Problem 64 in Chapter 7 introduced the model of a TCP/IP router whose packet-drop probability is controlled by using a random early detection (RED) algorithm (Hollot, 2001). Using Figure P8.3 as a model, a specific router queue’s open-loop transfer function is
The function e -0.2s represents delay. To apply the root locus method, the delay function must be replaced with a rational function approximation. A first-order Padé approximation can be used for this purpose.
Let e-sD ≈ 1 – Ds. Using this approximation, plot the root locus of the system as a function of L.
Data from Problem 64 in Chapter 7:
Packet information flow in a router working under TCP/IP can be modeled using the linearized transfer function
where
C = link capacity (packets/second)
N = load factor (number of TCP sessions)
Q= expected queue length
R = round trip time (second)
p =probability of a packet drop
The objective of an active queue management (AQM) algorithm is to automatically choose a packet-drop probability, p, so that the queue length is maintained at a desired level. This system can be represented by the block diagram of Figure P7.13 with the plant model in the P(s) block, the AQM algorithm in the G(s) block, and F(s) = H(s) = 1. Several AQM algorithms are available, but one that has received special attention in the literature is the random early detection (RED) algorithm. This algorithm can be approximated with G(s) = LK/s + K, where L and K are constants (Hollot, 2001). Find the value of L required to obtain a 10% steady-state error for a unit step input when C= 3750 packets/s, N=50 TCP sessions, R= 0.1 s, and K= 0.005.
