The effect of fading, channel inversion, and threshold scheduling on ad hoc networks

Steven Weber; Jeffrey G. Andrews; Nihar Jindal

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The effect of fading, channel inversion, and threshold scheduling on ad hoc networks

Steven Weber, Jeffrey G. Andrews, Nihar Jindal

arXiv:0707.3584v1Jul 24, 2007cs.IT

TL;DR

The paper asks how fading and two distributed uses of local CSI—channel inversion and threshold scheduling—change ad hoc network capacity. Using transmission-capacity analysis and stochastic-geometry bounds for Poisson networks, it finds that inversion reduces network capacity while threshold scheduling can substantially mitigate fading. The conclusions are subject to assumptions including narrowband, slotted, interference-limited operation without retransmissions or explicit multihop routing.

Problem
The paper examines how fading, random link conditions, and local CSI through power control and scheduling affect ad hoc network transmission capacity.
Method
The authors use transmission capacity and stochastic-geometry analysis of Poisson networks to derive asymptotically tight bounds on SIR, outage probability, and capacity.
Results
Channel inversion lowers overall network capacity despite helping poor links, whereas threshold scheduling can significantly increase capacity by activating only sufficiently strong channels.
Takeaways & Limitations
Simple transmitter-receiver threshold scheduling can exploit local CSI in a fully distributed manner and reduce fading's effect without coordination among transmitters.
Takeaways & Limitations
The analysis assumes narrowband fading, slotted transmissions, no explicit multihop communication, no retransmissions, and interference-limited operation.

Abstract

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This paper addresses three issues in the field of ad hoc network capacity: the impact of i)channel fading, ii) channel inversion power control, and iii) threshold-based scheduling on capacity. Channel inversion and threshold scheduling may be viewed as simple ways to exploit channel state information (CSI) without requiring cooperation across transmitters. We use the transmission capacity (TC) as our metric, defined as the maximum spatial intensity of successful simultaneous transmissions subject to a constraint on the outage probability (OP). By assuming the nodes are located on the infinite plane according to a Poisson process, we are able to employ tools from stochastic geometry to obtain asymptotically tight bounds on the distribution of the signal-to-interference (SIR) level, yielding in turn tight bounds on the OP (relative to a given SIR threshold) and the TC. We demonstrate that in the absence of CSI, fading can significantly reduce the TC and somewhat surprisingly, channel inversion only makes matters worse. We develop a threshold-based transmission rule where transmitters are active only if the channel to their receiver is acceptably strong, obtain expressions for the optimal threshold, and show that this simple, fully distributed scheme can significantly reduce the effect of fading.

I. INTRODUCTION

The paper studies how fading and local CSI mechanisms affect ad hoc network transmission capacity, using stochastic-geometry analysis of Poisson networks. It finds that channel inversion reduces overall capacity, while threshold scheduling can significantly improve it.

I. INTRODUCTION: The paper characterizes fading, random link distances, and local CSI through pairwise scheduling and power control in ad hoc networks.It considers lognormal shadowing, Rayleigh fading, and nearest-neighbor transmissions in a Poisson field.
I. INTRODUCTION: Channel inversion can help individual links and promote fairness, but it lowers network-wide capacity by increasing the likelihood of a dominant interferer causing outage.The mechanism arises because compensating for weak channels creates undesirable diversity in interference power.
I. INTRODUCTION: Threshold scheduling activates a transmitter only when its desired channel exceeds a threshold, reducing outage and increasing transmission intensity.The rule requires coordination only between each transmitter and its intended receiver and can introduce multi-user diversity.
I. INTRODUCTION: The framework assumes narrowband fading, slotted transmissions, no explicit multihop routing, and no retransmissions.These assumptions bound the scope of the capacity results; ignoring retransmissions reduces effective network capacity.
I. INTRODUCTION: Transmission capacity is the maximum density of successful links per unit area subject to an outage-probability constraint at a target SIR.The analysis derives asymptotically tight bounds on outage probability and transmission capacity.
I. INTRODUCTION: Threshold scheduling with channel inversion has little impact on transmission capacity because thresholding excludes deep fades and limits the power needed for inversion.The paper analyzes all four combinations of threshold scheduling, random access, fixed power, and channel inversion.

II. RELATED WORK AND PRELIMINARIES

The paper situates its capacity analysis at the intersection of stochastic geometry, stable distributions, shot noise, spatial interference, and distributed channel-aware scheduling. It models fading and distance variation while contrasting independent threshold decisions with coupled scheduling approaches.

The analysis combines stable distributions, shot noise processes, and spatial co-channel interference models to characterize transmission capacity under general fading.
Stable distributions: Stable random variables are central because aggregate interference from Poisson-distributed transmitters can exhibit stable behavior, although their PDFs and CDFs generally lack closed forms.
Distributed scheduling: Unlike game-theoretic scheduling, the proposed scheme makes transmission decisions independently using only each transmitter’s channel to its intended receiver.
Distributed scheduling: The approach targets a practical alternative to centralized global-CSI selection by characterizing a simple threshold rule that increases capacity over channel-blind Aloha.
Channel model: The channel model uses path-loss exponent α > 2, independent distance-independent channel gains, and iid transmitter–receiver distances.

C. Performance metrics

The paper evaluates ad hoc networks using outage probability, spatial throughput, and transmission capacity, all derived from the SIR distribution. Transmission capacity emphasizes successful spatial reuse under an outage constraint.

Performance metrics: Outage occurs when the received SIR falls below the threshold β, with q(µ) denoting outage probability at attempted-transmission intensity µ.
Performance metrics: Spatial throughput equals attempted transmission intensity µ multiplied by the average success probability 1 − q(µ).
Transmission capacity is the maximum density of successful communication links per unit area subject to a specified outage constraint and SIR target.
Performance bounds: The analysis derives asymptotically tight bounds on the interference CCDF, outage probability, spatial throughput, and transmission capacity without channel inversion.
Performance bounds: Random fading and link distances enter the asymptotic bounds through the fractional moments E[Ψ^δ]E[Ψ^−δ]E[D^2].
Performance bounds: The lower bound is tight as y → ∞, particularly in low-outage regimes where dominant interferers account for large interference events.

B. Performance without threshold scheduling and with channel inversion

Without threshold scheduling, the paper compares randomized transmission with fixed power and channel inversion. Channel inversion normalizes desired received power and simplifies interference dependence, but can reduce network capacity and require impractically large power.

Channel inversion sets each transmitter’s power inversely to its effective channel coefficient, ensuring unit received signal power at its intended receiver.
Scope and feasibility: For Rayleigh fading, E[1/W] is infinite, making channel inversion impractical without additional constraints, although a minimum fading threshold can restore feasibility.
Interference structure: Without inversion, all normalized interference contributions depend on the reference link’s channel coefficient; inversion instead divides each contribution by its interferer’s own coefficient.
Performance bounds: The channel-inversion analysis provides bounds for outage probability, spatial throughput, and transmission capacity analogous to the no-inversion results.
Analysis: Channel inversion removes conditioning on received signal power and yields an unconditioned stable interference distribution with dispersion determined by κ.

C. Discussion

The discussion shows that fading and random link distances generally reduce transmission capacity, while channel inversion worsens network-wide performance despite helping individual links. Threshold-based scheduling and separate signal/interference fading analysis clarify when local CSI can improve capacity and how the bounds should be interpreted.

The effect of channel inversion: Channel inversion strictly lowers network capacity, although it can help individual links and promote fairness.It increases vulnerability to signal fades from nearby interferers, raising the likelihood of a dominant interferer causing outage.
The effect of channel inversion: Channel inversion requires greater average transmission power than an equivalent fixed-power system delivering the same average received power.This can push the system from the interference-limited regime toward the noise-limited regime; threshold scheduling reduces the difference by excluding small channel gains.
Effect of random distance and fading: Random transmitter–receiver distances reduce capacity by increasing the rate factor θ relative to fixed distances with the same mean.The bounds express the impact of fading and random distances through fractional moments such as E[Ψ^δ]E[Ψ^-δ]E[D^2].
Separating the effects of signal and interference fading: Fading of desired signals reduces transmission capacity, whereas fading of interfering signals increases it; with identical fading distributions, the net effect is negative.The capacity effects are captured through fractional moments of the signal and interference gains.
Separating the effects of signal and interference fading: Rayleigh fading can significantly reduce capacity because its desired-signal fractional moment diverges as δ → 1.More generally, substantial probability mass near zero in the desired channel gain makes E[Ψ^-δ] large.
Maximum achievable spatial throughput and TC: The upper-bound-optimal outage probability is ϵu,* = 1 − 1/e ≈ 0.63, meaning almost two thirds of attempted transmissions fail at that bound optimum.The stated optimality applies to the analytical bounds, although numerical and simulation results indicate the approximation is valid over most regimes studied.

D. Examples

The examples instantiate the paper’s bounds for lognormal shadowing, Rayleigh fading, and random nearest-neighbor distances, showing how fading and distance variability affect transmission capacity.

Three examples fix distances with lognormal shadowing, fix distances with Rayleigh fading, or fix fading while modeling nearest-neighbor distances.
Rayleigh fading uses exponentially distributed channel gains, while lognormal shadowing uses Ψ ∼LN(0, σ2).
κ = 1 E[Ψδ]E[Ψ−δ] captures fading’s multiplicative effect on transmission capacity across Rayleigh, lognormal, and combined fading.
Both fading distributions become less harmful as the path-loss exponent increases, but Rayleigh fading imposes a harsh penalty when α is near two.
Nearest-receiver analysis simplifies by treating transmitter–receiver distances as iid, despite possible receiver collisions and distance dependence.

V. PERFORMANCE WITH THRESHOLD SCHEDULING

Threshold scheduling activates transmitters only when their desired channels exceed a global threshold, then analyzes performance with fixed power and channel inversion.

Transmitters participate only when their channel strength to the intended receiver exceeds the global threshold t.
The policy is intended to improve performance over randomized transmission while requiring only local channel information, not global coordination.
Threshold scheduling is most feasible when fading varies faster than allowable packet delays; slow fading may require combining randomized and threshold scheduling.
Thresholding conditions the desired received-power distribution on W ≥ t while leaving interfering channel coefficients unaffected.
Under threshold scheduling with channel inversion, each active transmitter uses P_i = 1/W_i and the maximum transmit power is P_max = 1/t.
The optimal threshold is obtained by maximizing the concave throughput or capacity function, with κ(t) decreasing monotonically in t.

C. Examples

The paper extends its threshold-scheduling analysis to lognormal shadowing, Rayleigh fading, and nearest-receiver distances using conditional channel distributions and computable performance quantities.

The three threshold-scheduling examples revisit lognormal shadowing, Rayleigh fading, and nearest-receiver transmissions.
Threshold decisions replace the unconditional channel distribution with the distribution conditioned on W > t.
For Rayleigh fading, the conditional analysis expresses κ(t) using the incomplete Gamma function.
The threshold formulas reduce consistently to the unthresholded case when t approaches zero.
The optimal-threshold derivation uses the Lambert W function on its k = −1 branch.

VI. NUMERICAL AND SIMULATION RESULTS

Numerical and simulation results compare randomized and threshold transmissions across fading and random-distance examples, finding substantially higher capacity with threshold scheduling and lower performance with channel inversion.

The simulations compare randomized and threshold transmissions with and without channel inversion using outage probability, spatial throughput, and transmission capacity.
For lognormal shadowing, analytical outage bounds are reasonably tight, especially when outage probability is small.
Channel inversion reduces performance, producing larger outage probability and smaller spatial throughput and transmission capacity.
Threshold transmissions achieve over twice the peak capacity of randomized transmissions in the lognormal-shadowing results.
Nearest-receiver threshold scheduling achieves a three fold or greater transmission-capacity increase over randomized transmissions for all ϵ < 0.2.
Across the paper, local-information analysis is presented as realistic and analytically tractable, though suboptimal.
The conclusion reports capacity about three times higher with threshold scheduling than with no scheduling over many ranges of interest.

APPENDIX

The appendix derives asymptotically tight lower and upper bounds on outage-related interference distributions using Poisson-process decompositions, moment inequalities, and asymptotic expansions. It then extends the analysis to channel inversion and threshold-based transmission policies.

Proof strategy: The proof proceeds in three steps: derive a lower bound, derive an upper bound, and obtain asymptotic expansions.The expansions condition on received signal power and then integrate over its distribution.
Bounds: The lower-bound construction partitions interferers into dominant points capable of causing outage and non-dominant points.The dominant set is characterized through a Poisson-process void probability, with radial symmetry enabling polar-coordinate integration.
Bounds: The upper bound conditions on the dominant-interferer process and uses the Chebychev inequality to control the non-dominant interference.Campbell’s Theorem supplies the mean and variance needed for this bound.
Asymptotics: For asymptotics, the analysis expands conditional complementary distributions as y →∞ and then recovers unconditional distributions by integrating over received-signal power.The derivation also translates a two-dimensional Poisson shot-noise process into a one-dimensional representation.
Channel inversion: Under channel inversion, the normalized interference becomes a shot-noise process with random marks, eliminating the need to condition on received signal power.The same asymptotic procedure applies after replacing the relevant channel-gain variables with the inversion-induced marks.

CHERNOFF UPPER BOUND

This section compares Chernoff and Chebychev approaches for bounding non-dominant interference. Although Chernoff can provide a tighter bound, the paper uses Chebychev because it is explicit and substantially easier to compute.

Chernoff bound: The Chernoff upper bound uses the log moment generating function of the non-dominant interference and a rate function obtained by Legendre transformation.The log MGF is defined as g(θ)=log E[exp{θY^c_y}].
Chernoff bound: Evaluating the Chernoff bound requires the mark distribution induced by the transmission policy, including the channel-inversion case.The mark density depends on the unconditioned signal-power distribution.
Computational burden: The Chernoff calculation can require nested integrations and numerical optimization over θ, making it difficult to evaluate in practice.The paper describes double-integral evaluations for the mark PDFs and MGF, effectively producing a quadruple integral for each θ.
Chosen bound: The paper therefore expresses its results with the Chebychev bound, which is less tight but explicit and does not require integral evaluation.This choice is motivated by both computational tractability and clarity of exposition.
Threshold scheduling: Threshold scheduling preserves the bound framework while changing the received-signal distribution and the associated fractional moment.The threshold t determines the normalized transmission-attempt intensity γ(t)=θ(t)µ(t), which is monotone decreasing.
Numerical comparisons: The figures compare analytical bounds with simulations across randomized and threshold transmissions, with and without channel inversion.The Rayleigh-fading figure plots spatial throughput τ against transmission probability p, while the nearest-receiver figure plots capacity c against outage requirement ϵ.