Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Convolutional Coded Poisson Receivers

Cheng-En Lee, Kuo-Yu Liao, Hsiao-Wen Yu, Ruhui Zhang, Cheng-Shang Chang, Duan-Shin Lee

TL;DR

This work introduces convolutional coded Poisson receivers (CCPRs) by spatially coupling coded Poisson receivers (CPRs) to expand stability regions under multi-class traffic. It derives outer bounds for the stability region via affine capacity envelopes of φ-ALOHA receivers and proves a threshold-saturation phenomenon whereby the stability region grows with the coupling length, converging to a limit as L→∞. For single-class CCPRs, a potential-function framework yields three thresholds, Gs*, Gconv*, and Gup*, with Gs*<Gconv*<Gup*; a saturation theorem shows stability for G<Gconv* under suitable window size. Numerical results corroborate the theory, showing significant stability-region enlargement with larger W and L and near-tightness of the bounds for large windows, including multi-class IRSA scenarios. Overall, the CCPR framework provides a principled approach to enhancing multi-class access by leveraging spatial coupling and density-evolution analysis, with implications for network slicing and uplink management in 5G and beyond.

Abstract

In this paper, we present a framework for convolutional coded Poisson receivers (CCPRs) that incorporates spatially coupled methods into the architecture of coded Poisson receivers (CPRs). We use density evolution equations to track the packet decoding process with the successive interference cancellation (SIC) technique. We derive outer bounds for the stability region of CPRs when the underlying channel can be modeled by a $φ$-ALOHA receiver. The stability region is the set of loads that every packet can be successfully received with a probability of 1. Our outer bounds extend those of the spatially-coupled Irregular Repetition Slotted ALOHA (IRSA) protocol and apply to channel models with multiple traffic classes. For CCPRs with a single class of users, the stability region is reduced to an interval. Therefore, it can be characterized by a percolation threshold. We study the potential threshold by the potential function of the base CPR used for constructing a CCPR. In addition, we prove that the CCPR is stable under a technical condition for the window size. For the multiclass scenario, we recursively evaluate the density evolution equations to determine the boundaries of the stability region. Numerical results demonstrate that the stability region of CCPRs can be enlarged compared to that of CPRs by leveraging the spatially-coupled method. Moreover, the stability region of CCPRs is close to our outer bounds when the window size is large.

Convolutional Coded Poisson Receivers

TL;DR

This work introduces convolutional coded Poisson receivers (CCPRs) by spatially coupling coded Poisson receivers (CPRs) to expand stability regions under multi-class traffic. It derives outer bounds for the stability region via affine capacity envelopes of φ-ALOHA receivers and proves a threshold-saturation phenomenon whereby the stability region grows with the coupling length, converging to a limit as L→∞. For single-class CCPRs, a potential-function framework yields three thresholds, Gs*, Gconv*, and Gup*, with Gs*<Gconv*<Gup*; a saturation theorem shows stability for G<Gconv* under suitable window size. Numerical results corroborate the theory, showing significant stability-region enlargement with larger W and L and near-tightness of the bounds for large windows, including multi-class IRSA scenarios. Overall, the CCPR framework provides a principled approach to enhancing multi-class access by leveraging spatial coupling and density-evolution analysis, with implications for network slicing and uplink management in 5G and beyond.

Abstract

In this paper, we present a framework for convolutional coded Poisson receivers (CCPRs) that incorporates spatially coupled methods into the architecture of coded Poisson receivers (CPRs). We use density evolution equations to track the packet decoding process with the successive interference cancellation (SIC) technique. We derive outer bounds for the stability region of CPRs when the underlying channel can be modeled by a -ALOHA receiver. The stability region is the set of loads that every packet can be successfully received with a probability of 1. Our outer bounds extend those of the spatially-coupled Irregular Repetition Slotted ALOHA (IRSA) protocol and apply to channel models with multiple traffic classes. For CCPRs with a single class of users, the stability region is reduced to an interval. Therefore, it can be characterized by a percolation threshold. We study the potential threshold by the potential function of the base CPR used for constructing a CCPR. In addition, we prove that the CCPR is stable under a technical condition for the window size. For the multiclass scenario, we recursively evaluate the density evolution equations to determine the boundaries of the stability region. Numerical results demonstrate that the stability region of CCPRs can be enlarged compared to that of CPRs by leveraging the spatially-coupled method. Moreover, the stability region of CCPRs is close to our outer bounds when the window size is large.
Paper Structure (21 sections, 20 theorems, 188 equations, 3 figures, 3 tables)

This paper contains 21 sections, 20 theorems, 188 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Review of the framework of coded Poisson receivers
  3. Poisson receivers and ALOHA receivers
  4. Coded Poisson receivers with multiple classes of users and receivers
  5. Stability
  6. Outer bounds for the stability region
  7. Convolutional coded Poisson receivers
  8. Circular convolutional coded Poisson receivers
  9. Stability and threshold saturation
  10. Percolation thresholds of systems with a single class of users
  11. The single-system threshold
  12. The saturation theorem
  13. The upper bound by the area theorem
  14. Numerical results
  15. Convolutional coded Poisson receivers with one class of users and one class of receivers
  16. ...and 6 more sections

Key Result

Theorem 3

(Theorem 1 of chang2020Poisson) As $T \to \infty$, the system of CPRs after the $i$-th SIC iteration converges to a Poisson receiver with the success probability function for the class $k$ traffic $k=1,2, \ldots, K$, where $q^{(i)}=(q_{1}^{(i)}, q_{2}^{(i)}, \ldots, q_{K}^{(i)})$ can be computed recursively from the following equation: with $q^{(0)}=(1, 1, \ldots, 1)$.

Figures (3)

  • Figure 1: The plot of the potential function $U(p;G)$ as a function of $p$ for various values of $G$ in Example \ref{['ex:IRSA_regular']} with $d=3$.
  • Figure 2: The boundaries of the stability region of the convolutional IRSA system with two classes of users and two classes of receivers under the complete sharing policy. The gray area presents the outer bound (\ref{['eq:IRSA2class2222']}) for the complete sharing policy.
  • Figure 3: The boundaries of the stability region of the convolutional IRSA system with two classes of users and two classes of receivers under the receiver reservation policy. The gray area is the intersection of the outer bounds (\ref{['eq:IRSA2clas5555']}) and (\ref{['eq:IRSA2class9999']}) for the receiver reservation policy.

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Definition 7
  • Definition 8
  • Theorem 9
  • Lemma 10
  • ...and 29 more