[Amps] Are Amps Slow on the uptake? Or is it just me?

[Jeff in Mali:]

I have been noticing in my pileups that I have a hard time copying the first couple of letters of people call signs in pileups on CW..
After listening more carefully, what I seen to find is that the strength of the signal as the amps fire up on CW is up to 2 S-units lower than when it gets to full power.
Is this the case that amps take a few milliseconds to come up to full power?  And therefore it may be putting out less than full power on the first few letters of a call sign than on latter letters?
73, 
Jeff TZ4AM