S2_G5_Network

Project on Coverage probability in wireless networks with determinantal scheduling

Stochastic Optimization of Spatial Throughput in Wireless Networks

Course: CSE 400: Fundamentals of Probability in Computing
Group: 5 Team Members: Yagnik, Prakshal, Devshree, Hiranshee, Dhruv

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📌 Project Overview

In wireless network design, there is a fundamental trade-off between Reliability (probability of successful transmission) and Capacity (network throughput).

• Standard Aloha protocols schedule transmissions randomly (Poisson Point Process), leading to clustering, severe interference, and network crashes at high densities.
• Conservative scheduling strictly spaces out users, guaranteeing high reliability but resulting in very low overall data throughput.

Our Objective: We aim to mathematically find the "sweet spot" by maximizing the Spatial Throughput ($T$) function:
T(λ) = λ × P_cov(λ)
(Where λ is the density of active transmitters and P_cov is the probability of coverage/success).

We achieve this by using Determinantal Point Processes (DPP) to introduce probabilistic "Negative Dependence" (geometric repulsion) among active nodes, inherently reducing interference and allowing for a higher optimal network density.

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🚀 Milestone 1: Problem Formulation

In M1, we established the system context and identified the primary sources of uncertainty in a wireless environment:

1. Physical Uncertainty: Signal strength fluctuates unpredictably due to multipath propagation (Rayleigh Fading).
2. Geometric Uncertainty: The exact locations of active transmitters are random.
3. Interaction Uncertainty: Because locations and fading are random, the Aggregate Interference ($I$) acting on any receiver is a highly volatile stochastic process.

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📊 Milestone 2: Mathematical Modeling

In M2, we deconstructed the network into its foundational Random Variables (RVs) and simulated them to build our probabilistic intuition.

1. The Physics (Channel Uncertainty)

• Random Variable: Fading Coefficient ($H$)
• Model: Exponential Distribution (μ = 1)
• Rationale: Models Rayleigh Fading in a non-line-of-sight environment. Our simulations confirmed that "deep fades" (signal power near zero) are highly probable, necessitating a robust scheduling algorithm.

2. The Geometry (Node Configuration)

• Random Variable: Spatial Point Process ($\Phi$)
• Baseline Model: Poisson Point Process (PPP) - Independent, prone to clustering.
• Proposed Model: Determinantal Point Process (DPP) - Negatively dependent.
• Rationale: We utilize a Kernel Matrix to mathematically penalize close proximity. Our visual simulations prove that DPP enforces "social distancing" among nodes, structuring the randomness.

3. The Interaction (Stochastic Interference)

• Random Variable: Aggregate Interference ($I$)
• Model: Stochastic Shot Noise Process (I = Σ P_i * H_i * r_i^(-α))
• Rationale: By plotting the Cumulative Distribution Function (CDF) of $I$, we mathematically proved that DPP geometry significantly reduces the variance and extreme upper bounds of network noise compared to Aloha.

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💻 Codebase & Execution (Milestone 2 Artifacts)

The repository contains the base simulation code provided by B. Błaszczyszyn and H.P. Keeler. For Milestone 2, we extracted specific logic to generate our diagnostic probability plots.

Generating the M2 Plots

We have created a script to visualize the Random Variables defined in Milestone 2. Run the following Python script to generate the artifacts:

python milestone2_plots.py