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Throughput and Energy Efficiency Analysis of Small Cell Networks with Multi-antenna Base Stations
C. Li, J. Zhang, K. B. Letaief
TL;DR
Irregular small cell deployments make throughput and energy-efficiency evaluation difficult. The paper develops a random spatial analytical framework with independent BS and user PPPs and a tractable outage expression, then shows that densification and additional antennas always increase throughput while energy efficiency may peak or always decline depending on BS power-consumption conditions.
Problem
Irregular BS deployments make achievable throughput and energy efficiency difficult to quantify, while prior models often ignore user distribution and BS-user association.
Method
The paper uses a random spatial network model with multi-antenna BSs and derives a simple, tractable outage-probability expression for evaluating throughput and energy efficiency.
Results
More BSs or antennas always increase network throughput, whereas energy efficiency either first increases then decreases or always decreases, depending on BS power-consumption conditions.
Takeaways & Limitations
BS density and antenna count should be selected with the BS-user density ratio and power-consumption components in mind; optimal values exist in the peak-efficiency case.
Abstract
from arXiv · showhide
Small cell networks have recently been proposed as an important evolution path for the next-generation cellular networks. However, with more and more irregularly deployed base stations (BSs), it is becoming increasingly difficult to quantify the achievable network throughput or energy efficiency. In this paper, we develop an analytical framework for downlink performance evaluation of small cell networks, based on a random spatial network model, where BSs and users are modeled as two independent spatial Poisson point processes. A new simple expression of the outage probability is derived, which is analytically tractable and is especially useful with multi-antenna transmissions. This new result is then applied to evaluate the network throughput and energy efficiency. It is analytically shown that deploying more BSs or more BS antennas can always increase the network throughput, but the performance gain critically depends on the BS-user density ratio and the number of BS antennas. On the other hand, increasing the BS density or the number of transmit antennas will first increase and then decrease the energy efficiency if different components of BS power consumption satisfy certain conditions, and the optimal BS density and the optimal number of BS antennas can be found. Otherwise, the energy efficiency will always decrease. Simulation results shall demonstrate that our conclusions based on the random network model are general and also hold in a regular grid-based model.
I. INTRODUCTION
Small cell densification can expand capacity, but irregular deployments make throughput and energy-efficiency evaluation difficult. The paper motivates a tractable random-network framework that explicitly accounts for user distribution, BS-user association, and multi-antenna transmission.
- Irregularly deployed BSs make achievable throughput gains from network densification difficult to evaluate.
- BS energy efficiency requires careful study because BSs consume more than 60% of cellular-network energy.
- The paper develops an analytical framework for evaluating small-cell throughput and energy efficiency and providing deployment guidelines.
- Simulation can illuminate specific settings, but its results may not extend across scenarios and its computational complexity is high.
- Prior analyses often ignore user distribution and BS-user association by assuming every BS always serves a user.
- That assumption is unsuitable when user and BS densities are comparable, as in small cell networks.
B. Contributions
The paper combines a random spatial model with a tractable outage expression to analyze throughput and energy efficiency in multi-antenna small cell networks. Its results characterize how BS density and antenna count affect performance under different density ratios and power-consumption conditions.
- System model: The framework models BSs and users with independent PPPs and associates each user with its nearest BS in an irregular network.
- Analytical framework: A new outage-probability expression is simpler and more tractable for multi-antenna transmissions.
- Throughput results: For fixed user density, throughput grows linearly with BS density when λb ≪λu but logarithmically when λb ∼λu.
- Throughput results: More BS antennas increase throughput, but the gain diminishes as antenna count increases.
- Energy-efficiency results: Energy efficiency may first rise to a maximum and then decline, or always decline, depending on BS power-consumption components.
- Energy-efficiency results: When energy efficiency has an interior maximum, the analysis derives an optimal BS density and antenna count.
B. Network Performance Metrics
The paper evaluates network and user throughput together with energy efficiency using outage probability under a fixed-rate downlink model. These metrics respond differently to BS and user densities because throughput reflects successful transmissions while energy efficiency also includes total BS power consumption.
- The analysis focuses on network throughput, user throughput, and energy efficiency as the principal performance metrics.
- Outage occurs when a typical user's SINR falls below threshold ˆγ, with pout = Pr (SINR ≤ˆγ).
- Network throughput is the average successfully transmitted bits per sec·Hz·unit-area and measures area spectral efficiency.
- User throughput is given by Ru = ρpa (1 −pout) R0 under the fixed-rate model.
- Energy efficiency uses total BS power consumption, including transmit, circuit, and non-transmission power from active and inactive BSs.
III. OUTAGE PROBABILITY ANALYSIS
The paper derives a simpler, analytically tractable outage-probability expression for multi-antenna small-cell transmissions. The derivation recursively handles a Laplace-transform derivative and converts it into a lower triangular Toeplitz-matrix form.
- A new outage-probability expression is derived in a much simpler form than existing results.The expression is intended to facilitate subsequent throughput and energy-efficiency analysis.
- The analysis assumes a dense, interference-limited network, so additive noise is ignored.
- The derived bounds use quantities whose dependence on antenna count, pathloss exponent, and SINR threshold supports later performance analysis.
- The main derivation challenge is simplifying the nth derivative of the interference Laplace transform.
- The proposed method expresses the derivative recursively and transforms it into a lower triangular Toeplitz matrix using linear algebra.
- The resulting outage expression is more mathematically tractable because Toeplitz-matrix and matrix-norm properties support further analysis.
B. Key Properties of the Outage Probability
The outage analysis shows that denser base-station deployments and more transmit antennas improve successful transmission, while antenna gains diminish with antenna count and depend on deployment density.
- The outage probability decreases as BS density increases.This follows because signal power grows faster than interference power when BS activity decreases with density.
- The successful transmission probability is bounded and can be approximated using terms that separate BS-density and antenna effects.BS activity depends on the BS-user density ratio, whereas the bounds also depend on antenna count, pathloss exponent, and SINR threshold.
- Adding one transmit antenna always increases successful transmission probability, but the incremental gain diminishes as antenna count increases.
- For large antenna counts, outage probability decreases linearly with antenna number on a logarithmic scale.The paper states that this linearity is also observed for small antenna counts in Fig. 2.
- The performance gain from adding antennas is greater at higher BS density.Fig. 2 reports faster outage reduction for λb = 5 × 10−4 than for λb = 10−4 per square meter.
IV. THROUGHPUT AND ENERGY EFFICIENCY ANALYSES
The paper analytically evaluates small-cell network throughput and energy efficiency, focusing on BS density and the number of BS antennas.
- Throughput and energy efficiency are analytically evaluated with emphasis on BS density and the number of BS antennas.
A. Throughput Analysis
Network throughput increases with BS density and transmit antennas, but its scaling depends critically on the BS-user density ratio. Energy efficiency instead depends on the BS power-consumption model and may have an interior optimum.
- Throughput Analysis: Increasing the number of BS antennas increases network throughput, with the same antenna effect across the two analyzed scenarios.
- Throughput Analysis: In the medium-density regime, where BS and user densities are comparable, aggregate throughput increases logarithmically with BS density.
- Throughput Analysis: The BS-user density ratio is critical for evaluating network throughput under explicit BS-user association.
- Throughput Analysis: With fixed BS-user density ratio, aggregate throughput grows linearly with BS density while typical-user throughput stays unchanged.The analysis connects this scaling to deploying more small BSs while maintaining user QoS.
- Energy Efficiency Analysis: Energy efficiency can decrease monotonically with BS density or first increase and then decrease, depending on the power-consumption condition.The paper identifies an approximated optimal BS density in the non-monotonic case.
- Energy Efficiency Analysis: The non-transmission power consumption P0 plays a critical role in energy efficiency.When PBS < γP0, a non-zero BS density can maximize energy efficiency.
C. Network Energy Efficiency Analysis – The Effect of the Number of BS Antennas
The number of BS transmit antennas has a nonmonotonic effect on energy efficiency: an optimal antenna count may exist, while circuit-power conditions determine whether multi-antenna BSs are preferable.
- An optimal number of BS transmit antennas M* maximizes energy efficiency.The proposition identifies M* as the maximizing antenna count.
- When M > M*, increasing the antenna count decreases energy efficiency, whereas when M < M*, additional antennas improve it.
- Lower BS circuit power Pc leads to a larger optimal number of transmit antennas, increasing both spectral and energy efficiency.
- Single-antenna BSs can provide higher energy efficiency than multi-antenna BSs under the stated condition, for any BS and user densities.
- Multi-antenna BSs are preferable in energy efficiency only when circuit power consumption is below the threshold γPc.
V. NUMERICAL RESULTS
Simulations show that increasing BS density or transmit antennas improves successful transmission probability, while energy-efficiency gains depend on antenna count and BS power-consumption conditions. Random-network trends generally match grid-based results, with the hexagonal model providing an upper bound.
- Successful transmission probability: Increasing BS density or transmit antennas increases successful transmission probability, with a significant gain from M = 1 to M = 3 and smaller gains from M = 3 to M = 5.The analytical results fit the simulations, indicating negligible additive-noise influence and an accurate approximation of ˜Φb.
- Model comparison: The hexagonal-cell network provides an upper bound over the random network model, but both models exhibit the same performance trend.The analytical random-network result becomes closer to the grid-based model as M increases.
- Energy efficiency: For the considered micro-BS power model, condition (31) is satisfied, making single-antenna BSs more energy efficient.The simulation uses η = 0.32, Pt = 6.3W, Pc = 35W, and P0 = 34W.
- Energy efficiency: When M > 2, energy efficiency decreases as BS density increases.These analytical conclusions are confirmed by the simulations in Fig. 5.
- Energy efficiency: For M = 1, energy efficiency has a non-zero optimal BS density of about 0.3 × 10−3m−2 in both random and grid-based models.The optimal density is reported as b = 0.32λu in the associated analysis.
- Conclusions: The study develops analytical results for random spatial networks and applies them to network throughput and energy-efficiency evaluation.The results characterize scaling with BS density and the number of BS antennas.
APPENDIX
The appendix derives a tractable outage-probability expression by transforming interference, solving a derivative recurrence, and averaging over the nearest-BS distance.
- Outage-probability derivation: The outage probability is derived from the Laplace transform of interference using the probability generating functional of a Poisson point process.Independent exponential fading terms permit the interference transform to be expressed before applying the PPP functional.
- Outage-probability derivation: Derivatives of the interference Laplace transform satisfy a recursive form that yields a linear recurrence for the outage-probability terms.The recurrence is solved explicitly through linear algebra and matrix representation.
- Closed form: The matrix formulation produces a closed-form outage-probability expression, which is subsequently simplified using lower-triangular matrix properties and Taylor expansion.The resulting expression is stated as equivalent to equation (10).
- Distance averaging: A typical user's nearest-BS distance follows from the null probability of a PPP, and expectation over that distance completes the outage-probability derivation.This averaging step connects the conditional expression to the final result.
B. Proof of Lemma 1
The proof analyzes a lower triangular Toeplitz matrix and bounds its induced norm using recurrence relations, matrix inequalities, and monotonicity of the coefficients.
- Matrix structure: The proof uses the lower triangular Toeplitz structure of QM to establish the corresponding structure of TM.This structural property supports the subsequent recurrence analysis.
- Recurrence solution: The recurrence for tn is solved in closed form using established results for the associated matrix sequence.The initial value is t0 = 1/(1+k0pa), with c = pa.
- Norm bounds: The upper bound is obtained by rewriting the induced norm and applying the triangle inequality.The proof separately derives the remaining bound for ∥TM∥1.
- Norm bounds: A lower bound on the induced norm follows by defining x as an all-ones vector, setting y = Ax, and applying ∥x∥1 ≤ ∥A−1∥1∥y∥1.The argument uses nonsingularity of A.
- Monotonicity: Because ki > ki+1 for i ∈ N, the relevant quantity decreases with M once M exceeds a certain value.The monotonicity statement supplies the lemma's large-M behavior.
C. Proof of Property 4 of the Outage Probability
The proof characterizes the energy-efficiency behavior as the number of transmit antennas increases, establishing that it cannot decrease and then increase. It identifies two possible cases: either single-antenna BSs are most efficient, or multi-antenna BSs yield higher efficiency up to an optimum.
- Convergence analysis: The radius of convergence r_u lies in (1, 1 + ˆγ^-1), approaching 1 + ˆγ^-1 as the BS activity probability p_a approaches zero.This follows from the defining equation for r_u and the limiting behavior of G(1 + ˆγ^-1).
- Energy-efficiency monotonicity: Energy efficiency cannot first decrease and then increase as the number of transmit antennas M grows.The proof rules out simultaneous inequalities that would produce this pattern.
- Energy-efficiency monotonicity: As M approaches infinity, energy efficiency approaches zero, while energy efficiency at M = 1 remains positive.These boundary facts constrain the possible energy-efficiency behaviors.
- Energy-efficiency cases: In one case, energy efficiency decreases with M, making single-antenna BSs more efficient than multi-antenna BSs.This is the first of the two cases established by the proof.
- Energy-efficiency cases: In the other case, multi-antenna BSs achieve higher energy efficiency than single-antenna BSs, with an optimal antenna count M.The optimal value is constrained by the inequalities analyzed in the proof.