An agent-based decentralized threshold policy finding the constrained shortest paths

TL;DR

This work considers a problem where autonomous agents enter a dynamic and unknown environment described by a network of weighted arcs, and shows that this policy ensures path-length optimality in the sense that, in a finite time, all new agents entering the network reach the closer sinks by the shortest paths.

Abstract

We consider a problem where autonomous agents enter a dynamic and unknown environment described by a network of weighted arcs. These agents move within the network from node to node according to a decentralized policy using only local information, with the goal of finding a path to an unknown sink node to leave the network. This policy makes each agent move to some adjacent node or stop at the current node. The transition along an arc is allowed or denied based on a threshold mechanism that takes into account the number of agents already accumulated in the arc's end nodes and the arc's weight. We show that this policy ensures path-length optimality in the sense that, in a finite time, all new agents entering the network reach the closer sinks by the shortest paths. Our approach is later extended to support constraints on the paths that agents can follow.

Figures (8)

• Figure 1: The mechanism with threshold value $\gamma=3$. The transition between two nodes is controlled by a threshold mechanism: no transition is possible if the difference between the number of tokens in the buffers of the two nodes is not above a threshold equal to the arc cost (left). A token can pass only when its arrival breaks such a threshold (right).
• Figure 2: A simple network $\mathcal{G}$. $\gamma_{ij}$ and $\sigma_{ij}$ are shown for each arc $(i,j)$.
• Figure 3: The expanded network $\mathcal{G}_E$ corresponding to network $\mathcal{G}$ in Fig. \ref{['fig:incidenceMatrixNetwork']}, when $C_{max}=2$. The value $\gamma_{ij}$ is indicated for each arc $(i,j)$.
• Figure 4: (Example 1). Time evolution of the states of the nodes. Dotted line: $\bar{\bar{x}}_1(k)$ for the source of the unconstrained system.
• Figure 5: (Example 2). The map of the network. Red circles: source nodes; blue circles: sink nodes, colored pixels: nodes of the network; color of each pixel: the corresponding value of $h_i$, $\bar{\bar{x}}_i$ or $\bar{x}_i(k)$, respectively, with gray representing 0. In Fig. \ref{['fig:sim2:map:val']}, there are hills in the yellow areas, and the altitude decreases as the color becomes bluer: arc costs $\gamma_{ij}$ are negative for the arcs directed downhill, and non-negative otherwise. In Figs. \ref{['fig:sim2:map:barx']}-\ref{['fig:sim2:map:xcon']}, there is a higher accumulation of tokens in the yellow areas; the number of tokens is below the zero reference level (negative states) in the violet areas. In Fig. \ref{['fig:sim2:map:barx']}, by the definition of $\bar{\bar{x}}_i$, colors also represent the length of the shortest path from each node to the closer sink. In Figs. \ref{['fig:sim2:map:xuncdet']}-\ref{['fig:sim2:map:xcon']}, the lines represent the paths traveled by $10000$ injected tokens; from white thin lines, when few tokens traversed each arc, to thicker black lines, when almost all the tokens traversed them.

Theorems & Definitions (21)

• Example 1
• Definition 1: Admissibility
• Lemma 1
• Remark 1
• Remark 2
• Definition 2
• Lemma 2: Boundedness of the admissibility states
• Theorem 1: Well posedness and positive invariance
• Definition 3
• Theorem 2