Prediction of Period-1 oscillations of the state-dependent delayed compound TCP model with PIE queue management policy via HDHBM

State-dependent time-delayed systems are more realistic delayed systems in nature and human society. Their dynamic problems are hot issues and difficult questions due to the fact that the classical RFDE theory is not suitable for them any more. And its corresponding rigorous theory is to be established. The study on the existence and stability of their periodic solutions is an important topic and a difficult question. Compound TCP is a widely used transport control protocol, and proportional integral enhanced (PIE) queue policy is an effective and simple queue policy in the Internet. Thus it is significant to study the dynamics and control of the compound TCP-PIE system. In this paper, Period-1 oscillations of the state-dependent delayed TCP models with PIE queue management policy are predicted by the high-dimensional harmonic balance method (HDHBM). Compared with the traditional harmonic balance method, HDHBM avoids the complex integration of the Fourier coefficients of the nonlinear right hand of the systems by solving the time domain variables of the Fourier coefficients at equally spaced sub-time over the period of one cycle of motion. Thus it transforms the problem into a relatively simple nonlinear algebraic equations, which greatly reduces the computation. By HDHBM, the approximate expressions of the Period-1 solutions for this state-dependent delayed compound TCP-PIE system are obtained. We compare their amplitudes and phase portraits with those of the DDE-BIFTOOL. They agree well with each other. It suggests the validity, effectiveness and simplicity of the employed method. It also turns out that many parameters of the state-dependent delayed TCP model with PIE queue management policy play a very important role in its dynamic behaviours. It follows that PIE parameter μ and update time T may play the most important roles in the Compound TCP-PIE congestion control model and packet loss probability p(t˜) may dominate the dynamics and performances of the network. As a result, we should control the parameters in a proper range to achieve our desired network performances.