Ergodic-Rate Formulation

• Ergodic-rate formulation is a method that leverages long-term average metrics to quantify system performance and tradeoffs in the presence of randomness.
• In communication networks, it underpins interference management, MIMO, and rate–delay tradeoffs by providing closed-form expressions and design guidelines.
• It offers rigorous ergodic convergence rates in iterative algorithms, ensuring reliable performance benchmarks for optimization and resource allocation.

An ergodic-rate-based formulation formalizes performance metrics, convergence guarantees, and resource tradeoffs in random or time-varying systems by considering long-run average (ergodic) rates, rather than instantaneous or worst-case quantities. Such formulations are central to information and communication theory, algorithmic optimization, and large-deviation analysis, enabling precise quantification and optimization of throughput, convergence speed, or capacity in the presence of randomness or ergodicity.

1. Foundational Models and Definitions

The ergodic-rate framework presumes an underlying stochastic or ergodic process (e.g., fast-fading channels, random iterates, Markov chains) over which system quantities fluctuate. The ergodic (long-term average) rate of a process $\{X_t\}_{t=1}^\infty$ is typically of the form

$$R_{\text{erg}} = \lim_{N\to\infty} \frac{1}{N} \sum_{t=1}^N r(X_t)$$

for a rate function $r(\cdot)$. In finite settings, the ergodic rate is estimated by sample averages, expectations, or, in the presence of randomness, by $\mathbb{E}[\cdot]$ with respect to the stationary distribution.

In networks and coding, such as $n$-user fast-fading interference channels over $\mathbb{F}_q$, each transmitter seeks to encode information at rate

$R_i = \frac{m_i}{N} \log q$

where $N$ is the blocklength and $m_i$ the number of field symbols. The channel output is random due to fading matrices and noise processes, so the ergodic-capacity or sum-rate is the key metric (Johnson et al., 2010).

For optimization algorithms (e.g., Proximal Point or ADMM), ergodic convergence rates refer to the decay of averaged primal/dual gaps or violations over iterates: $$\bar{x}_N = \frac{1}{N} \sum_{k=1}^N x_k,\qquad G_N(w) = (\bar{x}_N - w)^T F(w)$$ where $F$ is monotone and $G_N(w)$ measures stationarity or dual gap at reference $w$.

2. Ergodic-Rate in Communication Networks

The ergodic-rate-based formulation is intrinsic to interference management, MIMO, and random-access regimes. In “Delay–Rate Tradeoff in Ergodic Interference Alignment” (Johnson et al., 2010), the primary goal is to maximize long-term per-user or sum throughput given statistical channel variations and user objectives. The key constructs include:

• Degrees of Freedom (DoF): The per-user DoF is $\mathsf{DOF} = R/C$ where $C$ is single-user capacity; e.g., for fast-fading distributed $n$-user interference, ergodic schemes achieve DoF up to $1/2$ per user with NGJV alignment.
• Rate–Delay Tradeoff: Alignment schemes are parametrized by rate per channel use and expected alignment delay, i.e., mean number of fading instances until desired channel structure occurs. For NGJV, delay scales as $q^{n^2}$. For generalized block schemes JAP$(\mathbf{a})$ and JAP-B$(\mathbf{a})$, rate and delay exponent ($T$) are parametrically linked:

$$R_{\rm user} = \frac{C}{K+1},\quad T = \max_k a_k(n-k-1)$$

with beamforming reducing the exponent further.

• Unified Toolkit: The framework interpolates between maximal DoF (with large delay) and minimal delay (with DoF approaching time-division). It supports arbitrary tradeoff points via “child” TDMA schemes.

Such formulations underpin the design and analysis of ergodic interference alignment, distributed coding, and link-layer scheduling for wireless systems, both theoretically and algorithmically.

3. Ergodic-Rate Formulations in Optimization and Convergence Analysis

In monotone operator theory and convex optimization, ergodic-rate formulations provide worst-case convergence benchmarks for iterative algorithms. For the relaxed proximal point algorithm (RPPA) and its variants (Gu et al., 2019, Goncalves et al., 2016), the ergodic rate quantifies the decay of stationarity or primal-dual gap via

$$\bar{w}_N = \frac{1}{N+1} \sum_{k=0}^N \tilde{w}_k,$$

with guarantees of the form

$$(\bar{w}_N - w)^T F(w) \leq \frac{1}{2(\lambda N + 2)} \|w-w_0\|_H^2,$$

where $\lambda$ is the relaxation parameter. This $O(1/N)$ rate is tight; worst-case construction and performance estimation frameworks establish its optimality.

For operator splitting schemes such as (proximal) ADMM, ergodic-rate theorems extend guarantees to averaged iterates and, crucially, apply even at “boundary” stepsizes where pointwise bounds fail (Goncalves et al., 2016). Typical results assert that all ergodic primal-dual residuals and errors in KKT conditions decay as $O(1/k)$.

These ergodic-rate results form the analytic backbone for assessing algorithmic complexity when the solution is approached through averaged dynamics, particularly in nonsmooth or non-Euclidean settings.

4. Closed-Form and Tractable Expressions: Stochastic Networks and Massive MIMO

In wireless network analysis, closed-form expressions for ergodic-rate are vital in system design and optimization. Under various models (Poisson cellular networks, multiuser MIMO, RIS-aided NOMA, low-resolution ADCs), ergodic-rate is formulated as an expectation over random channel coefficients or interference:

• Cellular Networks: For BS/user PPPs, ergodic capacity is given by

$$C = \int_{0}^{\infty} \frac{\mathbb{P}[\mathrm{SINR}>t]}{1+t}\,\mathrm{d}t,$$

with $\mathbb{P}[\mathrm{SINR}>t]$ analytically evaluated via Laplace transforms and moment generating functions of the aggregate interference field (Aravanis et al., 2024).

• Massive MIMO: The ergodic achievable rate in multi-user downlink is

$$R_k = \mathbb{E}\left[ \log_2\left( 1 + \frac{|g_{k,k}|^2}{\sigma^2 + \sum_{j\neq k} |g_{k,j}|^2} \right) \right],$$

and multiple tight lower bounds are established, adapting to regimes where channel hardening, finite blocklength, or interference suppression is dominant (Caire, 2017).

• Reconfigurable Surfaces and Low-Resolution ADCs: Ergodic-rate is expressed via distributions (e.g., Gamma, generalized Weibull sums) of composite channel gains, leading to tractable, validated integrals and asymptotic formulas (Zhao et al., 2022, Choi et al., 2018).

These closed-form ergodic-rate formulations support system-level optimization: e.g., BS densification maps, resource allocation, energy management, and code design under hardware constraints.

5. Ergodic-Rate Tradeoffs: Delay, Throughput, and Complexity

A central use of ergodic-rate-based formulations is making explicit tradeoffs between achievable per-user or sum rates and system-level costs such as delay, synchronization overhead, power-budget, or algorithmic complexity.

• In interference alignment, maximizing per-user DoF (via extensive ergodic averaging and delay) is balanced against the delay exponent—alternative JAP and JAP-B schemes interpolate continuously between fast alignment (short delay, lower DoF) and full-rate but slow-alignment (high DoF, long delay) (Johnson et al., 2010).
• In cognitive radios or multi-access, ergodic rate optimization under QoS or interference constraints yields explicit resource allocation “laws” (e.g., water-filling, power-split functions) with pointwise or ergodic optimality (Yue et al., 2022, Kang et al., 2014). For both average and peak constraint regimes, ergodic-rate maximization specifies the optimal scheduling and power allocation rules, and identifies conditions (e.g., D-TDMA optimality) and computational shortcuts.
• In random iterative algorithms, only ergodic (averaged) rates exhibit robust convergence guarantees in non-strongly monotone or degenerate Markovian contexts (Grothaus et al., 2015).

Such explicit tradeoff characterizations enable precise system tuning and clarify the fundamental limits imposed by ergodicity, system architecture, and algorithmic design.

6. Ergodic-Rate in Large Deviation and Statistical Physics Frameworks

In ergodic large-deviation theory, rate functionals quantify the exponential likelihood of deviations from ergodic averages. The Donsker–Varadhan principle for empirical measure and the Bertini–Faggionato–Gabrielli functional for fluxes are expressed as variational (inf-convolution) rate functions involving ergodic occupation measures and fluxes (Renger, 2024).

The ergodic-rate-based approach here embeds empirical observables into a Markov-bridge framework, with the key result that the large-deviation rate function decomposes into a sum of “bridge-Cramér” and empirical-measure components: $$I(k, \theta) = \sum_{x,y} \theta_{xy} \phi^{xy*}\bigl(k_{xy}/\theta_{xy}\bigr) + \sum_{x,y} s(\theta_{xy}\mid (e^1\#\theta)_x P_{xy})$$ where $\theta$ is the empirical pair measure and $k$ the empirical block-average, thus providing both theoretical insight and practical tools for numerical evaluation and statistical mechanics.

7. Generalizations and Applications

The ergodic-rate-based formulation extends naturally to:

• Secrecy-rate analysis in Poisson satellite networks by ergodicizing the secrecy capacity over satellite/user/eavesdropper random locations and channel/fading random variables (Kim et al., 2023).
• Programmable environments (PWEs) and discrete antenna selection (PAS), where ergodic capacity integrates over discrete spatial and random user distributions, supporting hardware-software co-design (Tyrovolas et al., 3 Nov 2025).
• Multi-user coding (e.g., cyclic delay diversity, CDD), where ergodic sum-rate bounds quantify multiplexing efficiency and establish near-capacity achievability even with simple or sub-optimal coding (0707.2780).

These applications demonstrate the flexibility of the ergodic-rate methodology for yielding closed-form expressions, providing algorithmic and theoretical performance guarantees, and framing optimize-and-design strategies under random, ergodic, and practical constraints.