So if you call exec() from a sub called by SIGALRM, you need to unmask SIGALRM before calling the exec() or risk losing further alarms:

use POSIX qw(SIGALRM SIG_UNBLOCK sigprocmask);
...
sub handler {
    my $sigset = POSIX::SigSet->new( &POSIX::SIGALRM );
    sigprocmask(SIG_UNBLOCK, $sigset, undef);
    ...
    exec([stuff]);
}
$SIG{ALRM} = \&handler;

Without the sigprocmask, your alarm will fire exactly once, be masked from running again, and your subsequent processes won't have an alarm to use anymore.