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The Meta Distribution of the SIR in Poisson Bipolar and Cellular Networks
Martin Haenggi
TL;DR
Typical-link success probabilities provide limited information about individual links. The paper introduces the SIR meta distribution, derives moments and bounds, and shows accurate beta approximations alongside contrasting bipolar and cellular variance behavior.
Problem
Typical-link success probabilities and standard SIR distributions do not reveal how concentrated success probabilities are across individual links or users.
Method
The paper introduces the SIR meta distribution and derives closed-form moments for Poisson bipolar and cellular networks with Rayleigh fading.
Results
The beta approximation matches exact distributions extremely accurately, while bipolar variance vanishes as p→0 and cellular variance approaches a non-zero constant.
Takeaways & Limitations
The meta distribution characterizes complete conditional link-success-probability distributions, providing finer network-performance information than spatial averages.
Abstract
from arXiv · showhide
The calculation of the SIR distribution at the typical receiver (or, equivalently, the success probability of transmissions over the typical link) in Poisson bipolar and cellular networks with Rayleigh fading is relatively straightforward, but it only provides limited information on the success probabilities of the individual links. This paper introduces the notion of the meta distribution of the SIR, which is the distribution of the conditional success probability $P$ given the point process, and provides bounds, an exact analytical expression, and a simple approximation for it. The meta distribution provides fine-grained information on the SIR and answers questions such as "What fraction of users in a Poisson cellular network achieve 90% link reliability if the required SIR is 5 dB?". Interestingly, in the bipolar model, if the transmit probability $p$ is reduced while increasing the network density $λ$ such that the density of concurrent transmitters $λp$ stays constant as $p\to 0$, $P$ degenerates to a constant, i.e., all links have exactly the same success probability in the limit, which is the one of the typical link. In contrast, in the cellular case, if the interfering base stations are active independently with probability $p$, the variance of $P$ approaches a non-zero constant when $p$ is reduced to $0$ while keeping the mean success probability constant.
I. INTRODUCTION
The paper introduces the meta distribution of the SIR to characterize the complete distribution of conditional link success probabilities, beyond the typical-link success probability. It derives moments, bounds, exact expressions, and a beta approximation for Poisson bipolar and cellular networks.
- Motivation and definition: Typical-link success probabilities can hide substantial variability among individual link reliabilities and user experiences.Networks with the same mean success probability may have very different reliability ranges across links.
- Analytical framework: Closed-form moments enable analytical expressions and bounds for the meta distribution in Poisson bipolar and cellular networks.The standard success probability is the first moment, ps(θ) ≡ M1(θ).
- Motivation and definition: The meta distribution describes the distribution of conditional SIR success probabilities given the point process.It quantifies the fraction of links or users achieving reliability above a specified level for an SIR threshold.
- Analytical framework: A beta distribution matched to the first two moments provides a highly accurate approximation to the exact meta distribution.The approximation is proposed for both network types and is reported to match exact distributions extremely accurately.
- Network-model comparison: In dense bipolar networks with p → 0 at fixed concurrent-transmitter density, all links converge to the same success probability, unlike cellular networks with random base-station activity.For cellular networks, the variance remains bounded away from zero as activity probability tends to zero.
- Additional result: The paper also gives conditions on the SIR threshold and transmit probability for a finite mean local delay.
II. POISSON BIPOLAR NETWORKS
The Poisson bipolar model places transmitters as a PPP with dedicated receivers, ALOHA access, and Rayleigh fading, then analyzes conditional link success probabilities through their moments. It establishes exact moment formulas, bounds, and concentration results showing that sparse access with fixed concurrent-transmitter density makes link reliabilities increasingly uniform.
- System model: Transmitters form a PPP of intensity λ, each has a receiver at distance R, and nodes independently transmit with probability p under Rayleigh fading.
- Meta distribution: The conditional success probability is attached to each transmitter as a mark, producing a correlated marked point process because nearby links experience correlated interference.
- Moments: Closed-form moments M_b are derived for the conditional success probability, including a theorem valid whenever D_b(p, δ) is defined.
- Mean local delay: M_-1 is the mean number of transmission attempts needed for one success, and it becomes infinite when all nodes always transmit.
- Concentration: As p → 0 with τ = λp fixed, P_s(θ) converges to the typical-link success probability in mean square, probability, and distribution.
- Concentration: The variance is proportional to p for small p at fixed M_1, so it vanishes in the sparse-access limit and all links have the same success probability.
C. Exact expression
The paper obtains an exact integral representation of the meta distribution from purely imaginary moments. The derivation applies the characteristic function of log P_s(θ) and the Gil-Pelaez theorem, yielding an efficiently computable expression whose variance behavior matches the concentration result.
- The meta distribution has an exact integral expression based on purely imaginary moments M_jt.
- The derivation sets X = log P_s(θ), forms its characteristic function from M_jt, and applies the Gil-Pelaez theorem to obtain the ccdf.
- Because |M_jt| decreases essentially exponentially with t, the exact integral can be evaluated efficiently.
- For fixed λp = 1/4, the exact meta distribution shows reduced variance when p is smaller, as predicted by Corollary 1.
D. Classical bounds on the meta distribution
Classical inequalities provide bounds on the meta distribution from its moments. Markov, Chebyshev, and Paley-Zygmund bounds characterize reliability fractions, tighten as variance decreases, and can be near-optimal in some regions.
- Bound construction: The meta distribution is bounded using Markov and Chebyshev inequalities, with a Paley-Zygmund lower bound also available.
- Numerical illustration: For the Fig. 2 parameters, the mean success probability is M_1 = 0.735, while the variance depends on λ and p and is proportional to p for small p.
- Numerical illustration: Decreasing p reduces the variance and tightens the bounds, while some classical bounds are near-optimal and others are significantly looser across intervals.
- Bound interpretation: The upper Markov bound is useful for small M_1, whereas the lower bound is useful when M_1 is approximately 1.
- Bound interpretation: The Paley-Zygmund bound lower-bounds the fraction of links whose reliability exceeds a specified fraction x of the average performance.
- Sparse-access limit: As p → 0, the lower bound approaches 1, consistent with concentration of link success probabilities.
E. Best bounds given four moments
The paper derives tightest possible lower and upper bounds on the SIR meta distribution using the first four moments, finding that these bounds can substantially improve classical bounds in some intervals.
- Best bounds given four moments: The method finds minimum and maximum cdf values over all distributions with four prescribed moments.It applies a method from to obtain the best lower and upper bounds for each support value.
- Best bounds given four moments: The four-moment bounds are applied to the conditional success probability Ps(θ), whose moments are E(Y^k) = Mk.The formulation uses ccdfs and identifies Y with Ps(θ).
- Best bounds given four moments: In some intervals, the optimized bounds are significantly tighter than classical Markov and Paley-Zygmund bounds, while in others they are near-optimum.Figure 3 compares the exact meta distribution with these bounds.
- Best bounds given four moments: The method is not restricted to four moments but becomes considerably more tedious when additional moments are included.This limits the practical convenience of extending the procedure to higher moments.
F. Approximation with beta distribution
Because Ps(θ) is supported on [0, 1], the paper approximates its meta distribution with a beta distribution matched to its first two moments. The approximation closely matches exact distributions and quantifies reliability–SIR trade-offs.
- Approximation with beta distribution: The beta distribution is chosen as a simple approximation because Ps(θ) is supported on [0, 1].Its parameters are determined from the mean and variance.
- Approximation with beta distribution: The beta approximation provides an excellent match for link success probabilities across the evaluated parameter sets.For one parameter set, analytical moments from the −1-st and 3-rd through 8-th orders differ by less than 3%.
- Approximation with beta distribution: The approximation accurately captures skewness, kurtosis, and mean local delay.These quantities are approximated through the close agreement of higher moments.
- Illustrations of the meta distribution: For λ = 1, p = 1/4, α = 4, and R = 1/2, most links achieve an SIR of −10 dB with 80% probability, whereas virtually no links achieve 10 dB with that probability.The cross-sections provide more precise quantitative views of this distribution.
- Illustrations of the meta distribution: At θ = 0 dB, 60% of links have success probability at least 80%, while achieving 80% reliability for 80% of links requires θ no greater than −7.6 dB.These values illustrate the trade-off between SIR threshold and link reliability.
- Illustrations of the meta distribution: For a link fraction of 0.95, the contour plot gives reliability 0.6 at −5 dB and 0.31 at 5 dB.The contours quantify the trade-off between data rate, represented by θ, and reliability x.
III. POISSON CELLULAR NETWORKS
In Poisson cellular networks, the paper studies the conditional success probability given the base-station process and derives closed-form moments for its meta distribution. The resulting distribution describes user-level reliability beyond the typical-user success probability.
- Poisson cellular networks: Base stations form a PPP, users form a stationary point process, and users associate with their nearest base station for downlink service.Initially, all base stations are assumed to be always active.
- Poisson cellular networks: The conditional success probability is the probability that the typical user's SIR exceeds θ given the base-station process and a user at the origin.The meta distribution is the ccdf of this conditional probability.
- Poisson cellular networks: The cellular meta distribution gives the fraction of users achieving SIR at least θ with reliability at least x.The standard success probability instead describes the typical user and is independent of user and base-station densities.
- Poisson cellular networks: The moments Mb of the conditional success probability are derived in closed form for Poisson cellular networks.For integer b, Mb equals the joint success probability of b transmissions.
- Poisson cellular networks: The mean local delay is finite when θ < 1/δ − 1 and infinite when θ ≥ α/2 − 1 because of correlated interference.The condition can also be written as θ MISR < 1.
- Poisson cellular networks: For α = 3, the variance is maximized near θ = 1, and for α = 3 and α = 4 the corresponding success probability is ps = 0.38.The variance tends to zero as θ approaches either 0 or infinity.
C. Exact expression, bounds, and beta approximation
The paper obtains an exact cellular SIR meta distribution through the Gil-Pelaez theorem and evaluates its numerical tractability. The beta approximation fits across thresholds, while contour plots expose reliability trade-offs for low-performing users.
- Exact expression, bounds, and beta approximation: The exact cellular meta distribution is derived from the moments using the Gil-Pelaez theorem.The result is stated as an analytical expression for the meta distribution.
- Exact expression, bounds, and beta approximation: The inversion integral is efficient to evaluate because numerical investigations indicate |M_jt| = Θ(t^-1), giving an integrand that decays as t^-2.This decay supports efficient numerical evaluation.
- Exact expression, bounds, and beta approximation: For θ = 1, the meta distribution has almost constant slope, indicating that user success probabilities are essentially uniform between 0 and 1.Figure 10 compares the exact distribution with classical and best bounds.
- Exact expression, bounds, and beta approximation: The beta approximation provides an excellent fit over a wide range of θ values, including θ ∈ {−10, 0, 10} dB.It also illustrates which reliability and user-fraction combinations are achievable.
- Exact expression, bounds, and beta approximation: For the bottom 5% of users, an SIR of −10 dB is achievable with reliability 0.72, whereas −4.3 dB is achievable with reliability 0.3.These values come from the contour corresponding to F̄(θ, x) = 0.95.
D. Effect of random base station activity
Random base-station activity changes the success-probability contours and delay behavior in cellular networks, but does not eliminate link-experience disparities as activity becomes sparse.
- Theorem 3 expresses the b-th moment of the cellular success probability when interfering base stations are independently active with probability p.
- The critical transmit probability pc(θ) marks the transition from finite to infinite mean local delay.When θ < 1/δ − 1, pc(θ) = 1; for p < 1, larger θ can maintain finite mean local delay.
- As p → 0 and θ → ∞ with t = pθ^δ fixed, the cellular success probability remains asymptotically constant.The corresponding contour asymptotes are obtained from the same scaling relation.
- Decreasing p by 10 dB permits increasing θ by 5α dB while maintaining the same success probability in the small-p limit.
- The variance changes relatively little with p, especially at higher target success probabilities, so random base-station activation cannot significantly reduce user-experience disparity.The plotted variances follow contour lines for target success probabilities and are compared with their asymptotic values.
IV. CONCLUSIONS
The paper develops the SIR meta distribution to characterize conditional link reliability beyond spatial averages. It derives moment-based analyses and shows contrasting effects of sparse activity in bipolar and cellular networks.
- The meta distribution gives the distribution of conditional link success probabilities, providing finer-grained information than the usual mean success probability.
- Closed-form moments enable exact meta-distribution expressions, moment-based bounds, and a highly accurate beta approximation.
- In bipolar networks, the variance of Ps tends to zero as p → 0 at constant mean success probability, whereas cellular-network variance approaches a non-zero constant.
- These results demonstrate that stochastic geometry can derive complete spatial performance distributions, not only spatial averages.The distributions capture statistics across all links in a network realization.
APPENDIX
The appendix applies fading and ALOHA averaging to moment expressions and relates the resulting formulas to prior diversity-polynomial results.
- Averaging over fading and ALOHA yields the moment expression used in the analysis.
- For integer b, the resulting integral is the diversity polynomial derived in prior work.
- For general non-integer b, the moments are represented as Mb = e^−λFb before applying the modified derivation.