# No Switching Policy is Optimal for a Positive Linear System with a Bottleneck Entrance

## TL;DR

This paper proves that for a specific class of nonlinear systems with a bottleneck, maintaining a constant inflow rate is optimal for maximizing throughput, outperforming any switching strategies.

## Contribution

The study demonstrates that no switching policy can outperform a constant inflow in maximizing throughput in positive linear systems with bottleneck entrances.

## Key findings

- Constant inflow achieves maximum throughput asymptotically.
- Switching strategies cannot outperform constant inflow.
- Applicable to traffic, ribosome flow, and security scheduling.

## Abstract

We consider a nonlinear SISO system that is a cascade of a scalar "bottleneck entrance" and an arbitrary Hurwitz positive linear system. This system entrains i.e. in response to a $T$-periodic inflow every solution converges to a unique $T$-periodic solution of the system. We study the problem of maximizing the averaged throughput via controlled switching. The objective is to choose a periodic inflow rate with a given mean value that maximizes the averaged outflow rate of the system. We compare two strategies: 1) switching between a high and low value, and 2) using a constant inflow equal to the prescribed mean value. We show that no switching policy can outperform a constant inflow rate, though it can approach it asymptotically. We describe several potential applications of this problem in traffic systems, ribosome flow models, and scheduling at security checks.