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Spatial and Temporal Correlation of the Interference in ALOHA Ad Hoc Networks
Radha Krishna Ganti, Martin Haenggi
TL;DR
Interference correlations in wireless ad hoc networks matter for retransmissions and routing, but are often neglected analytically. This paper models nodes as a planar PPP using ALOHA, derives spatial and temporal interference correlations, and proves that link outages are temporally correlated. These results should inform ad hoc performance analysis and retransmission-strategy design.
Problem
Temporal and spatial interference correlations affect retransmissions and routing, yet are generally neglected for analytical tractability.
Method
The paper models a planar PPP network with ALOHA and derives spatial and temporal correlations of interference and link outages.
Results
The paper derives spatial and temporal interference correlations and proves that link outages are temporally correlated.
Takeaways & Limitations
Interference and outage correlations should be considered when analyzing ad hoc performance and designing retransmission strategies.
Abstract
from arXiv · showhide
Interference is a main limiting factor of the performance of a wireless ad hoc network. The temporal and the spatial correlation of the interference makes the outages correlated temporally (important for retransmissions) and spatially correlated (important for routing). In this letter we quantify the temporal and spatial correlation of the interference in a wireless ad hoc network whose nodes are distributed as a Poisson point process on the plane when ALOHA is used as the multiple-access scheme.
I. INTRODUCTION
Interference correlations arise from common randomness in transmitter locations and affect retransmission strategies and routing. The paper quantifies spatial and temporal correlations of interference and link outages in an ALOHA ad hoc network.
- Common randomness in interferer locations induces temporal and spatial interference correlations even when ALOHA is used.
- These correlations affect retransmission strategies and routing.
- Analytical work generally neglects these correlations for tractability and because they do not change ad hoc network scaling behavior.
- Extending transmission capacity from single-hop to multi-hop scenarios requires accounting for spatio-temporal correlations.
- The letter quantifies spatial and temporal correlations of interference and link outages.
II. SYSTEM MODEL
The system models radios as a planar PPP with unit-power transmissions and independent unit-mean fading, while ALOHA independently activates each node with probability p per slot. Interference is defined from the transmitting sets, fading, and path loss.
- Radio locations are modeled as a Poisson point process on R2 with density λ.
- All nodes transmit with unit power, and fading is spatially and temporally independent with unit mean.
- The fading coefficient between nodes x and y at time n is denoted by hxy(n).
- The model uses a large-scale path-loss function g(x), including a singular model obtained as the limit limε→0 gε(x).
- At each time, interference at location z is formed from the transmitting set, fading, and path loss.
- ALOHA lets each node decide independently whether to transmit with probability p in each slot.
III. SPATIO-TEMPORAL CORRELATION OF INTERFERENCE
The paper derives spatial and temporal interference correlation for a Poisson ad hoc network using ALOHA, including fading and regularized path-loss models. It shows that common transmitter-location randomness creates temporal dependence, while spatial dependence decreases with receiver separation.
- Temporal correlation: ALOHA selects transmitting sets from a common node-location randomness across time, producing temporally correlated interference.Nodes transmit independently in ALOHA, but the underlying transmitter locations are shared across slots.
- Path-loss model: The variance of interference under the regularized path-loss model gε(x) = 1/(ε + ∥x∥^α) is obtained from the first- and second-moment expressions.The regularization is used because the singular model’s integral of g²(x) is not defined.
- General correlation result: The spatio-temporal correlation coefficient is derived for interference observed at locations u and v in different time slots.The derivation uses Campbell’s theorem and the second-order product density of a Poisson point process.
- Temporal correlation: For Nakagami-m fading, the temporal correlation coefficient is ζt = p/m, increasing with m because stronger fading reduces correlation.The reported special cases are p/2 for m = 2 and p as m →∞ without fading.
- Path-loss model: The zero spatial correlation obtained with singular path loss is identified as an artifact of that model.The paper examines the singular case as the limit of the regularized path-loss model and plots spatial correlation versus ∥u −v∥ for different ε.
- Spatial correlation: Spatial correlation tends to zero for separated observation points because nearby transmitters dominate each location and Poisson transmitter locations are independent in disjoint neighborhoods.The paper notes that mutual information may better capture dependence under the singular path-loss model.
IV. TEMPORAL CORRELATION OF LINK OUTAGES
The analysis shows that link formation and outages are correlated across time under ALOHA, so retransmissions should account for prior transmission outcomes rather than proceed blindly. The correlation increases with the ALOHA parameter, transmitter density, and threshold.
- Figure 2 compares conditional P(A_k|A_l) and unconditional P(A_l) link-success probabilities as the ALOHA parameter p varies.The plotted setting is λ = 1, g(x) = ∥x∥^-4, z = 0.5, and θ = 1.
- Link formation is correlated across time, contrary to the standard assumption that link failures are temporally independent.
- A successful transmission makes a later success more likely, while an outage makes a later outage more likely.
- Retransmission strategies should reduce the transmission rate or transmitter density instead of retransmitting blindly.
- The unconditional link-success probability increases with θ, λ, and p, while larger λ and p also increase temporal correlation through repeated transmitting subsets.
V. CONCLUSIONS
The paper derives spatial and temporal interference correlations in an ALOHA wireless network and proves that link outages are temporally correlated. These correlations should be considered in ad hoc performance analysis and retransmission-strategy design.
- The paper derives the spatial and temporal correlations of interference in an ALOHA wireless network.
- Link outages are temporally correlated and should be considered when analyzing ad hoc performance.
- Interference and outage correlations should inform retransmission-strategy design.