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Fundamentals of Heterogeneous Cellular Networks with Energy Harvesting
Harpreet S. Dhillon, Ying Li, Pavan Nuggehalli, Zhouyue Pi, Jeffrey G. Andrews
TL;DR
The paper studies how unreliable, finite energy supplies affect multi-tier cellular networks with self-powered base stations. It develops a tractable stochastic-geometry and Markov-process framework to characterize achievable ON-time availabilities under uncoordinated operation. The analysis identifies an availability region and a fundamental limit on tier availabilities, while showing when energy-harvesting HetNets can match reliable-power performance.
Problem
Understanding cellular networks with multiple self-powered transmitters remains severely limited, despite their relevance to heterogeneous deployments with variable energy and traffic.
Method
The paper models K-tier self-powered BSs with heterogeneous harvesting and storage parameters, birth-death energy dynamics, stochastic-geometry utilization rates, fixed points, and hitting/stopping times.
Results
The paper characterizes the availability region achievable by general uncoordinated strategies and proves a fundamental limit on ρ_k that no such strategy can surpass.
Takeaways & Limitations
When the availability vector for optimal network performance lies in the availability region, energy-harvesting HetNets have the same performance as HetNets with reliable energy sources.
Abstract
from arXiv · showhide
We develop a new tractable model for K-tier heterogeneous cellular networks (HetNets), where each base station (BS) is powered solely by a self-contained energy harvesting module. The BSs across tiers differ in terms of the energy harvesting rate, energy storage capacity, transmit power and deployment density. Since a BS may not always have enough energy, it may need to be kept OFF and allowed to recharge while nearby users are served by neighboring BSs that are ON. We show that the fraction of time a k^{th} tier BS can be kept ON, termed availability ρ_k, is a fundamental metric of interest. Using tools from random walk theory, fixed point analysis and stochastic geometry, we characterize the set of K-tuples (ρ_1, ρ_2, ... ρ_K), termed the availability region, that is achievable by general uncoordinated operational strategies, where the decision to toggle the current ON/OFF state of a BS is taken independently of the other BSs. If the availability vector corresponding to the optimal system performance, e.g., in terms of rate, lies in this availability region, there is no performance loss due to the presence of unreliable energy sources. As a part of our analysis, we model the temporal dynamics of the energy level at each BS as a birth-death process, derive the energy utilization rate, and use hitting/stopping time analysis to prove that there exists a fundamental limit on ρ_k that cannot be surpassed by any uncoordinated strategy.
I. INTRODUCTION
The paper addresses the limited understanding of cellular networks with multiple self-powered transmitters by developing a tractable K-tier HetNet model and characterizing achievable BS availabilities under uncoordinated operation.
- Research gap: Multiple self-powered transmitters are identified as a comparatively underinvestigated extension of isolated energy-harvesting links and broadcast channels.The paper focuses on cellular networks, where energy and load vary across both space and time.
- System model: The model includes K classes of self-powered BSs that can differ in transmit power, deployment density, energy harvesting rate, and storage capacity.BSs may turn OFF to recharge while neighboring ON BSs handle their load.
- System model: Operational states are toggled independently across BSs, with long- and short-time-scale decisions separating ON/OFF control from scheduling and cell selection.This separation treats BS states as static over short intervals and cell-selection effects through long-term averages.
- Contributions: Availability ρ_k is the fraction of time a kth-tier BS remains ON, and the availability region comprises achievable K-tuples under general uncoordinated strategies.The analysis combines birth-death energy dynamics, stochastic-geometry energy utilization, fixed-point analysis, and hitting/stopping times.
- Contributions: If the availability vector for optimal network performance lies in the availability region, energy harvesting can achieve the same performance as reliable energy sources.The paper also notes that maximizing availability is not always optimal; some BSs may preferably remain OFF despite having sufficient energy.
III. AVAILABILITY ANALYSIS
The analysis models each BS’s energy level as a birth-death process and derives energy utilization from service-region geometry and control-channel coverage. Unavailable BSs transfer load to neighboring available BSs, while some users outside control-channel coverage create no additional BS energy expenditure.
- Energy dynamics: BS energy evolves as a CTMC birth-death process, increasing through harvesting and decreasing according to the number of served users.When OFF, a BS serves no users and therefore does not consume service energy.
- Service regions: Unavailable BSs transfer their coverage load to nearby available BSs, expanding the latter’s effective service regions.This increased effective load is central to calculating energy utilization.
- Energy utilization: Control-channel coverage Pc determines which users can enter the network and contribute to data-channel energy expenditure.Users outside control-channel coverage do not add energy expenditure at the BS.
- Energy utilization: The average service-region area yields the average number of users served by a typical tier-k BS and hence its energy utilization rate.The derivation uses the service-region definition and Palm calculus.
- Shadowing: The utilization rate is invariant to shadowing distributions when the relevant shadowing moments match across tiers.For lognormal shadowing, this occurs when mj = mk and σj = σk for all j,k.
- Limitations: The utilization expression depends on unknown tier availabilities, and the model omits control-signaling and backhaul energy usage.Fixed expenditure could be deducted from harvesting rates, but formal treatment is left for future work.
B. Availabilities for a Simple Operational Strategy
The simple strategy treats a BS as available whenever it has positive energy, producing coupled fixed-point equations for tier availabilities. The equations have a unique positive solution when total harvested energy exceeds effective demand, while battery capacity and other-tier availability improve availability.
- Strategy and fixed points: The simple strategy declares a BS available when its energy state is nonzero, forming one availability fixed-point equation per tier.The zero vector is always a trivial solution, but it represents complete service outage if no positive solution exists.
- Fixed-point properties: The fixed-point mapping is element-wise increasing and concave, enabling existence and uniqueness analysis for positive availabilities.The analysis applies a fixed-point theorem for increasing concave functions.
- Feasibility and outage: A positive availability solution exists if and only if the network’s total harvested energy exceeds its effective user energy demand.When the condition fails, energy outage can occur; the paper assumes the condition holds thereafter.
- Parameter effects: Availability increases with battery capacity because the fixed-point function is increasing in Nk.This establishes a direct monotonicity relationship between storage capacity and availability.
- Parameter effects: Increasing availability in any other tier also increases kth-tier availability by reducing its effective load.The coupled equations capture this cross-tier load-sharing effect.
- Availability limits: The paper next uses stopping and hitting times to show that the simple positive-energy strategy maximizes availability among general uncoordinated strategies.These results support characterization of the achievable availability region.
C. Availabilities for any General Uncoordinated Strategy
General uncoordinated strategies toggle each BS using only its own energy level. Their availability is determined by mean ON and OFF durations, which are analyzed through harvesting dynamics, birth-death processes, and hitting times.
- Operational strategies: A BS toggles OFF at a chosen minimum energy level and ON at a higher cutoff, using only its current energy level.The cutoff may be adjusted on a slower timescale, but the strategy remains uncoordinated across BSs.
- Operational strategies: Turning OFF at Nk min is equivalent to reducing storage capacity to Nk − Nk min, so analysis can set Nk min = 0 without loss of generality.The resulting strategy is denoted Sk(Nkc).
- Availability calculation: Availability equals the long-run ON fraction and depends on the means of the ON duration Jk1 and OFF duration Jk2.The expression follows by applying the law of large numbers to ON–OFF cycles.
- OFF-time analysis: Under Sk(Nkc), the OFF period is the time required to harvest Nkc energy units, represented as a sum of Nkc exponential arrivals with mean 1/µk.This gives a closed-form mean OFF-time expression for the fixed-point analysis.
- ON-time analysis: The ON period is derived from a birth-death generator and mean hitting times to energy level 0.The resulting mean ON time is substituted into the availability fixed-point equation.
- Fixed-point characterization: Substituting the mean ON time into the availability relation yields fixed-point equations in the tier availabilities.The derivation provides the general strategy framework used to characterize achievable availability vectors.
1) Policy 1 (Sk(1)):
Policy 1 turns a BS OFF when its energy is depleted and turns it ON after harvesting one energy unit. Its mean ON time leads to the same fixed-point equation as the earlier simple policy, making the two policies availability-equivalent.
- Policy definition: Policy 1 serves users until all stored energy is depleted, then turns OFF and returns ON after harvesting one energy unit.Its mean ON time is obtained using the hitting-time formulas.
- Availability derivation: The resulting mean ON time, substituted into the availability equation, produces the policy’s fixed-point relation.This connects the policy-level ON-duration calculation to tier availability.
- Equivalence: Policy 1 achieves the same availabilities as the policy studied in the previous subsection.The paper therefore establishes an operational equivalence between the two policies.
2) Policy 2 (Sk(Nk)):
The paper establishes that the full-charge policy is not availability-optimal: any positive use of policies Sk(i), i > 1, lowers availabilities, including under timer-based strategies.
- Policy operation: Under the considered full-charge behavior, a BS serves users until its energy is depleted, then remains OFF until harvesting Nk units and fully recharging.The resulting fixed-point equation is obtained by substituting the expected recharge-cycle quantity into the availability relation.
- Policy optimality: Strategy Sk(1) jointly maximizes the availabilities of all BS tiers among general uncoordinated strategies.The result applies to energy-based, timer-based, and combined uncoordinated strategies.
- Policy optimality: Any tier using Sk(i), i > 1, with non-zero probability has strictly lower availability than under Sk(1).The reduced availability also increases the effective load on other tiers and decreases their availabilities.
- Policy optimality: Timer-based strategies are represented as combinations of Sk(i), i > 1, and therefore cannot attain the maximum availabilities achieved by Sk(1).This extends the upper-limit argument beyond purely energy-based policies.
D. Availability Region
The exact availability region is the set of tier-availability vectors satisfying each tier’s maximum availability given the others. It is achievable by uncoordinated strategies, while constrained strategy classes yield smaller regions.
- Definition and characterization: The availability region is defined as the set of K-tuples achievable by uncoordinated strategies, with the overall region formed by their union.The paper denotes the strategy-specific set by R(UC) and the union over all uncoordinated strategies by R.
- Upper bound and exact region: The coordinate-wise maxima ρmax provide an orthotope upper bound, but the paper shows this bound is rather loose relative to the exact region.The exact region accounts for the coupling among tier availabilities through the fixed-point equations.
- Definition and characterization: The exact region consists of vectors satisfying ρk ≤ ρ∗k({ρj} \ ρk) for every tier k.Here ρ∗k is the fixed-point maximum for tier k given the availabilities of the other tiers.
- Achievability: Points inside the exact region are achievable through constructions such as time sharing, whereas points exceeding another tier’s corresponding maximum are not achievable.For the two-tier illustration, point E is achieved by time sharing, while point F violates the second-tier constraint given the first-tier availability.
- Constrained strategies: Constraining a tier to strategy Sk(Nk) produces a region strictly contained within the general uncoordinated availability region.Figure 5 compares the constrained and general regions for a two-tier setup.
- Performance implications: Maximum availability is not always performance-optimal, because downlink rate can improve when some BSs are intentionally unavailable and users are offloaded to small cells.This resembles almost blank subframes, although ABS operates at a smaller time scale and assumes coordination across BSs.
- Performance implications: If the availability vector optimal for a performance metric lies in the region, energy harvesting causes no performance loss relative to reliable energy sources.The paper contrasts this case with an optimal vector outside the region, where some performance loss occurs.
E. Coverage Probability and Downlink Rate
The paper derives coverage and rate expressions for the available-BS network. Coverage is insensitive to available-BS density, whereas rate coverage can depend on availabilities through user loads.
- Coverage probability: The coverage probability is Pc = 1 / (1 + F(β, α)) under the paper’s general cell-selection model.F(β, α) is expressed using a Gauss hypergeometric function.
- Coverage probability: Coverage probability in interference-limited HetNets is independent of available-BS densities and therefore independent of the availabilities {ρk}.This validates the paper’s stated invariance of coverage to BS availability.
- Downlink rate: Because user load enters the rate expression through Ψk, downlink rate distribution need not share coverage probability’s invariance to availability.The paper uses the rate result to examine cases where maximum availability is not rate-optimal.
- Downlink rate: Rate coverage is derived as the CCDF of downlink rate R in bps/Hz, incorporating active-BS densities ρkλk and effective active-user density Pcλu.The derivation uses an approximation for the number of users served by the connected BS and treats it as independent of SIR.
- Downlink rate: Rate coverage is invariant to the shadowing distribution under the condition stated in Theorem 6.The supplied passage refers to the theorem’s condition involving the shadowing moments E[…].
IV. NUMERICAL RESULTS AND DISCUSSION
The numerical results examine how battery capacity and modeling assumptions affect the availability region, using a two-tier HetNet setup. They also motivate evaluating rate coverage over availability choices rather than assuming maximum availability is optimal.
- Experimental setup: The experiments use a two-tier HetNet and examine availability regions under equal tier battery capacities and constrained operational strategies.The setup includes α = 4, K = 2, γ = 1.1, P = [1, 0.1], µ1 = 10, µ2 = 3, and λ2 = 10λ1.
- Modeling assumption: Energy utilization and rate distribution are invariant to common lognormal shadowing under the stated equal-mean and equal-standard-deviation assumption.
- Availability region: The availability region expands with battery capacity, but the numerical study also considers constraints from forcing one tier to use strategy Sk(N).The corresponding figures compare unconstrained and constrained availability regions.
A. Effect of Battery Capacity on Availability Region
Battery capacity enlarges the achievable availability region, but cannot remove the fundamental availability limit, whereas increasing over-provisioning can eventually cover the full unit square. Constraining a tier to strategy Sk(N) reduces the region, and rate coverage may favor intentionally lower availability.
- A. Effect of Battery Capacity on Availability Region: Increasing battery capacity enlarges the availability region, yet even infinite capacity cannot achieve every point in [0, 1] × [0, 1].The limiting region depends on the over-provisioning factor γ, while maximum tier availabilities approach one at modest battery levels.
- A. Effect of Battery Capacity on Availability Region: For equal battery capacity N, constraining one tier to strategy Sk(N) yields a smaller achievable region, especially when N is small.
- B. Effect of Over-Provisioning Factor: Constraining one tier to strategy Sk(Nk) considerably reduces the availability region as γ varies.
- C. Rate coverage: Rate coverage is strictly suboptimal at (ρ1, ρ2) = (1, 1), and higher second-tier density makes it optimal to keep first-tier BSs OFF more often.The results motivate characterizing metric-specific optimal availability vectors and designing harvesting modules so those vectors lie in the availability region.
- V. Conclusions: The framework characterizes regimes in which energy-harvesting HetNets can match the performance of HetNets with reliable energy sources.
- V. Conclusions: The paper identifies extensions involving more complete energy expenditure models, coordinated strategies, metric-specific optimal availabilities, MIMO, and unreliable-energy uplinks.
APPENDIX
The appendix develops analytical proofs for service-area expressions and structural properties of an auxiliary function. It uses Palm and PPP tools, PGFL evaluation, substitutions, and derivative arguments to establish continuity, monotonicity, and strict concavity.
- Service-area derivation: Palm probabilities and expectations are used to express the service area of a kth-tier BS and derive its average over network randomness.
- Service-area derivation: Stationarity, Slivnyak’s theorem, independence, and the PPP probability generating functional simplify the average service-area expression.
- Service-area derivation: The appendix evaluates the remaining integral to obtain the closed-form service-area result.
- Function properties: A substitution shifts the auxiliary function without changing its monotonicity or concavity, enabling element-wise analysis in one variable.
- Function properties: Derivative and limit arguments establish continuity at the apparent singular point and prove that the auxiliary function is monotonically increasing and strictly concave.
- Function properties: The second derivative is shown to be negative, including at the point requiring a limiting argument, completing the strict-concavity proof.
C. Proof of Lemma 3
The proof establishes equivalence between per-tier inequalities and their aggregate condition by algebraically transforming, summing, and reversing the inequalities. The reverse implication uses arbitrariness of λk to recover each tier condition.
- Proof of Lemma 3: The forward implication multiplies each per-tier inequality by λk, rearranges it, and sums the K resulting inequalities.
- Proof of Lemma 3: For the reverse implication, the aggregate condition is multiplied by a tier-dependent product and rearranged into per-tier terms.
- Proof of Lemma 3: Because λk is arbitrary, the bracketed term must be positive for every tier, yielding the original set of per-tier conditions.