Analyze a rate controller interacting with a bottleneck of capacity C by means of a fluid approximation. The rate controller changes the emitted packet rate λ(t), aiming at a target rate (t).Let...

Analyze a rate controller interacting with a bottleneck of capacity C by means of a fluid approximation. The rate controller changes the emitted packet rate λ(t), aiming at a target rate
(t).Let Q(t)denote the content of the bottleneck buffer at time t. Let
 be the feedback delay. The evolution equation of λ (t) is

The target rate
(t)is set to the current rate whenever the current rate is decreased because of negative feedback, and it increases at a constant rate:

With
(0) = λ(0) = r Then p(t) = I (Q(t) > K) for an assigned threshold K>0.

(a) Write the evolution equation of Q(t).

(b) Study the stability of the control eras a function of
 assuming C = 1,r = 4, R = 0.5, and K = 20. For this purpose, assume that the feedback delay is equal to the quantization time step when discretizing the differential equations for numerical integration.