Wireless Network Scheduling with Discrete Propagation Delays: Theorems and Algorithms

TL;DR

The dominance property of a special scheduling graph called the step- $T$ scheduling graph is investigated, which allows the utilization of specific subgraphs of the step- $T$ scheduling graph to characterize the scheduling rate region, achieving a reduction in both the number of cycles and their lengths.

Abstract

This paper focuses on the link scheduling problem in networks where signal delays between nodes are multiples of a time interval. To model such networks, a directed hypergraph is employed, along with an integer matrix that specifies the delays. The link scheduling problem is closely connected to the independent sets of the periodic hypergraph induced by the network model. However, due to the infinite number of vertices, it is impractical to enumerate the independent sets of the periodic hypergraph using generic graph algorithms. To tackle this challenge, a graphical approach is proposed in this paper. The link scheduling rate region is characterized using a finite directed graph called a scheduling graph, which is derived from the network model. A collision-free schedule of the network corresponds to a path in the scheduling graph, and the rate region is determined by the convex hull of the rate vectors associated with the cycles in the scheduling graph. Although existing cycle enumeration algorithms can be employed to calculate the rate region, their computational complexity becomes prohibitively high as the size of the scheduling graph grows exponentially with the number of network links. To address this issue, the dominance property of a special scheduling graph called the step-T scheduling graph is investigated. This property allows the utilization of specific subgraphs of the step-T scheduling graph to characterize the scheduling rate region, achieving a reduction in both the number of cycles and their lengths. For common problems such as calculating the rate region and maximizing a weighted sum of the scheduling rates, algorithms leveraging the dominance property are developed. These algorithms can be more efficient than using generic graph algorithms directly on the scheduling graphs.

Figures (6)

• Figure 1: The graphical representation of $\mathcal{N}_{4,1}^{\text{line}}$ and $\mathcal{N}_{4,2}^{\text{line}}$. In these graphs, as well as the following graphical representation of our discrete network models, the vertices in a graph represent links in the network. The number on an edge $(l,l')$ is the value of $D_{\mathcal{L}}(l,l')$.
• Figure 2: Illustration of the periodic graphs. (a) is the periodic graph induced by $\mathcal{N}_{4,1}^{\text{line}}$. (b) is the periodic graph induced by a network that shares the same link set and collision profile as $\mathcal{N}_{4,1}^{\text{line}}$, but has $\mathbf{0}$ as the delay matrix.
• Figure 3: An periodic graph with three components. Each component is illustrated by a different color (gray scale).
• Figure 4: Illustration of the associated part in the periodic graph of $S[T,Q,k]$.
• Figure 5: Illustration of the proof of Theorem \ref{['thm:cf-3']}. A thick tick indicts the start position of a submatrix $S[T,Q,k]$, and a thin tick indicts the time.

Theorems & Definitions (44)

• Example 1: Uniform line networks
• Example 2
• Example 3: Line network with the $K$-hop collision model
• Example 4
• Definition 1: Collision-free schedule
• Definition 2: Periodic hypergraph
• Theorem 1
• Definition 3: Scheduling rate region
• Example 5
• Definition 4: Periodic schedule