I think I would comment on our progress with this.
Although none of the SQL Server documentation / articles / blogs mentions that this error could be caused by server busyness , I found a forum where some experienced IT professionals named Matt Neerincx state that this could be the following:
Possible reasons for this error include: 1. Poor network link from client to server. 2. Server is very busy (meaning high CPU) and cannot respond to new connection attempts. 3. Server is running out of memory (so high memory usage for SQL). 4. tcp-ip layer on client is over-saturated with connection attempts so tcp-ip layer rejects the connection. 5. tcp-ip layer on server side is over-staturated with connection attempts and so tcp-ip layer is rejecting new connections. 6. With SQL 2005 SP2 and later there could be a custom login trigger that rejects your connection. You can increase the connect timeout to potentially alleviate issues
So, now we are going this way - reducing the number of requests that are executed simultaneously in some of our batch requests, optimizing some of our requests, etc.
In addition, in our application connection string, we increased the connection timeout and set the Min Pool Size to 20 (assuming that it is good to try to provide some existing, unused connections for the application to capture, instead of establishing a new connection )
At this point, it was almost 48 hours without receiving an error; which makes us very encouraging.
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