The SystemOut*.log files contained no WSVR0605W Hung Thread Detection Warning messages, so moved on to the Javacore thread dumps.
 
All of the sets of Javacores (8 per each PID) on (3) different ProcessIDs showed vast majority of Web Container threads slow moving or not moving at all within the (6) minute duration the dump sets were generated for, also all affiliated with the same Code Stacks or Frames.  
Index values were in the (300-400) upwards range, but this appears to be where problem focus should be, relevant partial extracts included below for your convenience and review, but full stack frames available in any Javacore*.txt file for any WebSphere Process (just search on "WebContainer :")
 
D:\IBM\WebSphere\AppServer\profiles\ePOSAppSrv01\javacore.20190516.103518.9368.0008.txt
 
"WebContainer : 440" J9VMThread:0x0000000005127200, j9thread_t:0x0000000025D78560, java/lang/Thread:0x0000000140BD0E68, state:CW, prio=5
(java/lang/Thread getId:0x643, isDaemon:true)
(native thread ID:0x1F00, native priority:0x5, native policy:UNKNOWN)
Java callstack:
at java/lang/Object.wait(Native Method)
at edu/emory/mathcs/backport/java/util/concurrent/TimeUnit.timedWait(TimeUnit.java:301(Compiled Code))
at edu/emory/mathcs/backport/java/util/concurrent/Semaphore$NonfairSync.attempt(Semaphore.java:275(Compiled Code))
at edu/emory/mathcs/backport/java/util/concurrent/Semaphore.tryAcquire(Semaphore.java:571(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/ThreadBlockingAdaptingServlet$ThreadBlockingRequestResponse.blockUntilRespond(ThreadBlockingAdaptingServlet.java:53(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/ThreadBlockingAdaptingServlet.service(ThreadBlockingAdaptingServlet.java:26(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/EnvironmentAdaptingServlet.service(EnvironmentAdaptingServlet.java:63(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/PathDispatcher.service(PathDispatcher.java:23(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/SessionDispatcher.service(SessionDispatcher.java:60(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/SessionVerifier.service(SessionVerifier.java:26(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/PathDispatcher.service(PathDispatcher.java:23(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/MainServlet.service(MainServlet.java:148(Compiled Code))
at javax/servlet/http/HttpServlet.service(HttpServlet.java:831(Compiled Code))
at sg/com/nhg/epos/ui/common/PersistentFacesServletWrapper.service(PersistentFacesServletWrapper.java:41(Compiled Code))
at com/icesoft/faces/webapp/xmlhttp/BlockingServlet.service(BlockingServlet.java:56(Compiled Code))
at com/ibm/ws/cache/servlet/ServletWrapper.serviceProxied(ServletWrapper.java:307(Compiled Code))
...
 
D:\IBM\WebSphere\AppServer\profiles\ePOSAppSrv01\javacore.20190516.103436.52420.0008.txt
"WebContainer : 369" J9VMThread:0x0000000005199400, j9thread_t:0x000000002FB465C0, java/lang/Thread:0x00000001092AE298, state:CW, prio=5
(java/lang/Thread getId:0x52E, isDaemon:true)
(native thread ID:0xA204, native priority:0x5, native policy:UNKNOWN)
Java callstack:
at java/lang/Object.wait(Native Method)
at edu/emory/mathcs/backport/java/util/concurrent/TimeUnit.timedWait(TimeUnit.java:301(Compiled Code))
at edu/emory/mathcs/backport/java/util/concurrent/Semaphore$NonfairSync.attempt(Semaphore.java:275(Compiled Code))
at edu/emory/mathcs/backport/java/util/concurrent/Semaphore.tryAcquire(Semaphore.java:571(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/ThreadBlockingAdaptingServlet$ThreadBlockingRequestResponse.blockUntilRespond(ThreadBlockingAdaptingServlet.java:53(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/ThreadBlockingAdaptingServlet.service(ThreadBlockingAdaptingServlet.java:26(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/EnvironmentAdaptingServlet.service(EnvironmentAdaptingServlet.java:63(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/PathDispatcher.service(PathDispatcher.java:23(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/SessionDispatcher.service(SessionDispatcher.java:60(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/SessionVerifier.service(SessionVerifier.java:26(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/PathDispatcher.service(PathDispatcher.java:23(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/MainServlet.service(MainServlet.java:148(Compiled Code))
at javax/servlet/http/HttpServlet.service(HttpServlet.java:831(Compiled Code))
at sg/com/nhg/epos/ui/common/PersistentFacesServletWrapper.service(PersistentFacesServletWrapper.java:41(Compiled Code))
...
 
D:\IBM\WebSphere\AppServer\profiles\ePOSAppSrv01\javacore.20190516.102736.44984.0008.txt
 
"WebContainer : 388" J9VMThread:0x000000000598C200, j9thread_t:0x000000004CD8A550, java/lang/Thread:0x00000001157978A0, state:CW, prio=5
(java/lang/Thread getId:0x588, isDaemon:true)
(native thread ID:0xEBAC, native priority:0x5, native policy:UNKNOWN)
Java callstack:
at java/lang/Object.wait(Native Method)
at edu/emory/mathcs/backport/java/util/concurrent/TimeUnit.timedWait(TimeUnit.java:301(Compiled Code))
at edu/emory/mathcs/backport/java/util/concurrent/Semaphore$NonfairSync.attempt(Semaphore.java:275(Compiled Code))
at edu/emory/mathcs/backport/java/util/concurrent/Semaphore.tryAcquire(Semaphore.java:571(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/ThreadBlockingAdaptingServlet$ThreadBlockingRequestResponse.blockUntilRespond(ThreadBlockingAdaptingServlet.java:53(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/ThreadBlockingAdaptingServlet.service(ThreadBlockingAdaptingServlet.java:26(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/EnvironmentAdaptingServlet.service(EnvironmentAdaptingServlet.java:63(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/PathDispatcher.service(PathDispatcher.java:23(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/SessionDispatcher.service(SessionDispatcher.java:60(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/SessionVerifier.service(SessionVerifier.java:26(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/PathDispatcher.service(PathDispatcher.java:23(Compiled Code))
at com/icesoft/faces/webapp/http/servlet/MainServlet.service(MainServlet.java:148(Compiled Code))
at javax/servlet/http/HttpServlet.service(HttpServlet.java:831(Compiled Code))
at sg/com/nhg/epos/ui/common/PersistentFacesServletWrapper.service(PersistentFacesServletWrapper.java:41(Compiled Code))
...
 
Please coordinate with App developers who own the code mentioned above listed in the partial stack frames, to help determine what the code is doing and why there might be a responsiveness issue resulting, which appears stuck in TimeUnit.timedWait() top function call instead of the more common socketRead() call to a remote resource.
 
I also confirmed that you have extended support contract in place for CN #622861 for WebSphere 7, which includes Java 6, through 4/30/2020, so good there.
JRE 1.6.0 Windows Server 2008 R2 amd64-64 build jvmwa6460sr16-20140416_196573 (pwa6460sr16ifix-20140425_01(SR16+IX90144))
 
Let us know what you find out with App developers and also confirm if there is a Remote Resource involved outside of the AppServers, and if so, check also with Network Admins for any related network latency during the timeframe of (10:21 - 10:35).