Does mail need to be running for email to be forwarded to other accounts

I need to have mail forwarded from specific email from one source with specific subject line etc to other people immediately on my Mac Mini. I believe I have it setup correctly but am wondering if I need to have my computer on and mail running; or does the computer and mail not need to be running for this to happen?, or is this handled by apple servers automatically somehow? Do I need to change certain settings in mail to automatically do this as there is a time limit of 9 minutes in which the recipients need to receive the forwarded email.