Transports Regulators of Networks with Junctions Detected by Durations Functions

Abstract

This study advocates a mathematical framework of ''transport relations'' on a network. They single out a subset of ''traffic states'' described by time, duration, position and other traffic attributes (called ''monads'' for short). Duration evolutions are non-negative, decreasing toward zero for incoming durations, increasing from zero for outgoing durations, allowing the detection of ''junction states'' defined as traffic states with ''zero duration''. A ''junction relation'' (crossroads, synapses, clearing houses, etc.) Is a subset of the transport relation made of junction states? The objective is to construct a ''transport regulator'' associating with traffic states a set of ''celerities'' that mobiles circulating in the network can use as velocities. In other word, a network is regarded as a ''provider of velocity information'' to the mobiles for travelling from one departure state to an arrival state across a junction relation (a kind of geodesic problem). This investigation assumes that a system governs the evolution of monads in function of time, duration and position using celerities as controls and provides the transport regulator, a feedback from transport states to celerities. The proposed mathematical framework can acclimate road or aerial networks, endocrine (hormonal) or synaptic (neurotransmitters) networks, financial or economic networks, which motivated this investigation. This framework could probably accommodate computer and even social networks. This investigation is restricted to junctions between two routes.