After the "heavy" activity update statistics and defragment indexes.
Make sure there are not missing indexes, even if you have even set "Automatic tuning" on the Azure Portal. Try the query shared on this article.
Your index strategy has to be good and the query need to be well tuned with Azure SQL Database or you will see high DTU usage, throttling and timeouts occurring.
When databases are busy scaling up/down may take long time, Queries are canceled when a scale up/down is initiated. Transactional queries, which modify your data or the structure of the data, may not be able to stop quickly. Transactional queries, by definition, must either complete in their entirety or rollback their changes. Rolling back the work completed by a transactional query can take as long, or even longer, than the original change the query was applying. For example, if you cancel a query which was deleting rows and has already been running for an hour, it could take the system an hour to insert back the rows which were deleted. If you run scaling while transactions are in flight, your scaling may seem to take a long time because scaling has to wait for the rollback to complete before it can proceed.