Test-and-test-and-set has much smaller traffic smaller to the just test-and-set. It generates one invalidation per waiting processor per LOCK RELEASE. Therefore, O(P^2) traffic if all processors have the lock cached. But it still much better than test-and-set because it generates one invalidation per waiting processor per TEST.

I agree test and test and set will be a better solution in the case a variable is shared by many processors, in which case we would be able to avoid a lot of write invalidation traffic on the bus.

where does the O(P^2) interconnect traffic number come from? I feel like there are only O(P) interconnect traffic per lock release