NC task Case study week 8

Question 1:

Start from an initial guess x0 = 1. Use the Newton Raphson method to find the

approximate load factor at which latency is approximately zero.

Answer:

https://drive.google.com/file/d/1xIQ127E3HKCsB9qw6SYkPF8H-5GFZhSn/view?usp=drive_link

Question 2:Explain why solving this equation is important in network modeling.

Answer:

In computer networks, latency is a key performance metric. The equation:

$$L(x) = e^{-3x} - \cot(x)$$

models latency as a combination of two factors:

• Exponential Decay (e^{-3x}): Represents how system responsiveness decreases under heavy load. As x increases (indicating higher load or delay), this term decreases rapidly, simulating degrading performance.

• Cotangent Function (cot(x)): Simulates periodic network congestion or routing delays which repeat due to protocol cycles or queue behaviors.

Finding the root of this equation (i.e., solving L(x) = 0) allows us to:

• Identify the optimal load factor x where the latency is zero or minimal.

• Tune network parameters to avoid instability and congestion.

• Optimize Quality of Service (QoS) by designing systems to operate near these critical points.

This makes the Newton-Raphson method a powerful tool in performance tuning and network optimization.

Question 3:Write a Python or C++ program to find the root of the equation using Newton’s

Method.

Answer:

Code:

import math

# Define the function and its derivative

def f(x):

    return math.exp(-3 * x) - 1 / math.tan(x)

def f_prime(x):

    return -3 * math.exp(-3 * x) + 1 / (math.sin(x) ** 2)

# Newton-Raphson Method

def newton_raphson(x0, tolerance=0.0005, max_iter=100):

    for i in range(max_iter):

        fx = f(x0)

        fpx = f_prime(x0)

        if fpx == 0:

            print("Derivative is zero. No solution found.")

            return None

        x1 = x0 - fx / fpx

        print(f"Iteration {i+1}: x = {x1}")

        if abs(x1 - x0) < tolerance:

            return x1

        x0 = x1

    print("Did not converge within the maximum number of iterations.")

    return None

# Initial guess

x0 = 1

root = newton_raphson(x0)

if root is not None:

    print(f"\nApproximate root (load factor where latency is zero): {root:.4f}")