# Impact of queue feedback on the stability and dynamics of a Rate Control Protocol (RCP) with two delays

## TL;DR

This paper analyzes the stability of the Rate Control Protocol (RCP) with two delays, showing that queue size feedback can destabilize the system and suggesting that feedback based only on rate mismatch is preferable.

## Contribution

The study provides a detailed bifurcation analysis of RCP with two delays, revealing that queue feedback can induce sub-critical Hopf bifurcations, which are undesirable for system stability.

## Key findings

- Queue feedback can destabilize RCP.
- RCP without queue feedback undergoes super-critical Hopf bifurcation.
- Queue feedback may cause large or unstable limit cycles.

## Abstract

Rate Control Protocol (RCP) uses feedback from routers to assign flows their fair rate. RCP estimates the fair rate using two forms of feedback: rate mismatch and queue size. An outstanding design question for RCP is whether the queue size feedback is useful or not. To address this, we analyze stability and the bifurcation properties of RCP in both the cases i.e., with and without queue size feedback. The model considers flows with two different round-trip times, operating over a single bottleneck link. By using an exogenous bifurcation parameter, we show that the system loses stability via a Hopf bifurcation and hence we can expect a limit cycle branching from the fixed point. We highlight that the presence of queue feedback can readily destabilize the system. Using Poincar{\`e} normal forms and the center manifold theorem, we show that the Hopf bifurcation is super-critical in the case of RCP without queue feedback. Whereas, in the presence of queue feedback, we show that the system can undergoes a sub-critical Hopf bifurcation for some parameter values. A sub-critical Hopf bifurcation can result in either large amplitude limit cycles or unstable limit cycles, and hence should be avoided in engineering applications. Thus, the presence of queue feedback would create adverse effects on the stability of the emerging limit cycles. In essence, the analytical results of RCP with two delays favor the design choice that uses feedback based only on rate mismatch. The theoretical analysis is validated with numerical computations and some packet level simulations as well.