One could use a buffer of the desired size of data, and then store the data to be averaged cycling through the buffer. As I cannot replicate your data, I am using random numbers to demonstrate the principle:

use strict;
use warnings;

# initialize averaging
use List::Util 'sum';
my $size = 10;
my @store;
my $i = -1;

while (1){
  my $line_count = int rand 100;

  # store data and output average
  $store[ ($i+=1) %= $size ] = $line_count;
  my $average = sum( grep defined, @store )/$size;
  print "Average line count: $average\n";
}
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This way, one would always print the average of the last $size numbers generated. It requires a run up period, so one cannot trust the first $size averages, which I would guess, should not be an issue if you want to run it for a long time.