[CentOS] lots of disk activity keep process from responding

hi all

I have something like 3000 log files in a directory (from data collection).
At 4am in the morning cron runs a script of mine that essentiallly trims 
all files in that directory back to a certain size.
Basically does a "tail -c XXXX filename > tmp_filename"

I have noticed that only around the 4am-4:15 time frame when the trim is 
happening does my
other process log connection attempts but my process (forking and 
opening databases) is not responding in time to give data back
to the connecting process. Not critically important as it just tries 
again and all ok...

My question is how do I tell the script that runs to run at a lower 
priority perhaps ???
Or is there another way to do this I am not aware of?

The trimmming of the log files is just so they they dont take a bunch of 
space on the HD.
It doesnt have to be done supper fast or anything.

Any ideas I might try? Thanks,

Jerry