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Outage and Local Throughput and Capacity of Random Wireless Networks

Martin Haenggi

arXiv:0806.0909v1cs.ITcs.NImath.IT

TL;DR

Quantitative evidence on outage and local throughput is limited, while network-wide scaling may not distinguish protocols for design. The paper uses an uncertainty cube and spatial contention to study performance across network models; for Rayleigh-fading ALOHA, contention bounds success probability as transmission probability rises.

  • Problem

    Network-wide scaling results may not distinguish protocols for design, and quantitative results on outage and local throughput are relatively scarce.

    Source: A. Background · p. 1

  • Method

    The paper classifies networks by uncertainty and uses spatial contention, the initial slope of outage probability with transmit probability, to characterize performance.

    Source: §VII · pp. 19–20

  • Results

    For Rayleigh-fading ALOHA, success probability satisfies 1 − pγ ≤ ps ≤ exp(−pγ).

    Source: §VII · p. 20

  • Takeaways & Limitations

    The spatial capacity is about 2.5× larger than throughput.

    Source: §VII · p. 21

Abstract

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Outage probabilities and single-hop throughput are two important performance metrics that have been evaluated for certain specific types of wireless networks. However, there is a lack of comprehensive results for larger classes of networks, and there is no systematic approach that permits the convenient comparison of the performance of networks with different geometries and levels of randomness. The uncertainty cube is introduced to categorize the uncertainty present in a network. The three axes of the cube represent the three main potential sources of uncertainty in interference-limited networks: the node distribution, the channel gains (fading), and the channel access (set of transmitting nodes). For the performance analysis, a new parameter, the so-called {\em spatial contention}, is defined. It measures the slope of the outage probability in an ALOHA network as a function of the transmit probability $p$ at $p=0$. Outage is defined as the event that the signal-to-interference ratio (SIR) is below a certain threshold in a given time slot. It is shown that the spatial contention is sufficient to characterize outage and throughput in large classes of wireless networks, corresponding to different positions on the uncertainty cube. Existing results are placed in this framework, and new ones are derived. Further, interpreting the outage probability as the SIR distribution, the ergodic capacity of unit-distance links is determined and compared to the throughput achievable for fixed (yet optimized) transmission rates.

I. INTRODUCTION

The paper addresses the scarcity of quantitative outage and per-link throughput results, which can offer design insight beyond network-wide scaling laws. It organizes network uncertainty and introduces spatial contention to compare performance across network models.

A. Background

  • Spatial contention is introduced to classify and extend prior results to stochastic wireless networks with uncertainty in node placement, channel characteristics, and channel access.The uncertainty cube represents these three sources of uncertainty.Source: A. Background · p. 2

B. The uncertainty cube

  • The uncertainty cube assigns coordinates for uncertainty in node placement, fading, and channel access; its fully uncertain corner corresponds to a Poisson network with Rayleigh fading and slotted ALOHA.The opposite (0, 0, 0) corner has deterministic positions, no fading, and deterministic scheduling.Source: B. The uncertainty cube · p. 2

C. Models, notation, and definitions

  • Spatial contention γ is the slope of outage probability at transmit probability p = 0; its inverse, spatial efficiency, measures how efficiently the network uses space.The parameter depends on the SIR threshold, network geometry, and path-loss exponent.Source: C. Models, notation, and definitions · p. 3

II. RELATED WORK

The related work covers interference characterization and outage analyses for several network types. The section also draws on interference transforms to obtain success probabilities in fading networks.

II. RELATED WORK

  • Prior studies address interference in large wireless systems and calculate outage for cellular and Poisson networks, including analyses with fading and shadowing,,, –.The paper draws on these results for its study of random and regular networks.Source: §II · p. 4

A. (1, 0, 1): Infinite non-fading random networks with α = 4 and slotted ALOHA

  • The studied case appears in, where the characteristic function of the interference is determined; notation is adapted here, and a small mistake in [3, Eqn. (18)] is corrected.Source: A. (1, 0, 1): Infinite non-fading… · p. 4

B. (0, 1, 1): Regular fading networks with α = 2 and slotted ALOHA

  • For regular fading networks, prior work derives interference distributions and Laplace transforms, which yield success probabilities in Rayleigh fading.In interference-limited settings the interference transform suffices; with fixed-variance noise, a noise factor is added,,.Source: B. (0, 1, 1): Regular fading networks… · p. 5

C. (1, 1, 1): Random fading networks with slotted ALOHA

The section develops outage results for random fading networks and a single interferer. It relates outage’s initial increase under ALOHA to spatial contention and compares the effects of fading on the desired and interfering links.

C. (1, 1, 1): Random fading networks with slotted ALOHA

  • For two-dimensional random networks with Rayleigh fading and slotted ALOHA, spatial contention is γ = θ^(2/α)C2(α), with C2(2) = ∞ and C2(4) = π²/2 ≈ 4.9.The cited prior analyses ignore noise,.Source: C. (1, 1, 1): Random fading networks… · p. 5

III. THE CASE OF A SINGLE INTERFERER

  • With one interferer, outage increases linearly with transmit probability p at slope γ, and desired-link fading is harmful while fading on the interferer’s channel is helpful.The ordering of spatial contention across fading cases is strict except when the interferer is at distance zero.Source: §III · p. 6
  • For a single interferer, fading effects vary with effective distance: at short distances desired-link fading matters little, while at longer distances interferer-channel fading matters little.For Nakagami-m fading, success probability changes monotonically with m in the two cases described.Source: §III · pp. 6–7

IV. NETWORKS WITH RANDOM NODE DISTRIBUTION

The section derives spatial-contention results for random node distributions under power and exponential path loss, with full, partial, and no fading. It also identifies conditions for finite contention and a case that requires numerical study.

A. (1, 1, 1): One-dimensional fading random networks with slotted ALOHA

  • For one-dimensional fading random networks, the spatial-contention expression motivates the conjecture γ = θ^(d/α)Cd(α) for general dimension d, with finite γ requiring α > d.The general-dimensional expression is presented as a conjecture.Source: A. (1, 1, 1): One-dimensional fading… · p. 7

B. (1, 1, 1): Partially fading random networks with slotted ALOHA

C. (1, 0, 1): Non-fading random networks with α = 4 and slotted ALOHA

  • For non-fading random networks with α = 4, success probability is bounded by 1 − γp < ps(p) < exp(−γp).The supplied passage identifies this as the outage probability for unit-distance links.Source: C. (1, 0, 1): Non-fading random… · p. 8

D. (1, 1, 1): Fully random networks with exponential path loss

  • With exponential path loss, spatial contention grows with log θ rather than θ^(2/α) at large θ, and its defining integral is finite for any δ > 0.Unlike power-law path loss, this case does not require α to exceed network dimension; the discussed mixed laws have no closed-form solution and require numerical study.Source: D. (1, 1, 1): Fully random networks… · p. 8

V. NETWORKS WITH DETERMINISTIC NODE PLACEMENT

For deterministic placements, spatial contention and effective distances organize outage and success-probability results. In infinite regular line networks, the paper gives bounds and approximations whose accuracy depends on the threshold and access probability.

V. NETWORKS WITH DETERMINISTIC NODE PLACEMENT

B. (0, 1, 1): Infinite regular line networks with fading and ALOHA

C. (0, 1, 1): Partially fading regular networks

The section compares partial fading and TDMA in regular line networks. It reports how fading assumptions affect spatial contention and gives success-probability results and bounds for TDMA.

C. (0, 1, 1): Partially fading regular networks

  • With fading only on the desired link, spatial contention is larger than with full fading, where fading in interferer channels helps.For regular line networks, the stated result is γ = θζ(α) and p_s = e^(−pθζ(α)).Source: C. (0, 1, 1): Partially fading regular… · p. 10

D. (0, 1, 0): Regular line networks with fading and TDMA

  • For one-sided infinite regular line networks with Rayleigh fading and m-phase TDMA, the paper gives success-probability formulas for α = 2 and α = 4.Having every m-th node transmit is equivalent to reducing θ by a factor m^α and setting p = 1; two-sided success probabilities are obtained by squaring the one-sided probabilities.Source: D. (0, 1, 0): Regular line networks… · pp. 10–11
  • The TDMA success-probability bounds approach the exact value as θ/m^α decreases.The paper also gives a tighter upper bound and notes that higher-order Taylor expansions can deviate from it.Source: D. (0, 1, 0): Regular line networks… · p. 11

E. Spatial contention in TDMA networks

The paper extends spatial contention to TDMA and uses it to characterize throughput in slotted ALOHA. For the ALOHA throughput bound, numerical calculations find a small maximum gap from the true optimum.

E. Spatial contention in TDMA networks

  • For TDMA, setting the transmission-time fraction to 1/m yields γ = ζ(α)θ, identical to spatial contention for the ALOHA line network.The section’s table summarizes spatial-contention parameters for slotted ALOHA networks and adds TDMA for comparison.Source: E. Spatial contention in TDMA networks · p. 12, Table III · p. 12

A. (ul, uf, 1): Networks with slotted ALOHA

  • The throughput analysis defines probabilistic throughput using unconditional success, accounting for whether the desired transmitter sends and, in half-duplex, whether the receiver listens.An approximate optimizing transmit probability is p_opt ≈ 1/(2 + γ).Source: A. (ul, uf, 1): Networks with slotted… · p. 13
  • The lower bound on maximum ALOHA probabilistic throughput is within 1.4% of the true maximum over γ ∈ R+.The proposition considers ALOHA with Rayleigh fading and p_s = e^(−pγ).Source: A. (ul, uf, 1): Networks with slotted… · p. 13

B. (0, 1, 0): Two-sided regular line networks with TDMA

For two-sided regular lines, the paper estimates the TDMA parameter maximizing probabilistic throughput and compares that estimate with numerical optimization. It also optimizes transmission rate and SIR threshold, reporting throughput differences between full- and half-duplex operation.

B. (0, 1, 0): Two-sided regular line networks with TDMA

  • The estimated m_opt equals the numerically found optimum for most θ values, with a negligibly small difference in maximum throughput.The figure compares m_opt and its estimate against θ for α = 2, and also shows success probability at the optimum m against θ.Source: B. (0, 1, 0): Two-sided regular line… · p. 14, Figure 2 · p. 14

C. Rate optimization

  • The paper defines throughput as probabilistic throughput times log(1 + θ), in nats/s/Hz, and gives an SIR threshold that maximizes it for full-duplex operation.The threshold expression uses the principal branch of the Lambert W function and network dimension d = 1 or 2.Source: C. Rate optimization · p. 15
  • The optimum normalized transmission rate is concave for α > d, with derivative 2 at α = d for d = 1 and 1 for d = 2.The stated upper bounds are R_opt(α) < α − 2 for d = 2 and R_opt(α) < 2(α − 1) for d = 1.Source: C. Rate optimization · p. 15
  • Full-duplex throughput is about 10% higher than half-duplex throughput over the entire practical range of α.The optimum transmit probabilities are around 0.105 for full-duplex and 0.08 for half-duplex; maximum throughput scales almost linearly with α − d.Source: C. Rate optimization · p. 15

D. (1, 1, 1): Ergodic capacity

For the two-dimensional (1, 1, 1) network, the paper derives ergodic-capacity results and relates capacity to the SIR distribution. It also examines spatial capacity under full- and half-duplex access.

D. (1, 1, 1): Ergodic capacity

  • For α = 4, Proposition 7 gives an ergodic-capacity expression and a lower bound for the two-dimensional (1, 1, 1) network.For rational α, pseudo-closed-form expressions are also available using the Meijer G function.Source: D. (1, 1, 1): Ergodic capacity · pp. 16–17
  • The 2/α-th moment of the SIR is exponentially distributed with mean 1/cp, making ALOHA-channel capacity equivalent to Rayleigh-fading capacity with mean SIR c^-1.Source: D. (1, 1, 1): Ergodic capacity · p. 16
  • An SIR-boost exponent α/2 > 1 does not by itself imply capacity above the standard Rayleigh case; that comparison holds only when average SIR is high.Source: D. (1, 1, 1): Ergodic capacity · p. 17
  • For general p and α, the capacity integral has no closed-form expression, while direct calculation is available for α = 4 and the lower bound follows by rewriting the integral.Source: D. (1, 1, 1): Ergodic capacity · p. 17
  • The spatial-capacity optimum uses pC for full duplex and p(1 − p)C for half duplex; the optimal p decreases with α for full duplex and is approximately 1/9 for half duplex, independent of α.Source: D. (1, 1, 1): Ergodic capacity · p. 17

E. TDMA line networks

For TDMA line networks, spatial reuse balances capacity against reuse, with the best reuse factor depending on the path-loss exponent. The concluding discussion presents spatial contention as a compact way to relate network uncertainty, outage, and throughput.

E. TDMA line networks

  • For α = 2, the specific lower bound tightens as m increases, and the lower bounds are compared with the actual TDMA line-network capacity.The discussion describes the bound comparison for the ergodic capacity as a function of reuse parameter m.Source: E. TDMA line networks · pp. 18–19
  • For α = 2, reuse factor m = 2 maximizes spatial capacity C/m; for α = 4, m = 3 yields a slightly higher C/m.The α = 2 ergodic capacity is also approximated by C ≈ log(1 + m^α/ζ(α)).Source: E. TDMA line networks · p. 19

VII. DISCUSSION AND CONCLUDING REMARKS

  • For large classes of networks, outage probability describes the SIR distribution for unit-distance links, and the uncertainty cube classifies networks by their underlying stochastic processes.The discussion also notes that extensions such as power control and directional transmissions are possible.Source: §VII · pp. 19, 21
  • Spatial contention γ is the initial slope of success probability, and for Rayleigh-fading ALOHA it bounds success probability between 1 − pγ and exp(−pγ).The authors conjecture the bound holds for general ALOHA networks.Source: §VII · p. 20
  • In Poisson networks, success probability corresponds to the probability that a disk of area γ around the receiver contains no concurrent transmitters.This gives an equivalent disk model whose interference radius is tied to γ.Source: §VII · p. 20
  • For full-duplex ALOHA, probabilistic throughput is p exp(−pγ), while transmission capacity is proportional to spatial efficiency at small outage constraints.The discussion gives p_opt = min{σ, 1} and, for small ϵ, p = ϵσ.Source: §VII · pp. 20–21
  • Spatial contention offers a single-parameter characterization of spatial-use capability even when channel access differs from ALOHA.The paper also uses success probabilities to determine optimal ALOHA transmission probabilities, TDMA reuse parameters, transmission rates, and ergodic capacity.Source: §VII · p. 21
  • Spatial capacity is about 2.5× throughput, half-duplex operation carries a 10–20% penalty, and the optimum transmit probability is around 1/9 for both cited optima.The cited optima are for throughput and maximum spatial capacity.Source: §VII · p. 21
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