From the comments so far, I gather that what you're looking for is the maximum number of 'start-to-end' time periods which are in progress within your dataset. This is easily calculated, given that the number of open time periods increments at each @start time and decrements at each @end time:

my @proc_start = sort @start;
my @proc_end = sort @end;
my $count = 0;
my $max = 0;

while (@proc_start && @proc_end) {
  if ($proc_start[0] lt $proc_end[0]) {
    # Next event is a start
    $count++;
    $max = $count if $count > $max;
    shift @proc_start;
  } else {
    # Next event is an end
    $count--;
    shift @proc_end;
  }
}
print "Peak was $max\n";
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(Untested code. Assumes that the time period covered by the dataset begins and ends with 0 open periods. Hour-to-hour transitions are left as an exercise for the reader.)