I believe I might have found the reason for all this. The Web site is configured to run on a specific account. The same goes with the application pool. The Web site also starts and stops Windows Services by the use of an account. We also have Task Scheduler items running from an account.
This account is a domain account and the same everywhere.
When looking at the Event Viewer in very details, I have been wondering if this is causing a conflict. For example, the Web site launches a request to start 26 Windows Services. All those services are defined with the same account. So, their will be authentication against that operation for every Windows Service, login in, login out. But, at the same time, if this takes effect, is this a potential treat against the Web site which is running off the same account? The same would then apply to Task Scheduler items which are scheduled to restart every minute. So, I have been wondering if this is the cause of this conflict.
For now, I have requested the creation of a dedicated account linked to the Windows Services and two more linked to the Task Scheduler items. Once available, I will be able to test more and see if I still have this issue.