After having established logging information on shutdown, I now have more info in regards to the situation where the server seems to fall in a sleep mode. Basically, the ShutDownMessage is as follow:

HostingEnvironment initiated shutdown
HostingEnvironment caused shutdown

The ShutDownStack is as follow:

at System.Environment.GetStackTrace(Exception e, Boolean needFileInfo)
at System.Environment.get_StackTrace()
at System.Web.Hosting.HostingEnvironment.InitiateShutdownInternal()
at System.Web.Hosting.HostingEnvironment.UnregisterRunningObjectInternal(IRegisteredObject obj)
at System.Web.Hosting.HostingEnvironment.UnregisterObject(IRegisteredObject obj)
at System.Web.Hosting.ISAPIRuntime.StopProcessing()

By looking at this, it is still unclear what happened exactly to force a shutdown. That situation is unique to two servers only out of about 7 I am dealing with.