AJAX - Two Related Selects

First of all, change will really only be triggered when you call the event.

Will fac_name have a default value on page load? If so, you have two options. You can either make an AJAX call on page load outside the current function you have. Or simple enough, you can trigger the change event after the function is defined, like

<script>
  $(document).ready(function () {
    $("#fac_name").change(function () {
      var id = $(this).val();
      var dataString = 'id=' + id;
      $.ajax({
        type: "POST",
        url: "qry/ajax_proceedures.php",
        data: dataString,
        cache: false,
        success: function (html) {
          $("#proceedure_type").html(html);
        }
      });
    });
    $('#fac_name').trigger('event'); // This will make the AJAX call with the value of the option that is currently selected.
  });
</script>