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Interference and Outage in Clustered Wireless Ad Hoc Networks
RadhaKrishna Ganti, Martin Haenggi
TL;DR
The paper extends wireless-network analysis beyond homogeneous Poisson node distributions by studying Poisson clustered processes. It derives interference and outage results under Rayleigh fading and finds that clustering benefits large link distances while transmission capacity equals that of homogeneous networks.
Problem
Wireless-network results largely assume completely spatially random, homogeneous Poisson node distributions, motivating analysis of other node distributions.
Method
The paper analyzes Poisson clustered transmitters using stochastic-geometry tools, deriving interference bounds and outage expressions for interference-limited Rayleigh-fading channels.
Results
For large link distances, clustering is always beneficial under Rayleigh fading, while transmission capacity equals that of homogeneous networks; smaller-distance gains depend on the path-loss model.
Takeaways & Limitations
The results characterize wireless-system performance under geographical or MAC-induced clustering and show that interference distributions depend heavily on the path-loss model.
Abstract
from arXiv · showhide
In the analysis of large random wireless networks, the underlying node distribution is almost ubiquitously assumed to be the homogeneous Poisson point process. In this paper, the node locations are assumed to form a Poisson clustered process on the plane. We derive the distributional properties of the interference and provide upper and lower bounds for its CCDF. We consider the probability of successful transmission in an interference limited channel when fading is modeled as Rayleigh. We provide a numerically integrable expression for the outage probability and closed-form upper and lower bounds.We show that when the transmitter-receiver distance is large, the success probability is greater than that of a Poisson arrangement. These results characterize the performance of the system under geographical or MAC-induced clustering. We obtain the maximum intensity of transmitting nodes for a given outage constraint, i.e., the transmission capacity (of this spatial arrangement) and show that it is equal to that of a Poisson arrangement of nodes. For the analysis, techniques from stochastic geometry are used, in particular the probability generating functional of Poisson cluster processes, the Palm characterization of Poisson cluster processes and the Campbell-Mecke theorem.
I. Introduction
Large wireless-network analyses commonly assume homogeneous Poisson node locations, but clustered or regular deployments can arise from geography or MAC protocols. This paper extends interference, outage, clustering-gain, and transmission-capacity analysis to Poisson clustered networks.
- Motivation: Homogeneous Poisson point processes model node counts independently across disjoint areas, but real deployments may instead be clustered or regularly distributed.The paper motivates extending PPP results to alternative node distributions.
- Motivation: Clustering can reflect geographical organization, such as nodes inside buildings or coordinated mobility, or logical organization induced by MAC protocols.
- Contributions: The paper models transmitters with a Poisson cluster process and derives interference properties using stochastic-geometry tools.The analysis uses generating functionals, Palm characterization, and related point-process techniques.
- Interference: Upper and lower bounds are obtained for the interference CCDF, including a heavy-tailed exponent 2/α for singular path loss g(x)=∥x∥^-α.For bounded path loss at the origin, the interference distribution depends heavily on fading.
- Performance: For large transmitter-receiver distances, clustering increases success probability relative to a PPP, while small-distance gain depends on path loss and transmission intensity.The clustering gain G(R) exceeds unity at large R and can diverge as R approaches infinity.
- Performance: The maximum transmitting-node intensity under an outage constraint, called transmission capacity, equals that of a Poisson arrangement.
II. System Model and Assumptions
This section introduces the system model and derives preliminary results for Poisson cluster processes.
- II. System Model and Assumptions: The paper introduces the system model and derives required preliminary results for Poisson cluster processes.
A. System model and notation
The system models transmitting nodes as a stationary isotropic Poisson cluster process with unit transmit power, fading, path loss, interference, and an interference-limited success criterion.
- A. System model and notation: Transmitting nodes are modeled as a stationary and isotropic Poisson cluster process on R^2.
- A. System model and notation: Each transmitter uses unit power, and received power from x at receiver z is modeled as h_x g(x−z).The coefficient h_x represents power fading, while g is the path-loss model.
- A. System model and notation: Fading coefficients are independent and identically distributed, with Rayleigh and Nakagami-m fading considered in the analysis.The corresponding power-fading variables are exponential and gamma distributed, respectively.
- A. System model and notation: The path-loss function is defined on R^2 excluding the origin and includes power-law, bounded, and truncated-power-law forms; the singular power law requires α>2.
B. Neyman-Scott cluster processes
The paper specializes Poisson cluster processes to Neyman-Scott, Matern, and Thomas models, then develops generating-functional and Palm-distribution tools for their analysis.
- B. Neyman-Scott cluster processes: Neyman-Scott processes are Poisson cluster processes formed by independently generated daughter clusters around stationary Poisson parent points.
- B. Neyman-Scott cluster processes: Matern and Thomas cluster processes use Poisson-distributed cluster sizes with mean c̄.Matern daughters are uniform within a ball, whereas Thomas daughters follow a symmetric normal distribution with variance σ^2.
- B. Neyman-Scott cluster processes: The generating-functional framework supports nearest-neighbor and interference calculations for the clustered processes.
- B. Neyman-Scott cluster processes: For Thomas and Matern processes, the paper derives a conditional generating functional for evaluating clustered-process quantities.The derivation uses representative-cluster functionals, Campbell-Mecke calculations, and reduced Palm distributions.
- B. Neyman-Scott cluster processes: The representative cluster is treated as a finite point set, and parent points themselves are excluded from the daughter-point process.
III. Interference and Outage Probability of Poisson Cluster Processes
The paper analyzes interference and outage probability for transmitters modeled by a Neyman–Scott Poisson cluster process. Cluster sizes are Poisson, with spatial locations governed by density function f(x).
- The analysis conditions interference characteristics on a transmitting node at the origin.
- Outage probability is evaluated for a transmit–receive pair whose transmitters follow a Neyman–Scott cluster process.
- Each cluster contains a Poisson number of points with mean c̄ and spatial density function f(x).
A. Properties of the Interference Iφ(z)
This section derives conditional interference transforms, moments, and CCDF bounds for clustered transmitter processes. It characterizes heavy-tailed behavior under singular path loss and shows that location and fading affect interference under broader models.
- Conditional interference: The conditional Laplace transform of interference is derived for clustered transmitters using the process generating functional.
- Moments: The mean interference in the clustered process is larger than the mean interference for a PPP.
- CCDF bounds: For stationary and isotropic transmitter processes, the interference CCDF is bounded using intensity, conditional generating functionals, and second-order product density.
- Singular path loss: For singular path loss g(x) = ∥x∥^-α, clustered interference has a heavy-tailed distribution with parameter 2/α, and its CCDF bounds scale accordingly for large y.
- Singular path loss: Because 2/α < 1, the mean and variance diverge under the singular channel model due to the path-loss singularity at the origin.
- Non-singular path loss: With non-singular path loss, interference depends more intricately on fading; Rayleigh fading yields exponential tail decay, whereas power-law fading yields power-law decay.
B. Success probability: P(success)
The paper derives success probability for clustered transmitters in an interference-limited Rayleigh channel, including integral expressions and closed-form bounds. Success depends only on transmitter-receiver distance: clustering can reduce success at short distances but improve it over PPP at large distances.
- The success probability is derived for a transmitter at the origin and receiver at distance R in a no-noise channel.The derivation uses Rayleigh fading and evaluates the interference transform at the relevant threshold.
- Success probability depends only on the distance R, not the receiver angle, and can be interpreted as an average over the circle of radius R.This follows from isotropy of the Palm distribution and the underlying motion-invariant process.
- For large R, the clustered process has higher success probability than a PPP of the same intensity because the receiver is likely to lie outside clusters.The comparison reverses at small distances: the PPP can have higher success probability, including for R < 0.8 in the Matern example.
- For small distances, DS-CDMA performs better with a Poisson node distribution, while large distances require spreading gain to increase approximately like g(z).The outage probability has a spreading-gain scaling law with exponent −2/α under the stated finite-moment condition.
- Closed-form upper and lower bounds for P(success) are derived, with the clustered process compared against a PPP of intensity λ = λp¯c.The bounds use separate contributions from interference outside and within the transmitter’s cluster.
- Conditioning on a transmitter at the origin can make Neyman-Scott success probability lower than PPP, because the transmitter’s cluster causes the greatest interference damage near the receiver.As the receiver moves away, the clustered process’s success probability becomes better than the PPP’s.
C. Clustering Gain G(R)
The clustering gain G(R) compares clustered and Poisson transmitter arrangements at equal intensity, and its sign depends on link distance, path loss, and operating conditions. Clustering can underperform at short distances but outperform uniformly distributed transmitters for sufficiently long links.
- G(R) compares the success probability of clustered and Poisson networks with the same transmitter intensity.The gain reflects whether scheduling transmitters in clusters performs better than spreading them uniformly.
- For large transmit-receive distances, G(R) > 1 and limR→∞G(R) = ∞, so the clustered process has higher success probability.Interference from the intended transmitter’s cluster becomes small, while interference from other clusters contributes more to outage.
- For singular path loss g(x) = ∥x∥−α, G(R) is initially below 1 at small R and eventually exceeds 1, producing a crossover point R∗.Thus, uniform deployment is preferable below R∗, whereas logical clustering is preferable above it.
- With fixed total transmitter intensity λ = λp¯c, G(R) decreases as the average cluster size ¯c increases.The result characterizes how the clustering gain changes when density is redistributed between parent nodes and nodes per cluster.
- For nonsingular path loss, the small-distance comparison depends on total intensity: G(0) < 1 below λ∗(0, T ), while clustering benefits long-hop transmissions.For the example g(x) = (1 + ∥x∥4)−1 and σ = 0.25, λ∗(0, 0.5) ≈1.26.
- For DS-CDMA and FH-CDMA, spreading gain shifts conditions toward uniform scheduling at shorter distances and clustering at longer distances.DS-CDMA changes T, whereas FH-CDMA reduces total transmission intensity λp¯c.
IV. Transmission Capacity of Clustered Transmitters
The paper defines transmission capacity for Poisson clustered transmitters under an outage constraint and derives bounds for both unrestricted and constrained optimization. Unrestricted transmission capacity equals the Poisson benchmark, while constrained capacity remains below it and approaches it as parent intensity grows.
- The unrestricted transmission capacity optimizes λp¯c over parent intensity and average cluster size subject to an outage constraint.This definition does not individually constrain parent-node density or the average number of nodes per cluster.
- The equal capacity is attained as λp →∞ and ¯c →0, when the clustered process degenerates to a PPP.The product λp¯c remains fixed at the Poisson capacity level in this limiting construction.
- For ǫ ≤1 −e−ρ(T ), the transmission capacity of Poisson clustered processes equals the Poisson capacity Cp(ǫ, T ).The equality follows from matching lower and upper bounds.
- Constrained transmission capacity fixes λp and optimizes ¯c under the outage constraint.It is defined as C∗(ǫ, T ) := λp(1 −ǫ) sup{¯c : ¯c > 0, outage-constraint}.
- The constrained capacity increases slowly with λp but remains below the Poisson capacity and approaches Cp(ǫ, T ) as λp →∞.Figure 8 provides upper and lower bounds for C∗(ǫ, T ).
- For clustered transmitters, the constrained DS-CDMA capacity exhibits a similar M 1−2/α gain with spreading gain M.This behavior is reported for g(x) = ∥x∥−α and ǫ = 0.01.
V. Conclusions
The paper extends wireless-network analysis from homogeneous Poisson node distributions to clustered processes, deriving interference and outage characterizations. Clustering improves success probability at large link distances, while transmission capacity equals that of homogeneous networks but requires optimizing two parameters.
- V. Conclusions: The analysis extends prior homogeneous-Poisson results to geographically or MAC-induced clustered node processes.The paper uses stochastic geometry and Palm probabilities to obtain the conditional Laplace transform of interference.
- V. Conclusions: Upper and lower bounds are derived for the interference CCDF under any stationary node distribution and fading.
- V. Conclusions: Interference distribution depends heavily on the path-loss model, particularly whether the model contains a singularity.
- V. Conclusions: Clustering is always beneficial for large transmitter-receiver distances under Rayleigh fading, while gains at shorter distances depend on the path-loss model.
- V. Conclusions: Transmission capacity of clustered networks equals that of homogeneous networks, but defining it requires optimizing over two clustered-process parameters.
- V. Conclusions: Conditional generating functionals are identified as analytical techniques likely to have wider applicability.
Appendix
The appendix develops conditional generating-function and success-probability results for Poisson cluster processes. It treats fixed cluster sizes, Rayleigh fading, and Nakagami-m fading, including consistency with the Rayleigh special case.
- Appendix: The appendix derives asymptotic behavior of the conditional generating functional using the representative-cluster distribution.
- Appendix: The conditional generating-functional analysis uses product densities and the distance distribution between two independently scattered cluster points.
- Appendix: For fixed cluster size c̄ ∈ N, points are independently distributed with density f(x), yielding a conditional generating-functional formulation.
- Appendix: Under Rayleigh fading, the conditional Laplace-transform framework yields the success probability for clustered networks.
- Appendix: For integer Nakagami parameter m ≥ 1, success probability can be evaluated from the derived expressions.
- Appendix: When m = 1, the Nakagami-based probability matches the result obtained for Rayleigh fading.
D. Proof of Lemma 7
The proof establishes bounds relating clustered-network success probability and transmission capacity to the homogeneous Poisson case. It uses inequalities, asymptotic bounds, and a contradiction argument to show equality of transmission capacities.
- D. Proof of Lemma 7: The proof bounds the clustering term η(c̄,R) using Young’s inequality and shows its small-R behavior is proportional to R^2.
- D. Proof of Lemma 7: The proof controls the relevant interference terms using independent points with density f and the convolution density f * f.
- D. Proof of Lemma 7: For small link distance R, clustered-network success probability is bounded above by the corresponding Poisson-network probability.
- D. Proof of Lemma 7: A contradiction argument shows that the supremum clustered transmission capacity cannot exceed the Poisson transmission capacity.
- D. Proof of Lemma 7: The proof restricts the cluster-size parameter to be finite because success probability tends to zero as c̄ approaches infinity.
- D. Proof of Lemma 7: The Poisson transmission-capacity value is achieved as the cluster parameter approaches the dispersed limiting construction.