The search service is not able to connect to the machine that hosts the administration component

I have discovered a problem with the Search Service Application in my DR farm.  Any query in the Enterprise Search Centre produces an error:

Search has encountered a problem that prevents results from being returned. If the issue persists, please contact your administrator.

When I attempt to manage the service application within Central Administration I see the following message:

The search service is not able to connect to the machine that hosts the administration component. Verify that the administration component '7c8a54c7-0c9c-4afd-9442-b99c08852982' in search application 'Search Service Application' is in a good state and try again.

When I ran the Get-SPEnterpriseSearchServiceInstance cmdlet it showed the status as online.  I also found Event ID 1365 logged in Application log so after a quick Google I followed the steps on this TechNet wiki article but to no avail.  I had a dig around the logs (using ULS Viewer) and spotted this:

SearchServiceApplication::Execute--Exception: Microsoft.SharePoint.SPException: Tried IMS endpoints for operation Execute: Operation sent to IMS failed: Resource saturation, try again later.

I thought this may be due to under-spec’d hardware so I increased the RAM but this didn’t help.

After a few hours of looking through logs and trying different things I resorted to completely removing the Search Service Application (using this script) and creating it again (using AutoSPInstaller).  I can only assume that some configuration became corrupt but I didn’t really have enough time to continue investigating.