++ to Marto, and one more idea: are you perhaps using the mpm_worker Apache module? That one is said to cause problems with threaded code because it does its own threading and the two implementations don't coexist too peacefully. Completely dead processes point to more serious problems than either Perl or Apache usually cause on their own, so this could be one.
All,
Thank you very much for your comments, suggestions and ideas. I upgraded the VPS RAM to 1GB. Problem solved. So, it looks like threads died when lack of memory.
Since you didn't post any code we've know way of knowing if there could be any changes made to fix the "problem" rather than simply throwing more resources at it.