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

\[T(s)[X_1 - Ke^{-s\tau}Q_1+Ke^{-s\tau}Q_2] = Q_1 \\ T(s)[X_1 - Ke^{-s\tau}Q_2+Ke^{-s\tau}Q_1] = Q_2\]

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

   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",