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It's a nice solution, but doesn't seem to solve the problem they present at the start of the article. "What we wanted was a way to keep at least 24 hours worth of entries at all times"; but their solution only stores the last MAX_CIRCULAR_LOG_ROWS entries.
Ah yes, good point. Luckily we have a regular rate of messages and we knew that 2 million would be enough to always have roughly 24 hours worth.
Wouldnt it be faster to have the log_id in a variable so you don't have to do a select and an update? In fact wouldnt it be faster to just store 48 hours and delete the oldest 24? Is it going to kill you to have 47:59 hours of logs?
That's exactly my thought. You wouldn't even need a database then and you could use a flat file for each day.
they probably have this in SQL because of the ease-of-use to run stats on it though.
If you don't mind having a more static table size, you can also do:

  create table test_table( id integer identity(1), val varchar(max) )
  -----
  create trigger auto_updater on test_table after insert as
  begin
   declare @id integer = (select id from inserted)
   declare @val varchar = (select val from inserted)
   
   update test_table 
   set val=@val
   where id=@id%10+1
   
   if @id >= 10
      delete from test where id=@id
  end
Then to insert you just insert. It'll auto-override the correct entry.

  insert into test_table(val) values ('hooray')
insta-looping for free (syntactically). Though I did write this in t-sql, I'd assume one can pull the same trick in MySQL (identity => autoincrement). You can also have it timestamp while you update.
This helped me find another really cool way of handling this. INSERT ... ON DUPLICATE KEY UPDATE