We have found the reason for the latency. Something causes our Web site and Web Service worker process to reload, practically at every hit. As the initial hit of a worker process loads everything in memory, which takes about 2.5 seconds, this explains everything. Also, when this happens, it creates a queue of all other hits hitting the same worker process. They are released when the initial hit completes its loading process.

As this unexplained phenomemon is occuring several times per minute, I let you imagine the kind of situation this causes.

The reload of a worker process is scheduled to only happen at 6h00 am in the morning, thus once a day. We have it thousands of times per day. This is why it causes the latency.

Now, here is a message I am getting on all the shutdown events:

Change Notification for critical directories.
File Change Notification Error in App_Browsers
HostingEnvironment initiated shutdown
Change Notification for critical directories.
File Change Notification Error in App_Code
Change Notification for critical directories.
File Change Notification Error in App_WebReferences
Change Notification for critical directories.
File Change Notification Error in App_GlobalResources
CONFIG change
CONFIG change
File Change Notification Error in \\MyNasDevice\MyNasShare\WebService
File Change Notification Error in
File Change Notification Error in
Change Notification for critical directories.
File Change Notification Error in App_LocalResources
HostingEnvironment caused shutdown

Here is another one:

Directory rename change notification for '\\MyNasDevice\MyNasShare\MyWebSiteDirectory\www'.
File Change Notification Error in www
HostingEnvironment initiated shutdown
Change Notification for critical directories.
File Change Notification Error in bin
Change Notification for critical directories.
File Change Notification Error in App_GlobalResources
Change Notification for critical directories.
File Change Notification Error in App_Code
Change Notification for critical directories.
File Change Notification Error in App_Browsers
Change Notification for critical directories.
File Change Notification Error in App_WebReferences
CONFIG change
File Change Notification Error in \\MyNasDevice\MyNasShare\MyWebSiteDirectory\www
File Change Notification Error in
File Change Notification Error in
Change Notification for critical directories.
File Change Notification Error in App_LocalResources

Can someone explain this?

I have scheduled a reboot at 23h00 tonight to clear out of the equation a memory bloating. I doubt this would change something.