I have website, but no Alexa rank after 30 days.
According to the answers on How to get new site indexed and ranked by Alexa:
Once you do that, Mohit Bumb says that it should only take two or three days before your site is ranked by Alexa.
John Conde also says that you may be able to speed up the process by installing the Alexa Toolbar and visiting your site every day.
Alexa Rank is junk. I explain it using Statistical Analysis on my site: http://www.closetnoc.org/?i=definitive-proof-that-alexa-sucks Here you will see that Alexa cannot even come close to estimating traffic or rank.
It is absolutely impossible for Alexa to extrapolate performance and ranking of any site unless the site receives a large amount of traffic. Even then, the accuracy would be rather suspect. This is because there are so many factors that falsely influence Alexa.
[Update: This question changed so I am updating the answer somewhat.]
I realize that there is pressure for Alexa Rank by advertisers and so forth. This is unfortunate. Alexa can be influenced for rapid increases in ranking if this is the case. On the page I linked to, there are a few ways that Alexa can be influenced. I would say that you should follow Stephen Ostermiller's answer to get indexed by Alexa. Then look at the page I linked to to influence Alexa. Here is a quick run-down. Create a blog, install the tool bar, encourage your users to install the toolbar, install the Alexa bug for performance, participate in social media that drives traffic such as Twitter, create links from other blogs and forums though this can be bad for SEO, and finally, create an e-mail newsletter that drives traffic to your site. Certain activities such as blog and forum traffic, twitter traffic, and e-mail traffic tend to contain a higher number of Alexa toolbar users.
I do want Alexa to die so I really do not want to encourage promoting Alexa's importance with advice. This is because Alexa is such junk that it cannot be relied upon for any value and warps the good behavior we have been striving to promote rather than influencing rank with tricks.
You can't predict that when will your website's rank will appear on Alexa.com.
We all know that Alexa rank is very important to any blogger or website builder.
My blog was ranked by Alexa.com in just 14 days after I have created.
I have noted that on my blog, there was 90% traffic from Facebook.
I don't think that we can predict Alexa rankings of our website or blog.