The customer mentioned that the replication from the Child SMP to the Parent SMP stopped working few days ago.
The NS log showed that it is trying to reach "could not connect to the local server http://CHILD01.example.net/Altiris/NS/".
 

The Child and Parent SMP should be using HTTPS links to connect and not HTTP.

Under the SMP Console>Settings>Notification Server >Hierarchy>Topology tab, it showed that the proper HTTPS link reference is right.

The NS logs on the Child SMP showed the following entry while trying to replicate up:

Unable to schedule replication to the specified server (Server: PARENT01.EXAMPLE.NET, Exception: Altiris.NS.Exceptions.AeXException: The local credential [username] could not connect to the local server http://CHILD01.example.net/Altiris/NS/ at Altiris.NS.Replication.ReplicationJobHelper.CreateReplicationJob(String jobId, String name, Guid serverGuid, String serverName, String serverUrl, WebServiceCredential destinationCredentials, WebServiceCredential sourceCredentials, ReplicationRuleCollection rules, ReplicationMode mode, ReplicationPriority priority, Boolean hierarchy)
at Altiris.NS.Replication.Hierarchy.HierarchyRuleManager.CreateReplicationJob(String jobId, String name, HierarchyNodeItem node, ReplicationRuleCollection rules, ReplicationMode mode, ReplicationPriority priority)
at Altiris.NS.Replication.Hierarchy.HierarchyRuleManager.ResolveRules(IItem item, HierarchyRuleScope scope, ReplicationMode mode, ReplicationPriority priority, HierarchyRuleCollection rules, HierarchyNodeItem node, Boolean isEvent, Guid hierarchyJobId)).
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Date: 4/17/2018 1:15:01 PM, Tick Count: 1035826687 (11.23:43:46.6870000), Size: 1.31 KB
Process: AeXSvc (1336), Thread ID: 79, Module: Altiris.NS.Replication.Hierarchy.dll
Priority: 1, Source: HierarchyRuleManager.ResolveRules
 

ITMS 8.1 RU6

Unknown.

1. Run the following queries against the affected child SMP:
   --Check if there is the proper reference under ForwardServer table for this child SMP:
   select * from ForwardServer
   
   --If it shows the url as HTTP, run the following to update it:
   update ForwardServer
   set Web = 'https://CHILD01.example.net/Altiris/NS/'
   where Web = 'http://CHILD01.example.net/Altiris/NS/'
   
   update ForwardServer
   set AgentWeb = 'https://CHILD01.example.net/Altiris/'
   where AgentWeb = 'http://CHILD01.example.net/Altiris/'
   
    
2. Check the Parent SMP database and see if there is a http reference in the following hierarchy table:
   
   select i.name, cast (i.state as xml), i2.name, i3.name, hn.*
   from hierarchynode hn
   join item i on i.guid = hn.itemguid
   join vitem i2 on i2.guid = hn.parentguid
   join vitem i3 on i3.guid = hn.childguid
    
3. It is probably an entry in the state column in the item table. Adjust the reference that with this query:

update item
set state = replace (cast (state as nvarchar (max)), 'http://CHILD01.example.net/Altiris/','https://ACHILD01.example.net/Altiris/')
where guid = 'F241DB2F-F30D-4C19-AC4E-A15DA7B5347A'