# Continuity and monotonicity of solutions to a greedy maximization problem

## TL;DR

This paper studies the properties of solutions to a greedy maximization problem related to resource sharing networks, focusing on their continuity and monotonicity over time, which impacts fluid network models.

## Contribution

It provides new insights into the continuity and monotonicity of solutions to a network-based greedy maximization problem, considering the network topology.

## Key findings

- Solutions are continuous with respect to time under certain conditions.
- Monotonicity of solutions depends on network topology and constraints.
- Results inform the design of optimal fluid network models.

## Abstract

Motivated by an application to resource sharing network modelling, we consider a problem of greedy maximization (i.e., maximization of the consecutive minima) of a vector in $R^n$, with the admissible set indexed by the time parameter. The structure of the constraints depends on the underlying network topology. We investigate continuity and monotonicity of the resulting maximizers with respect to time. Our results have important consequences for fluid models of the corresponding networks which are optimal, in the appropriate sense, with respect to handling real-time transmission requests.