This paper modelled traffic as a time-delayed feedback control system.

The paper assumed a simple model of two networking devices, whose input is $x_1$ and $x_2$ respectively. The input will influent the queue length $q_1$ and $q_2$. Their relationship is modeled as a feedback system: Queue length variable $q_1$ and $q_2$ increases the input $x_2$ and $x_1$ respectively (with time delay $\tau$) because when a queue is too long, rerouting will be triggered. Then the system can be modelled as

The block diagram is presented in the paper.

The paper then set $x_1$ to zero and eliminate $Q_2$ in the above equation, then it deduced the notation of $Q_1/X_2$ and deduced that the rerouting network is stable iff $Ke^{-s\tau}T(s) \ne -\tfrac{1}{2}$. By setting $T(s)$ to be $1/s$, i.e. assume queue length is the integration, it found the bound for $K$ (amplifying factor of the feedback signal) to stabilize the network.

## Bibliographic data

@article{
title = "Stability of Traffic Patterns in Broadband Networks",
author = "R. M. Goodman and B. E. Ambrose",
journal = "Journal of Network and Systems Management, Special Issue on Routing in Broadband Networks",
howpublished = "J Network Sys Management",
volume = "3",
number = "4",
pages = "371--380",
month = "December",
year = "1995",
}