One duration is one data point. 

I'm i entitled to collect rent for the month of November or just for the duration of stay in November 2.

But I think there's a way structurally to make the allocations easier.When you sell a split use property the allocations are not always clear such as when you've used part as a primary residence for a period of time and then change the allocations, use and duration. 

If you or your property manager are on top of things and hitting all the eviction milestones on time, 3-4 weeks should be the maximum duration of the process from late rent to vacant. 

- Search the MLS/Realtor/Zillow and find properties that might be on the market for long durations?

Make sure you define who is paying for all of these costs in the contract.Whatever duration they tell you it will take to build, I would plan on it doubling.

The amount of time may well, be more than the amount of pet fees you are looking to get compensated for, for the duration of time. 

My concern is the duration.

Would it be a good idea to purchase the property with the cash I have and then offer them free rent to make up the difference for an agreed duration?  

Net monthly profit: 572, for an almost 50% yearly return, for the duration of the note.