Facebook Graph - What is the limit to send private messages from page to users?

Facebook Graph - What is the limit to send private messages from page to
users?

I am developing a chat robot that works with private messages on facebook.
The person sends a private message to a page that I own, and then I will
send an answer for each message.
Everything is working, but I need to be sure facebook won't complain about
the amount of messages I will send. This application will receive a lot of
interactions at the same time, but in some early tests one of my messages
were received like that:
http://cl.ly/image/1C1n0Z2L0R05
I am now using Batch Requests to send all messages, on an interval of 15s.
Do someone know some way to test it with multiple users and multiple
messages at the same time? How the process of identification of spam
messages work on facebook? How much messages can I send at the same time
and in what time range to prevent that kind of behavior?
Thanks.