Brain Teaser: The Scaling Paradox
A startup’s AI system performance follows this pattern:
1-100 users: 99% uptime
101-1000 users: 95% uptime
1001-10000 users: 90% uptime
10001+ users: 85% uptime
If downtime costs $100 per user per hour, and they gain 100 users per week, starting with 50 users, when does the weekly downtime cost first exceed $10,000?
Answer:
The weekly downtime cost first exceeds $10,000 after 1 week (when they reach 150 users and drop to 95% uptime).
The Math:
Let the weekly hours be (24×7). If uptime is , then expected downtime hours per week is:
Downtime cost is $100 per user per downtime hour, so weekly downtime cost at N users is:
Week 0 (start): 50 users, 99% uptime
C(50)=50 \cdot 1.68 \cdot 100 = 50 \cdot 168 = $8{,}400]
So it’s below $10,000 at the start.
After 1 week: 150 users (gain 100), now in the 101–1000 tier at 95% uptime
C(150)=150 \cdot 8.4 \cdot 100 = 150 \cdot 840 = $126{,}000]
That’s above $10,000.