Run Linux process at very very low priority?

I have a regular process that's not so important but will consume very much CPU power. I have another process which is really important, but it spends most of the time idle, but when it gets a job it really needs high computing power. I tried running with nice -20 ./low_priority_process and nice --20 ./high_priority_process but still the lower priority process consumes significant amount of CPU when the high priority process is in need. How can I run a process that will really yield or even auto-suspend when another process is using CPU power?