Back-of-envelope calculations for system design. Use when estimating QPS, storage, bandwidth, or latency for capacity planning. Includes latency numbers every programmer should know and common estimation patterns.
This skill provides frameworks for back-of-envelope calculations essential for system design and capacity planning.
Keywords: back-of-envelope, estimation, QPS, storage calculation, bandwidth, latency, capacity planning, scale estimation
Use this skill when:
Estimation is not about precision, it's about order of magnitude.
Getting within 10x is usually good enough for architectural decisions. The goal is to identify if you need:
| Power | Value | Approximate |
|---|---|---|
| 2^10 | 1,024 | ~1 Thousand (KB) |
| 2^20 | 1,048,576 | ~1 Million (MB) |
| 2^30 | 1,073,741,824 | ~1 Billion (GB) |
| 2^40 | 1,099,511,627,776 | ~1 Trillion (TB) |
| Unit | Seconds | Useful For |
|---|---|---|
| 1 minute | 60 | Short operations |
| 1 hour | 3,600 | Batch jobs |
| 1 day | 86,400 (~100K) | Daily aggregations |
| 1 month | 2,592,000 (~2.5M) | Monthly calculations |
| 1 year | 31,536,000 (~30M) | Annual projections |
| Availability | Downtime/Year | Downtime/Month | Downtime/Day |
|---|---|---|---|
| 99% (two 9s) | 3.65 days | 7.31 hours | 14.4 min |
| 99.9% (three 9s) | 8.76 hours | 43.8 min | 1.44 min |
| 99.99% (four 9s) | 52.6 min | 4.38 min | 8.64 sec |
| 99.999% (five 9s) | 5.26 min | 26.3 sec | 864 ms |
See full reference: references/latency-numbers.md
Quick reference:
| Operation | Latency | Relative |
|---|---|---|
| L1 cache reference | 0.5 ns | 1x |
| L2 cache reference | 7 ns | 14x |
| Main memory reference | 100 ns | 200x |
| SSD random read | 16 us | 32,000x |
| HDD seek | 2 ms | 4,000,000x |
| Round trip same datacenter | 0.5 ms | 1,000,000x |
| Round trip CA to Netherlands | 150 ms | 300,000,000x |
QPS = (Number of Users) x (Actions per User per Day) / (Seconds per Day)
Given:
- 300 million monthly active users
- 50% are daily active = 150M DAU
- Average user reads 20 tweets/day
QPS = 150M * 20 / 86,400
= 3 billion / 100,000
= 30,000 QPS
Peak load (typically 2-3x average):
Peak QPS = 30,000 * 3 = 90,000 QPS
Storage = (Number of Items) x (Size per Item) x (Replication Factor) x (Time Period)
Given:
- 100 million users
- 10% upload daily = 10M uploads/day
- Average photo size = 2 MB
- Keep 5 years of data
- Replication factor = 3
Daily storage = 10M * 2 MB = 20 TB
Yearly storage = 20 TB * 365 = 7.3 PB
5-year storage = 7.3 * 5 = 36.5 PB
With replication = 36.5 * 3 = ~110 PB
Bandwidth = (QPS) x (Request Size or Response Size)
Given:
- 1 million concurrent viewers
- Average bitrate = 5 Mbps
- Peak hours: 8 PM - 11 PM
Bandwidth = 1M * 5 Mbps = 5 Tbps
With 20% overhead: ~6 Tbps
CDN egress cost (rough):
$0.02/GB * 6 Tbps * 3 hours * 3600 sec/hour / 8 bits/byte
= massive cost (hence why Netflix built their own CDN)
Cache Size = (QPS) x (Cache TTL) x (Response Size) x (Unique Ratio)
Given:
- 10,000 QPS
- Cache TTL = 5 minutes = 300 seconds
- Average response = 10 KB
- 20% of requests are unique
Cache entries = 10,000 * 300 * 0.20 = 600,000 entries
Cache size = 600,000 * 10 KB = 6 GB
With overhead (keys, metadata): ~10 GB
DB Size = (Number of Rows) x (Row Size) x (Index Overhead) x (Replication)
Given:
- 500 million users
- Average profile = 1 KB (name, email, settings, etc.)
- Index overhead = 30%
- Primary + 2 replicas = 3x
Data size = 500M * 1 KB = 500 GB
With indexes = 500 GB * 1.3 = 650 GB
With replication = 650 GB * 3 = ~2 TB
Memory for hot data (20%): ~400 GB
Requirements:
- 100M new URLs/month
- 10:1 read:write ratio
Writes:
- 100M / (30 * 24 * 3600) = ~40 writes/second
- Peak: ~100 writes/second
Reads:
- 40 * 10 = 400 reads/second
- Peak: ~1000 reads/second
Storage (5 years):
- 100M URLs/month * 60 months = 6 billion URLs
- Average URL = 100 bytes (short) + 500 bytes (long) = 600 bytes
- 6B * 600 bytes = 3.6 TB
- With indexes and overhead: ~5 TB
Requirements:
- 10M daily active users
- Average 50 messages sent/day
- Average 200 messages received/day
Message throughput:
- Sends: 10M * 50 / 86,400 = ~6,000 messages/second
- Peak: ~20,000 messages/second
Connections:
- Each user maintains 1-3 connections (phone, laptop, tablet)
- Peak concurrent: 10M * 0.1 (10% online) * 2 = 2M connections
Storage (1 year):
- 10M users * 50 msgs/day * 365 days = 182B messages/year
- Average message = 200 bytes
- 182B * 200 bytes = 36.4 TB/year
Requirements:
- 100M monthly active users
- 30% watch daily = 30M DAU
- Average 1 hour/day viewing
Concurrent viewers (peak):
- 30M DAU / 24 hours * 3 (peak factor) = ~4M concurrent
Bandwidth:
- Average stream: 5 Mbps
- 4M * 5 Mbps = 20 Tbps peak bandwidth
Storage (library of 10K titles):
- Average video = 2 hours
- Multiple qualities: 480p (1GB), 720p (3GB), 1080p (5GB), 4K (20GB)
- Per title: ~30 GB
- Library: 10K * 30 GB = 300 TB
Instead of: Use:
86,400 seconds ~100,000 (10^5)
2.5 million ~3 million
7.3 petabytes ~10 petabytes
Think in powers of 10:
Always verbalize:
After calculating, ask:
Sizing for average load.
Average QPS: 10,000
Peak QPS: 30,000 (often 2-3x average)
If you size for 10,000, you'll fail at peak.
Calculating raw storage without copies.
Data: 1 TB
With 3 replicas: 3 TB
With backups: 4-5 TB
Sizing for current, not future.
Current users: 10M
Expected growth: 50%/year
Year 3: 10M * 1.5^3 = 34M users
Size for at least 2x current to avoid near-term issues.
Calculating to 3 decimal places.
Bad: "We need exactly 3,456,789 IOPS"
Good: "We need roughly 3-4 million IOPS"
| Need | Formula |
|---|---|
| QPS | users * actions/day / 86400 |
| Storage/day | items/day * size/item |
| Bandwidth | QPS * response_size |
| Cache hit rate | 1 - (DB_QPS / total_QPS) |
| Servers needed | QPS / QPS_per_server |
| Shards needed | data_size / max_shard_size |
design-interview-methodology - Overall interview frameworkquality-attributes-taxonomy - NFR definitions (scalability, performance)database-scaling - Database capacity planning (Phase 3)caching-strategies - Cache sizing and hit rates (Phase 3)/sd:estimate <scenario> - Calculate capacity interactivelycapacity-planner - Guided estimation with calculationsreferences/latency-numbers.md - Complete latency reference tableDate: 2025-12-26 Model: claude-opus-4-5-20251101