Sure, I can help you write a test case generator for the "Conveyor Belt" problem. Here is the Python code for generating random test cases based on the problem statement:

```python
import random

def generate_test_case():
    # Generate random values for N and Q
    N = random.randint(2, 10**5)
    Q = random.randint(1, 10**5)
    
    # Initialize the list of requests
    requests = []
    for _ in range(Q):
        a = random.randint(1, N-1)  # a should be at least 1 and less than N
        b = random.randint(a+1, N)  # b should be at least a+1 and at most N
        p = random.randint(1, 10**9)
        requests.append((a, b, p))
    
    # Print the generated test case
    print(N, Q)
    for req in requests:
        print(req[0], req[1], req[2])

# Example usage
generate_test_case()
```

This function generates random values for \( N \) and \( Q \) within their specified ranges. For each request, it ensures that \( a \) is less than \( b \), and both are within the range from 1 to \( N \). The number of products \( p \) is also generated randomly within its specified range.

You can run this function to generate random test cases for the "Conveyor Belt" problem. Each call to `generate_test_case` will output a random test case, which you can use to test your solution.