How often should I message this girl on facebook?

[Mike]

Well, there's this girl I recently became friends with as we have mutual friends. We used to talk daily or on alternate days. She sometimes gives long replies and sometimes just short replies. She sometimes used used to be the first one to talk and I used to chat with her. It's a month since we're chatting and I am starting to like her. But nowadays I don't see her online much at her usual time but she replies to messages once in a while. So I was thinking if she is getting irrirated with my daily chat? I like this girl very much and always talk politely with her but her offline status really makes me sad. If I don't see her on facebook I don't feel like remaining online any longer. What should I do? I am not sure if I should wait for her to message which may never come or message her myself which I doubt will make her feel irritated....Suggestions are welcome...