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Ultra Dense Small Cell Networks: Turning Density into Energy Efficiency

Sumudu Samarakoon, Mehdi Bennis, Walid Saad, Mérouane Debba, Matti Latva-aho

arXiv:1603.03682v2cs.NIcs.GTcs.IT

TL;DR

The paper addresses joint power control and user scheduling in ultra-dense small cell networks under uncertain queue and channel states, where severe interference complicates resource optimization. It combines mean-field analysis with Lyapunov-based scheduling and reports considerable energy-efficiency gains and massive outage reductions versus a baseline.

  • Problem

    Resource optimization in ultra-dense small cell networks is challenging because many devices and severe coupling make classical small-network approaches inadequate, while queue and channel uncertainties evolve over time.

  • Method

    The paper formulates joint power control and user scheduling as a mean-field game and analyzes its equilibrium using coupled HJB-FPK partial differential equations with Lyapunov-based QSI- and CSI-aware scheduling.

  • Results

    The proposed method provides considerable gains in energy efficiency and massive reductions in outage probability compared with a baseline model.

  • Takeaways & Limitations

    The resulting equilibrium control policy per SBS maximizes time-average energy efficiency while ensuring users’ quality of service concerning queue-capacity outages.

  • Takeaways & Limitations

    The proposed method is not applicable to highly sparse networks because solving the coupled HJB-FPK equations can be challenging for small numbers of SBSs.

Abstract

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In this paper, a novel approach for joint power control and user scheduling is proposed for optimizing energy efficiency (EE), in terms of bits per unit energy, in ultra dense small cell networks (UDNs). Due to severe coupling in interference, this problem is formulated as a dynamic stochastic game (DSG) between small cell base stations (SBSs). This game enables to capture the dynamics of both the queues and channel states of the system. To solve this game, assuming a large homogeneous UDN deployment, the problem is cast as a mean-field game (MFG) in which the MFG equilibrium is analyzed with the aid of low-complexity tractable partial differential equations. Exploiting the stochastic nature of the problem, user scheduling is formulated as a stochastic optimization problem and solved using the drift plus penalty (DPP) approach in the framework of Lyapunov optimization. Remarkably, it is shown that by weaving notions from Lyapunov optimization and mean-field theory, the proposed solution yields an equilibrium control policy per SBS which maximizes the network utility while ensuring users' quality-of-service. Simulation results show that the proposed approach achieves up to 70.7% gains in EE and 99.5% reductions in the network's outage probabilities compared to a baseline model which focuses on improving EE while attempting to satisfy the users' instantaneous quality-of-service requirements.

I. INTRODUCTION

The paper addresses energy-efficient resource management in ultra-dense small-cell networks, where interference, scale, and evolving queue and channel states make power control and user scheduling difficult. It combines mean-field game analysis with Lyapunov optimization to derive coordinated control and scheduling policies while preserving users’ quality of service.

  • Motivation: Queue-state and channel-state uncertainties evolve over time and play a pivotal role in resource optimization.These dynamics motivate a stochastic treatment of network control and scheduling.
  • Motivation: UDN resource management is difficult because many devices have severely coupled control parameters, making classical small-network approaches inadequate.The paper highlights power control, UE scheduling, interference mitigation, and deployment as key energy-efficiency challenges.
  • Contribution: UE scheduling is formulated as a stochastic optimization problem and addressed with Lyapunov DPP to simplify decisions into per-time-slot subproblems while ensuring queue stability.The scheduling process accounts for dynamic channels and arbitrary arrivals.
  • Contribution: The proposed mechanism jointly controls transmit power and user scheduling to maximize SBS time-average energy efficiency measured in bits per joule.SBSs compete under severe mutual interference while attempting to ensure UE quality of service.
  • Contribution: The DSG among SBSs is approximated in the mean-field regime, where a homogeneous policy and coupled HJB-FPK PDEs characterize the mean-field equilibrium.The approach analyzes equilibrium existence and convergence of SBSs to that equilibrium.
  • System model: The system model represents each SBS through transmit power, scheduling, queue, arrival, rate, and channel variables, with scheduling decisions based on QSI and CSI.Transmit power is treated as a fast process, whereas UE scheduling remains fixed over a duration T for stable transmission.

A. Resource management as a dynamic stochastic game

The paper models joint power control and user scheduling as a dynamic stochastic game among SBSs, with states capturing queue and channel information. Because the coupled game is difficult to solve directly, it establishes a closed-loop Nash-equilibrium condition and motivates a mean-field reduction for large deployments.

  • Game formulation: The objective is to find a control policy maximizing each SBS's utility over a finite horizon while accounting for state transitions and scheduled UEs.The utility is defined over the evolution from x(0) to x(T).
  • Game formulation: The DSG models SBSs as players whose actions jointly specify transmit power and scheduled UEs, while each SBS state contains queue-state and channel-state information.The game accounts for evolving network states and interference from other SBS control vectors.
  • Equilibrium analysis: A closed-loop Nash equilibrium requires each SBS to use its own state and feedback about the other SBSs' optimal strategies.The equilibrium is characterized through mutually coupled Hamilton–Jacobi–Bellman equations.
  • Equilibrium analysis: A sufficient equilibrium condition is 2v⋆(p⋆ + p0) + β ln(1 + βp⋆) ≠ 0 for all feasible SBS and scheduled-UE choices.The condition follows from existence of solutions to the HJB equation.
  • Scalability: Directly solving the |B| coupled HJB equations is impractical for large UDNs because it requires substantial network-wide information exchange.The paper therefore motivates a mean-field approach as the number of SBSs becomes very large.

III. OPTIMAL POWER CONTROL: A MEAN-FIELD APPROACH

For a large homogeneous SBS population, the paper replaces the multi-player DSG with a mean-field game whose interference depends on the limiting state distribution. The resulting equilibrium is obtained from coupled HJB and FPK equations, substantially reducing the solution complexity for dense deployments.

  • Mean-field formulation: As the SBS population grows, bounded interference and homogeneous local-state policies make SBSs indistinguishable and yield a continuum-player mean-field model.The individual impact of one SBS becomes negligible from the macroscopic perspective.
  • Mean-field formulation: Mean-field interference at a generic UE becomes independent of individual SBS states and policies and depends on time and the limiting distribution ρ(t).This is the interference interpretation used in the mean-field formulation.
  • Equilibrium solution: The mean-field equilibrium is computed by solving one HJB equation backward and the FPK equation forward for the limiting state distribution.The resulting solution determines a generic SBS's transmission power, utility, and state distribution.
  • Complexity: The mean-field solution replaces B coupled HJB PDEs with two coupled HJB-FPK PDEs, reducing complexity and scaling well with large player populations.The mean-field equilibrium is equivalent to the B-player DSG equilibrium only in the large-B regime.
  • Complexity: For large SBS populations, the proposed method reaches the mean-field equilibrium in a small number of iterations, so its complexity remains fixed as B increases.The paper illustrates this behavior using iteration counts over a fixed 0.5625 km2 area.
  • Scope boundary: The mean-field method can be challenging for small SBS populations and is not applicable to highly sparse networks.Its equivalence to the original DSG depends on a sufficiently large number of SBSs.

IV. UE SCHEDULING VIA LYAPUNOV FRAMEWORK

Time-scale separation decouples scheduling from the mean-field game, while Lyapunov drift-plus-penalty optimization enables distributed per-SBS scheduling that accounts for queue and channel dynamics.

  • Time-scale separation decouples scheduling variables from the mean-field game so they can be optimized separately.
  • Conventional proportional-fair, best-CSI, and highest-QSI schedulers do not properly account for CSI and QSI dynamics in UDNs.
  • The relaxed scheduling variables use a convex feasible set and represent the served rate as ˆr_bm(t) = λ_bm(t)˜r_bm(t).
  • Lyapunov drift-plus-penalty decomposes each SBS’s stochastic optimization problem into distributed sub-policies suitable for large SBS populations.Each SBS locally solves a copy of the scheduling problem as the number of SBSs grows.
  • Virtual queues are introduced to satisfy scheduling constraints, alongside a quadratic Lyapunov function for controlling queue evolution.
  • The DPP parameter |V| controls the tradeoff between queue length and solution accuracy, with larger |V| bounding the optimality gap more tightly.The gap between time-average penalty and the optimum is bounded by K/|V|.

A. Evaluation of auxiliary variables

The auxiliary-variable subproblem is solved over a convex-hull feasible set, whose affine objective places an optimum at a vertex.

  • A. Evaluation of auxiliary variables: The auxiliary variables are evaluated separately from scheduling variables after the DPP decomposition.
  • A. Evaluation of auxiliary variables: Because the feasible set is a convex hull and the objective is affine, the optimal auxiliary-variable solution lies at a vertex.

B. Determining the scheduling variables

The scheduling subproblem is optimized over a convex hull, yielding a vertex solution that is effectively boolean despite the continuous relaxation.

  • B. Determining the scheduling variables: Optimal SBS scheduling is obtained by solving a subproblem based on the scheduling term in the DPP control policy.
  • B. Determining the scheduling variables: The affine scheduling objective over the convex hull implies that the optimum lies at a vertex.
  • B. Determining the scheduling variables: Although scheduling variables are relaxed from boolean to continuous values, the resulting optimal scheduling vector is boolean.
  • B. Determining the scheduling variables: The Lyapunov DPP method bounds the gap between time-average penalty and the optimum through the parameter |V|.The paper states that sufficiently large |V| ensures solution optimality.

V. NUMERICAL RESULTS

The numerical study solves simplified coupled PDEs and evaluates energy efficiency under specified queue, arrival, power, and noise assumptions against a proportional-fair baseline. It also examines the mean-field queue-state distribution and notes that fixed-channel simulations can represent ergodic behavior.

  • The simulations use a finite-element solution of the coupled PDEs through MATLAB PDEPE, with SBS utility defined by energy efficiency.
  • The simulation model assumes time-invariant channels, so the system state is defined solely by queue-state information.
  • UE arrivals follow a Poisson process with mean 200 kbps, while transmit power ranges from 0 to 1 Watts and circuit power is 1 Watt.
  • The proposed method is compared with a baseline using proportional-fair scheduling and a myopic adaptive transmission policy that optimizes instantaneous energy efficiency.
  • Fixed-channel simulations do not affect the theoretical contribution and can model ergodic behavior even without fading.
  • At the mean-field equilibrium, the QSI distribution evolves toward queues close to zero by t = T during the transmission period T = 1.The decrease in scheduled UEs with high QSI allows a new set of UEs to be scheduled in the next phase.

A. Mean-field equilibrium of the proposed model

At the mean-field equilibrium, transmit power adapts over time and queue-state information: it is moderated for low queue state and increased for high queue state as the scheduling period ends.

  • Mean-field queue dynamics: The MF distribution ρ⋆(t, q) tracks the evolution of scheduled users’ queue-state information over time at equilibrium.During T = 1, transmission reduces high-QSI scheduled users toward QSI close to zero by t = T.
  • Mean-field queue dynamics: Queue fractions oscillate at q(t) = 0.4 while the fraction at q(t) = 0 increases monotonically over time.The oscillation follows queue emptying based on prior transmission rates and subsequent arrivals.
  • Equilibrium transmit-power policy: Transmit power is higher for high QSI and lower for low QSI at the MF equilibrium.At t = 0, moderate power for high-QSI users limits unnecessary interference; later power increases as the scheduling period approaches its end.
  • Equilibrium transmit-power policy: For high-QSI users, increasing transmit power near t = T provides high data rates to empty queues and prevent outages.For low-QSI users, moderate power is used to maximize EE.

B. Energy efficiency and outage comparisons

The proposed method compares energy efficiency and outage probabilities with a baseline across SBS densities and loads. Its benefits are generally larger in denser, more heavily loaded networks, although the baseline can be better in lightly loaded sparse settings.

  • Energy efficiency versus SBS density: 48.8% EE improvement occurs at k = 5 in the dense network, compared with 4.8% at ISD = 6.5 and k = 5.At ISD = 6.5 and k = 2, the baseline is about 14.1% better than the proposed method.
  • Outage probability versus SBS density: 99.5% outage reduction is achieved for high loads in sparse networks, compared with 92.2% for low loads.In UDNs, outage reductions are 91.8% and 41.8% for low and high loads, respectively.
  • Energy efficiency versus load: 70.7% EE gain occurs at k = 6 UEs per SBS and ISD = 3.5, while the gain reaches 20.3% at ISD = 5.75.The proposed method shows higher EE gains for dense networks when load increases.
  • Energy efficiency versus load: 12.6% EE loss occurs for the proposed method at ISD = 5.75 and k = 2, where the baseline exceeds it.In the dense network with ISD = 3.5 and k = 2, the proposed method’s EE gain decreases to 4.8%.
  • Outage probability versus load: Outage reductions are 91.8% and 92.2% at k = 2, and 33.7% and 87.6% at k = 6, for ultra-dense and sparse networks respectively.Outages increase with load for both methods because longer scheduling waits raise the chance of queue-capacity overflow.
  • Evaluation settings: The evaluation covers 60 SBSs/km2 at ISD = 6.5 and 250 SBSs/km2 at ISD = 3.5 across varying numbers of UEs per SBS.These settings represent sparse-like, intermediate, and ultra-dense deployments.

C. Transmit power and UE rate comparisons

The proposed method adapts transmit power and scheduling to network load and density, improving the balance between energy consumption and UE rates over the baseline. Gains are strongest in highly loaded or dense settings.

  • Sparse scenario: 20.5% lower transmit power is achieved than the baseline in highly loaded sparse networks.The proposed method maintains its EE objective with only a small transmit-power increase as load rises.
  • Sparse scenario: UE rates improve by up to 43.6% for k = 2 and 12.3% for k = 5 UEs per SBS in the sparse scenario.The comparison jointly indicates improved rate performance under both tested loads.
  • Dense scenario: In the dense scenario, average transmit power is 0.381 W versus 0.397 W at low load and 0.49 W versus 0.78 W at high load for the proposed and baseline methods, respectively.The corresponding reductions are about 4% and 37.1%.
  • Dense scenario: Dense-scenario UE rates are 0.41 versus 0.33 bits/s/Hz at low load and 0.18 versus 0.17 bits/s/Hz at high load for the proposed and baseline methods, respectively.These correspond to gains of 24.4% and 2.4%.
  • Overall comparison: 70.7% higher EE and up to 33.7% fewer outages are reported for UDNs compared with the baseline.The proposed method adapts to network dynamics while maintaining a balance between EE and data rate.

D. Impact of the boundary conditions

Boundary conditions determine how strongly the solution penalizes final queue state, creating a trade-off between emptying queues, energy efficiency, and outage probability. Relaxed conditions favor EE, while tighter conditions favor lower final queues and outages.

  • Boundary-condition design: The exponential boundary condition makes the terminal cost grow with final QSI, encouraging smaller queues at t = T.Its cost is minimum for an empty queue and increases exponentially as QSI at T increases.
  • Boundary-condition design: The uniform bound relaxes the terminal constraint, whereas the linear bound is tighter and forces QSI at t = T closer to zero.The three choices are exponential, uniform, and linear bounds.
  • Mean-field distributions: The uniform bound leaves a larger fraction of users with non-zero queues than the exponential bound, while the linear bound encourages almost empty queues by t = T.These distributions reflect how each terminal utility treats final QSI.
  • EE and outage trade-off: The uniform bound provides higher EE at the price of higher outage probabilities than the exponential bound.Relaxing final-QSI requirements lowers transmit power but accumulated queues prevent further arrivals.
  • EE and outage trade-off: The linear bound transfers all queued data by the end of the scheduling period, requiring more energy and reducing EE relative to the exponential bound.The tighter terminal condition lowers outage probabilities over the other boundary conditions.

VI. CONCLUSIONS

The paper formulates joint power control and user scheduling in ultra-dense small-cell deployments as a mean-field game under queue and channel uncertainty. It analyzes the equilibrium using two tractable PDEs while targeting energy efficiency and outage-related QoS.

  • Conclusion: Joint power control and user scheduling is formulated as a mean-field game for ultra-dense small-cell deployments under QSI and CSI uncertainty.The objective is time-average energy efficiency measured in bits per unit power while ensuring QoS concerning queue-capacity outages.
  • Conclusion: The MFG equilibrium is analyzed using two low-complexity, tractable partial differential equations.This provides the analytical basis for the proposed equilibrium solution.
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