I have been checking my test FCGI script from time to time in the last couple hours. I just saw that the counter reset to 0 after some time of inactivity - more than 30 mins I think.
If I click on it continuously after the first click, the counter increments by one each time. I can now better appreciate a little more what persistence means.
This ability to respond to a request without spawning a new process each time is indeed a very attractive feature to have. But until today, I didn't know it comes with the cost of using up server resources. Will have to try it out to see how it works on a shared server.