I found an agent to apply for a patent for utility model. It's been 20 days and nothing has happened. They replied that the teacher who wrote the manuscript was too busy. Is this normal? ...

Find a famous one and sign a confidentiality agreement with him; Generally, the time for a simple patent is about 3 months, and the time for a complex patent is more than 6 months. You will give the patent a lot of time if you write the principle clearly yourself. After the application is submitted, you can ask for a patent acceptance letter.