Backlog growth during a mismatch is simply arrival − service, sustained over the mismatch duration. Recovery time after scaling is backlog ÷ (new service rate − arrival rate) — and if the new service rate barely exceeds arrival, recovery takes far longer than intuition suggests.
This is not Little's Law. Little's Law (L = λW) describes a stable queue's average in-flight items; this arithmetic describes how fast an unstable queue (arrival > service) grows. See the "Little's Law vs queue accumulation" section on the backpressure page for the distinction.
If service rate after scaling is less than or equal to arrival rate, the backlog never drains — it just stops growing (or keeps growing).
This is a planning estimate, not a live measurement: real consumers rarely sustain a perfectly constant rate, and recovery competes with ongoing arrivals.
The same arithmetic applies to Kafka consumer lag, a Flink operator's backpressured buffer, an API queue, or a thread pool — see Backpressure & Queueing.