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Computing the Saturation Throughput for Heterogeneous p-CSMA in a General Wireless Network

Faezeh Dehghan Tarzjani, Bhaskar Krishnamachari

TL;DR

This work addresses the problem of exactly computing the saturation throughput $S_i$ for heterogeneous $p_i$-CSMA in wireless networks with arbitrary conflict graphs, where renewal-theory based estimates fail for non-complete topologies. It introduces a global Markov-chain model that tracks the backoff state via $S(t)=(a_0(t),\dots,a_{n-1}(t))$ and yields exact per-node throughputs through the stationary distribution $\pi(s)$. A closed-form solution for the special case $T=2$ provides a compact consistency check, while a general algorithm computes the transition matrix and $S_i$ for any $T$. The paper also shows how to optimize weighted sums of utility functions of the saturation throughputs using gradient ascent, enabling topology-aware tuning of transmit probabilities to improve overall performance. These results offer a rigorous, exact framework for analyzing and optimizing CSMA in realistic networks where interference is governed by a general conflict graph, with potential applicability to IoT mesh deployments and beyond.

Abstract

A well-known expression for the saturation throughput of heterogeneous transmitting nodes in a wireless network using p-CSMA, derived from Renewal Theory, implicitly assumes that all transmitting nodes are in range of, and therefore conflicting with, each other. This expression, as well as simple modifications of it, does not correctly capture the saturation throughput values when an arbitrary topology is specified for the conflict graph between transmitting links. For example, we show numerically that calculations based on renewal theory can underestimate throughput by 48-62% for large packet sizes when the conflict graph is represented by a star topology. This is problematic because real-world wireless networks, such as wireless IoT mesh networks, are often deployed over a large area, resulting in non-complete conflict graphs. To address this gap, we present a computational approach based on a novel Markov chain formulation that yields the exact saturation throughput for each node in the general network case for any given set of access probabilities, as well as a more compact expression for the special case where the packet length is twice the slot length. Using our approach, we show how the transmit probabilities could be optimized to maximize weighted utility functions of the saturation throughput values. This would allow a wireless system designer to set transmit probabilities to achieve desired throughput trade-offs in any given deployment.

Computing the Saturation Throughput for Heterogeneous p-CSMA in a General Wireless Network

TL;DR

This work addresses the problem of exactly computing the saturation throughput for heterogeneous -CSMA in wireless networks with arbitrary conflict graphs, where renewal-theory based estimates fail for non-complete topologies. It introduces a global Markov-chain model that tracks the backoff state via and yields exact per-node throughputs through the stationary distribution . A closed-form solution for the special case provides a compact consistency check, while a general algorithm computes the transition matrix and for any . The paper also shows how to optimize weighted sums of utility functions of the saturation throughputs using gradient ascent, enabling topology-aware tuning of transmit probabilities to improve overall performance. These results offer a rigorous, exact framework for analyzing and optimizing CSMA in realistic networks where interference is governed by a general conflict graph, with potential applicability to IoT mesh deployments and beyond.

Abstract

A well-known expression for the saturation throughput of heterogeneous transmitting nodes in a wireless network using p-CSMA, derived from Renewal Theory, implicitly assumes that all transmitting nodes are in range of, and therefore conflicting with, each other. This expression, as well as simple modifications of it, does not correctly capture the saturation throughput values when an arbitrary topology is specified for the conflict graph between transmitting links. For example, we show numerically that calculations based on renewal theory can underestimate throughput by 48-62% for large packet sizes when the conflict graph is represented by a star topology. This is problematic because real-world wireless networks, such as wireless IoT mesh networks, are often deployed over a large area, resulting in non-complete conflict graphs. To address this gap, we present a computational approach based on a novel Markov chain formulation that yields the exact saturation throughput for each node in the general network case for any given set of access probabilities, as well as a more compact expression for the special case where the packet length is twice the slot length. Using our approach, we show how the transmit probabilities could be optimized to maximize weighted utility functions of the saturation throughput values. This would allow a wireless system designer to set transmit probabilities to achieve desired throughput trade-offs in any given deployment.

Paper Structure

This paper contains 8 sections, 2 theorems, 25 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Problem Definition
  4. Renewal Theory Expressions and Shortcomings
  5. Proposed Exact Markov Chain-based Approach
  6. Application to Network Utility Maximization
  7. Numerical Results
  8. Conclusions

Key Result

Lemma 1

Let $X_0, U_1, U_2, \ldots$ be independent discrete random variables. Consider the process $\{X_n\}_{n \geq 0}$ defined by: where each $h_n$ is a deterministic function. Then $\{X_n\}_{n \geq 0}$ is a Markov chain.

Figures (7)

  • Figure 1: 3-Node Non-Complete Topology.
  • Figure 2: Idle, Successful Transmission, and Collision outcomes in a p-persistent CSMA time slot for T=2.
  • Figure 3: State Transition Diagram of the Markov Chain for the 3-Node Non-Complete Graph with $T = 2$ and $\sigma = 1$
  • Figure 4: Illustrating convergence of Utility optimization for the topology in figure \ref{['fig:3nodes_non_com']}
  • Figure 5: Theoretical Models and simulation differences across edge probabilities
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Theorem 1
  • proof