives the outage probability for a typical user associated with tier k and the corresponding outage for a randomly chosen network user.The theorem provides the paper’s most general outage result for arbitrary SINRs and association settings considered in the section.
- Main Result: Although the general theorem is not closed-form, its integral is straightforward to compute and simplifies in plausible special cases.The paper presents the general result as the basis for the remaining outage results.
B. Special Case: Interference-Limited Network
Under no noise and equal path-loss exponents, the interference-limited HCN admits simplified outage expressions. With unbiased association, outage is invariant to tier, BS power, BS density, and the number of tiers, while aggregate throughput grows with infrastructure.
- Equal path-loss exponents: When αj = α, the outage probability expressions for a tier and the overall network simplify to practically closed-form calculations.The expressions require only a computation or lookup of the remaining function.
- Unbiased association: When {bBj} = 1, unbiased association makes outage probability independent of BS transmit power Pj, BS density λj, and the number of tiers K.Adding infrastructure does not increase outage probability under this association policy.
- Equal path-loss exponents: For αj = 4, the relevant hypergeometric function collapses to an arctangent, yielding a simpler outage expression.Under no biasing, the expression further reduces to a single trigonometric function of one variable.
- Unbiased association: With equal path-loss exponents and unbiased association, every tier has the same outage probability as the overall network.The increase in interference is counter-balanced by the increase in signal power.
- Unbiased association: With equal path-loss exponents and unbiased association, network and per-tier sum throughput increase linearly with the number of BSs.This follows because each tier cell retains the same SINR while additional BSs contribute more cells.
IV. SPECTRAL EFFICIENCY
The paper derives average ergodic rates for typical users and the overall network, together with minimum average user throughput, using tractable integrations over association and serving distance. The special interference-limited unbiased case simplifies the rate expressions and removes dependence on several network parameters.
- Rate derivation: The section derives average ergodic rate and minimum average user throughput to measure network spectral efficiency.Both metrics are computed in nats/sec/Hz, with 1 nat/s = 1.443 bps.
- Rate derivation: The average ergodic rate of a kth-tier user averages link rate over the serving distance while accounting for thermal noise, BS density, bias, and path-loss exponent.The overall-network rate is then obtained from the per-tier rates and association probabilities.
- Rate derivation: Theorem 2 gives average ergodic rates for users associated with a tier and for the overall network.The resulting expression is efficiently evaluated numerically rather than by repeated Monte Carlo sampling.
- Special case: When thermal noise is ignored, association is unbiased, and all tiers share path-loss exponent α, the overall and per-tier average ergodic rates simplify substantially.The double integration in Theorem 2 becomes a single integration with an especially simple integrand.
- Special case: In this special case, average ergodic rate is unaffected by BS transmit power Pj, BS density λj, and the number of tiers K.Adding BSs or increasing power raises desired signal and interference by the same amount, while network sum rate grows in direct proportion to total BS count.
B. Minimum Average User Throughput
Minimum average user throughput captures the weakest per-tier user throughput under orthogonal transmission and round-robin scheduling. The metric is intended to measure the network’s minimum QoS and the effect of biasing.
- Definition: Under orthogonal transmission, equal time or frequency slots are allocated to users in a cell through round-robin scheduling.The per-tier average user throughput depends on the average number of users per cell.
- Definition: Minimum average user throughput is the minimum among the K average user throughputs supported by one cell in each tier.It is computed from the per-tier average user throughputs.
- Interpretation: The metric represents the minimum quality of service that the network can provide.Because it depends strongly on cell loading, it measures how biasing affects QoS in the HCN.
V. NUMERICAL RESULTS
Numerical results show that the PPP analysis closely matches simulated and grid-based outage behavior across SINR thresholds. They also show how path loss, biasing, and BS density differently affect tier-level and overall performance under full queues.
- Validation: The tier-1 PPP deployment is nearly as accurate as a hexagonal grid model, while the grid provides a lower bound with less than 1 dB gap from actual deployment.The PPP model gives an upper bound with less than a 1.5 dB gap from actual deployment.
- Validation: Analytic outage curves are remarkably close to simulations for all considered SINR thresholds.The framework therefore retains accuracy across the evaluated SINR range.
- BS density and path loss: With no biasing and equal path-loss exponents, adding BSs with different powers produces no change in outage or rate because inter-BS interference remains dominant.This confirms the interference-limited invariance behavior in the evaluated two-tier scenario.
- BS density and path loss: When low-tier BSs have higher path loss, adding them improves outage and average rate by reducing interference between pico and macro networks.The results indicate that new BSs are better deployed in areas with higher path loss when deployment location is selectable.
- Biasing: Increasing picocell bias improves picocell outage and rate but degrades macrocell metrics, while unbiased association outperforms biasing for the overall network.Biasing moves more low-SINR macro users to picocells and can connect users to BSs that do not offer the strongest received signal.
- Minimum average user throughput: Under full traffic loads, increasing bias first raises minimum user rate and then lowers it when picocell connections become excessive.For sufficiently large bias, picocell average user throughput falls below macrocell throughput due to a massive number of connections.
- BS density and biasing: Adding more pico BSs improves both tier-level outage and rate, but changes overall-network outage and rate only slightly.The increased macrocell interference and reduced macrocell association probability largely cancel, leaving overall rate weakly affected by BS density.
- Loading assumption: Biasing trends are preliminary and heavily dependent on cell loading; in lightly loaded HCNs, biasing can improve whole-network rate by increasing resource shares.The reported numerical trends assume all cells have full queues.
VI. CONCLUSIONS
The paper develops an analytical framework for flexible cell association in HCNs and evaluates outage probability, average ergodic rate, and minimum average user throughput. Under full queues, adding tiers or BSs need not reduce outage or rate without biasing, whereas biasing can worsen overall-network SINR metrics.
- The framework evaluates outage probability, spectral efficiency, average ergodic rate, and minimum average user throughput in flexibly associated HCNs.Minimum average user throughput is the smallest average user throughput supported by one cell in each tier.
- The number of tiers and BS density at most weakly affect outage probability and average ergodic rate, and under certain assumptions do not affect them.
- Adding pico and femto BSs for capacity improvement need not decrease network quality under the stated assumptions.
- With full queues at all BSs, biasing deteriorates overall-network outage and rate by lowering SINR.
- Further work is needed on biasing and related effects beyond the paper's full-queue assumption.
APPENDIX
The appendix derives outage probabilities and average ergodic rates by conditioning on tier association and distance, evaluating interference transforms, and integrating over SINR distributions. It then combines per-tier quantities into network-wide metrics.
- Tier association is determined by comparing biased received powers across tiers, with the selected tier indexed by n.
- The association-distance distribution follows from Poisson-process void probabilities and tier-specific exclusion distances.
- The per-tier SINR complementary CDF is obtained by conditioning on serving distance and using Laplace transforms of tier interference.
- A change of variables and the Gauss hypergeometric function yield the interference expression used for per-tier outage probability.
- Average ergodic rate is derived by integrating the SINR tail probability and then combining per-tier rates into the network-wide rate.
- The resulting per-tier average rates are combined to obtain the average ergodic rate of the entire network.