Resolving the Error During Attempt to Connect to External Data Source for PowerPivot

When you initially launch an Excel file stored in the PowerPivot Gallery, you see the last saved state.  Upon interacting with a slicer, filter, or otherwise trying to refresh the data being presented, Excel will return an error if it cannot query the PowerPivot data source.  Specifically, the error I’m referring to is:

An error occurred during an attempt to establish a connection to the external data source.  The following connections failed to refresh:  PowerPivot Data

    

There’s a number of possibilities for this error, as your favorite search engine will tell you.  One common resolution:  start the SQL Server Browser service.  The ULS Log (usually found at C:\Program Files\Common Files\Microsoft Shared\Web Server Extensions\14\LOGS) confirms this:

01/13/2013 15:01:06.03     w3wp.exe (0x1940)                           0x1B7C    PowerPivot Service                Background Jobs                   99    High        EXCEPTION: Microsoft.AnalysisServices.ConnectionException: A connection cannot be made to redirector. Ensure that 'SQL Browser' service is running. ---> System.Net.Sockets.SocketException: No connection could be made because the target machine actively refused it

Services pane in Control Panel: