As of January 2017, an AWS RDS environment does not offer administrators an easy option to update statistics or rebuild indexes.  If not addressed, the lack of database maintenance can result in publishing failures and performance issues. Content editors may also see errors such as the below, although there may be additional contributing factors.

A database error ocurred while executing Stored Procedure "EDA_ITEMS_GETPUBLISHEDITEMS".

The attached two scripts may be manually run, or scripted to run on a scheduled basis.  Note the following regarding the attached two scripts, with the caveat that the scheduling and duration estimates are a baseline only and may vary greatly in a production environment.  Also, the scripts should be tested thoroughly in a dev RDS database before being deployed to a production RDS database.
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The index script is usually scheduled for one day of the week, preferably on a weekend and after hours. This is because, depending on how much data and how many indexes you have, it can be a performance hit during peak times.
The update statistics job can be scheduled daily or every two days on every other day the index job isn’t running and is usually set to run at the same time. This is usually ligtweight in terms of performance.

Alternatively, the solution from Ola Hallengren can also be used for an AWS RDS database.
https://ola.hallengren.com/

If an administrator sees problems with running these scripts on a large database such as the broker DBs, it is possible that the tempdb for the AWS hosted DBs need to be tuned.  Below are some possible suggestions, but note that RWS Support does not have expertise in AWS RDS maintenance.   Administrator should raise a ticket with AWS Support for assistance on this if needed.
https://aws.amazon.com/blogs/database/best-practices-for-configuring-performance-parameters-for-amazon-rds-for-sql-server/