To provide resilience and to share performance load, the recommended approach is to have multiple (well, more than one anyway!) SharePoint Servers publishing your SSRS reports (in SharePoint mode).
The introduction of SQL 2012 certainly enhanced the load-balancing capabilities which weren’t as stable and effective in SQL 2008 R2, but regardless of which version of SQL you are leveraging, there is no quick way of identifying which server was publishing your report. But here’s the trick, instead of trawling through the ULS logs, put SharePoint to one side for a minute and kindly ask a DBA to open SQL Management Studio for you.
And run the following SQL query (uses the ‘Execution_log’ View)
/****** Script for SelectTopNRows command from SSMS ******/ SELECT TOP 100 [InstanceName], [ReportID], [UserName], [RequestType], [Format], [Parameters], [TimeStart], [TimeEnd], [TimeDataRetrieval], [TimeProcessing], [TimeRendering], [Source], [Status], [ByteCount], [RowCount] FROM [My_SSRS_ReportServerdb].[dbo].[ExecutionLog]