Why the ArcGis return logs for the deleted services! Since this service deleted from the ArcGis manager does it take from server performance, and how to prevent such logs.
Are those service URLs still being used by any webmap or other applications? These logs will come up whenever the layers within the webmaps/apps are being accessed. Please ensure all the web maps and applications are also updated (i.e. layers pointing to the deleted service are either removed or updated).
Typically the level of logging, e.g., WARNING, FINE, INFO, DEBUG, etc..., has the larger impact than one or two services generating a lot of SEVERE messages. As Jayanta Poddar already pointed out, those errors are showing up in your logs because someone is still trying to use the deleted services. If you are unable to track down where the requests are coming from, and you just can't stand to see these errors in the logs, you can setup URL filtering on your IIS server that hosts the web adaptor.
We have had this issue and resolved it by reviewing IIS (MS web server) logs.
We had a service that we shutdown, but still received SEVERE logs during a specific hour of the day every day. I opened IIS logs for that hour and the service name is in the logs.
The logs can show some valuable information. In our case, the IIS logs showed "ArcGIS+Hub+Indexer" for the problematic service. Turns out the hub instance runs a daily check or index and it was causing the errors since the service had been removed. It's possible IIS logs can at least point you in the right direction.