Draining While It Fills

An outage left 2,400,000 messages in a work queue. New messages keep arriving at a steady 900 per second, and each consumer sustains 6.5 messages per second. You want the backlog empty within 40 minutes.

How many consumers do you need?

1900, because the fleet has to move 1,900 messages per second and you need one consumer per message per second

293, because the fleet has to serve the arrival rate plus the drain rate: $\lceil (900 + \frac{2{,}400{,}000}{2400}) / 6.5 \rceil$

154, because the backlog is what you are draining: $\lceil \frac{2{,}400{,}000}{2400 \times 6.5} \rceil$

139, because you size the fleet to the arrival rate: $\lceil \frac{900}{6.5} \rceil$