A Min-Plus Algebra System Theory for Traffic Networks

In this article, we introduce a comprehensive system theory based on the min-plus algebra of 2x2 matrices of functions. This novel approach enables the algebraic construction of traffic networks and the analytical derivation of performance bounds for such networks. We use the term "traffic networks" or "congestion networks" to refer to networks where high densities of transported particles lead to flow drops, as commonly observed in road networks. Initially, we present a model for a segment or section of a link within the network and demonstrate that the dynamics can be expressed linearly within the min-plus algebra. Subsequently, we formulate the linear system using the min-plus algebra of 2x2 matrices of functions. By deriving the impulse response of the system, we establish its interpretation as a service guarantee, considering the traffic system as a server. Furthermore, we define a concatenation operator that allows for the combination of two segment systems, demonstrating that multiple segments can be algebraically linked to form a larger network. We also introduce a feedback operator within this system theory, enabling the modeling of closed systems. Lastly, we extend this theoretical framework to encompass two-dimensional systems, where nodes within the network are also taken into account in addition to the links. We present a model for a controlled node and provide insights into other potential two-dimensional models, along with directions for further extensions and research.