Back of the Envelope Estimations¶

A reference guide for quick calculations in system design interviews. These estimations help validate architecture decisions and identify bottlenecks.

Powers of 2¶

Power	Exact Value	Approx	Bytes
2^10	1,024	1 Thousand	1 KB
2^20	1,048,576	1 Million	1 MB
2^30	1,073,741,824	1 Billion	1 GB
2^40	1,099,511,627,776	1 Trillion	1 TB
2^50		1 Quadrillion	1 PB

Quick Conversions¶

1 KB = 1,000 bytes (for estimation, actually 1,024)
1 MB = 1,000 KB = 1 million bytes
1 GB = 1,000 MB = 1 billion bytes
1 TB = 1,000 GB = 1 trillion bytes
1 PB = 1,000 TB = 1 quadrillion bytes

Latency Numbers Every Programmer Should Know¶

Operation	Latency	Notes
L1 cache reference	0.5 ns
Branch mispredict	5 ns
L2 cache reference	7 ns	14x L1 cache
Mutex lock/unlock	25 ns
Main memory reference	100 ns	20x L2, 200x L1
Compress 1KB with Zippy	3 µs
Send 1KB over 1 Gbps network	10 µs
Read 4KB randomly from SSD	150 µs	~1 GB/sec SSD
Read 1MB sequentially from memory	250 µs
Round trip within same datacenter	500 µs
Read 1MB sequentially from SSD	1 ms	4x memory
HDD disk seek	10 ms	20x datacenter RT
Read 1MB sequentially from HDD	20 ms	80x memory, 20x SSD
Send packet CA → Netherlands → CA	150 ms

Key Takeaways¶

Memory is fast, disk is slow
SSD is 4-20x faster than HDD
Network within datacenter is fast (~500µs)
Cross-continent is slow (~150ms)
Avoid disk seeks when possible
Compress data before sending over network

Availability Numbers¶

Availability %	Downtime/Year	Downtime/Month	Downtime/Week
99% (two 9s)	3.65 days	7.31 hours	1.68 hours
99.9% (three 9s)	8.77 hours	43.83 minutes	10.08 minutes
99.99% (four 9s)	52.60 minutes	4.38 minutes	1.01 minutes
99.999% (five 9s)	5.26 minutes	26.30 seconds	6.05 seconds
99.9999% (six 9s)	31.56 seconds	2.63 seconds	0.60 seconds

Availability in Series vs Parallel¶

Series (both must work):

Availability = A1 × A2
Example: 99.9% × 99.9% = 99.8%

Parallel (either can work):

Availability = 1 - (1 - A1) × (1 - A2)
Example: 1 - (0.001 × 0.001) = 99.9999%

Time Conversions¶

Unit	Seconds	For Estimation
1 minute	60	~100
1 hour	3,600	~4,000
1 day	86,400	~100,000
1 week	604,800	~600,000
1 month	2,592,000	~2.5 million
1 year	31,536,000	~30 million

Quick Rules¶

Seconds in a day: ~100,000 (actually 86,400)
Seconds in a month: ~2.5 million
Seconds in a year: ~30 million

Traffic Estimations¶

Daily Active Users (DAU) to Requests¶

If 100M DAU, each user makes 10 requests/day:
= 100M × 10 = 1 billion requests/day
= 1B / 100,000 seconds = 10,000 RPS

General formula:
RPS = (DAU × requests_per_user) / 86,400

Peak vs Average¶

Peak traffic: Usually 2-3x average
Rule of thumb: Design for 3x average load

Storage Estimations¶

Common Object Sizes¶

Object	Size
UUID/GUID	16 bytes
Unix timestamp	4-8 bytes
Integer (32-bit)	4 bytes
Integer (64-bit)	8 bytes
Tweet (text only)	~300 bytes
Metadata per record	100-500 bytes
User profile (basic)	1 KB
Small image (thumbnail)	10-50 KB
Medium image	100-500 KB
Large image (HD)	1-5 MB
Short video (1 min)	5-10 MB
Long video (1 hour, compressed)	500 MB - 2 GB

Database Row Sizes (Typical)¶

Simple record (id, timestamps, few fields): 100-500 bytes
User profile: 1-5 KB
Product listing: 2-10 KB

Bandwidth Estimations¶

Formula¶

Bandwidth = (Data size × Requests) / Time

Example: 1 million requests/day, 100 KB each
= 1M × 100 KB / 86,400 sec
= 100 GB / 86,400 sec
≈ 1.16 MB/sec
≈ 10 Mbps

Common Bandwidths¶

Connection	Bandwidth
3G mobile	1-5 Mbps
4G mobile	10-50 Mbps
5G mobile	100-1000 Mbps
Home broadband	50-200 Mbps
Datacenter server	1-10 Gbps
AWS instance (typical)	1-25 Gbps

Common System Numbers¶

Platform	Metric	Value
Twitter/X	DAU	~250 million
Twitter/X	Tweets/day	~500 million
Facebook	DAU	~2 billion
Instagram	DAU	~500 million
YouTube	Videos watched/day	~5 billion
WhatsApp	Messages/day	~100 billion
Google	Searches/day	~8.5 billion

URL Shortener (like bit.ly)¶

New URLs/month: 100 million
Reads per write: 100:1
Total reads: 10 billion/month
Storage per URL: 500 bytes
Monthly storage: 100M × 500 = 50 GB

Chat Application¶

Messages/user/day: 40
Average message size: 100 bytes
1 million DAU:
- Messages/day: 40 million
- Storage/day: 4 GB
- Storage/year: 1.5 TB

Database Estimations¶

Read/Write Ratios (Typical)¶

Application	Read:Write
Social media feed	100:1 to 1000:1
E-commerce catalog	100:1
Chat application	1:1 to 5:1
Analytics/logging	1:10 to 1:100
User registration	1000:1

Database Capacity¶

Database	Practical Limit
Single MySQL	1-5 TB comfortable
Single PostgreSQL	1-10 TB comfortable
DynamoDB partition	10 GB
MongoDB single node	2-5 TB comfortable
Redis single node	25-50 GB practical

Query Performance¶

Operation	Typical Latency
Primary key lookup	1-5 ms
Indexed query	5-50 ms
Full table scan (1M rows)	100ms - seconds
Join across large tables	seconds

Caching Estimations¶

Cache Hit Ratio¶

Good: 80-90%
Excellent: 95%+
Poor: < 70%

Memory Sizing¶

If 20% of data is accessed 80% of the time:
Cache size = 20% of total data size

Example: 100 GB database
Effective cache: 20 GB

Cache Performance¶

Cache	Latency	Throughput
In-process (local)	1-10 µs	millions/sec
Redis/Memcached	0.5-1 ms	100K-1M ops/sec
CDN edge cache	10-100 ms	varies

Estimation Framework¶

Step-by-Step Approach¶

Clarify requirements
Read-heavy or write-heavy?
Consistency requirements?
Latency requirements?
Estimate scale
DAU/MAU
Requests per user
Data per request

Calculate QPS (Queries Per Second)

Average QPS = DAU × queries_per_user / 86,400
Peak QPS = Average QPS × 3

Calculate storage

Daily storage = requests × data_per_request
Yearly storage = daily × 365
5-year storage = yearly × 5

Calculate bandwidth
```
Bandwidth = QPS × data_size
```

Estimate servers needed

Servers = QPS / queries_per_server
(Typical web server: 1,000-10,000 QPS)

Quick Reference Formulas¶

QPS Calculation¶

QPS = (DAU × actions_per_day) / 86,400
Peak QPS = QPS × 2 to 3

Storage Calculation¶

Storage = users × data_per_user × retention_period
or
Storage = requests × data_per_request × retention_period

Bandwidth Calculation¶

Bandwidth (bytes/sec) = QPS × average_request_size
Bandwidth (Mbps) = Bandwidth (bytes/sec) × 8 / 1,000,000

Servers Needed¶

Web servers = Peak QPS / 10,000 (conservative)
Database connections = Servers × connections_per_server

Example: Design Twitter¶

Requirements¶

300 million MAU, 150 million DAU
Each user: 2 tweets/day, reads 100 tweets/day
Tweet size: 300 bytes + metadata = 500 bytes
10% tweets have image (200 KB avg)

Calculations¶

Write QPS:

Tweets/day = 150M × 2 = 300 million
Write QPS = 300M / 86,400 ≈ 3,500 QPS
Peak Write QPS ≈ 10,000 QPS

Read QPS:

Reads/day = 150M × 100 = 15 billion
Read QPS = 15B / 86,400 ≈ 170,000 QPS
Peak Read QPS ≈ 500,000 QPS

Storage (1 year):

Text: 300M tweets/day × 500 bytes × 365 = 55 TB/year
Images: 30M images/day × 200 KB × 365 = 2.2 PB/year
Total: ~2.3 PB/year

Bandwidth:

Read: 500K QPS × 500 bytes = 250 MB/sec text
Plus images: significant additional bandwidth

Example: Design URL Shortener¶

Requirements¶

100 million new URLs/month
Read:Write = 100:1
URL stored for 5 years
Short URL: 7 characters

Calculations¶

Write QPS:

100M / (30 × 86,400) ≈ 40 QPS
Peak: 80 QPS

Read QPS:

40 × 100 = 4,000 QPS
Peak: 8,000 QPS

Storage (5 years):

URLs = 100M × 12 months × 5 years = 6 billion URLs
Storage = 6B × 500 bytes = 3 TB

Short URL combinations:

7 chars, base62 = 62^7 = 3.5 trillion combinations
More than enough for 6 billion URLs

Common Mistakes to Avoid¶

Forgetting peak traffic - Always design for 2-3x average
Ignoring read/write ratio - Caching strategies depend on this
Underestimating metadata - Indexes, timestamps add up
Forgetting replication - 3x storage for replication
Ignoring network overhead - Headers, protocols add bytes
Not considering growth - Plan for 2-5 years ahead
Precise calculations - Round numbers are fine in interviews

Cheat Sheet¶

Seconds/day:     ~100K (86,400)
Seconds/month:   ~2.5M
Seconds/year:    ~30M

1 KB = 1,000 bytes
1 MB = 1,000 KB
1 GB = 1,000 MB  = 1 billion bytes
1 TB = 1,000 GB  = 1 trillion bytes

Typical web server: 1K-10K QPS
Typical DB server:  1K-10K QPS
Redis/Memcached:    100K+ ops/sec

Memory access:      100 ns
SSD random read:    150 µs
HDD seek:           10 ms
Network RT (DC):    500 µs
Network RT (cross-continent): 150 ms

Back of the Envelope Estimations¶

Powers of 2¶

Quick Conversions¶

Latency Numbers Every Programmer Should Know¶

Key Takeaways¶

Availability Numbers¶

Availability in Series vs Parallel¶

Time Conversions¶

Quick Rules¶

Traffic Estimations¶

Daily Active Users (DAU) to Requests¶

Peak vs Average¶

Storage Estimations¶

Common Object Sizes¶

Database Row Sizes (Typical)¶

Bandwidth Estimations¶

Formula¶

Common Bandwidths¶

Common System Numbers¶

Social Media Scale (Reference)¶

URL Shortener (like bit.ly)¶

Chat Application¶

Database Estimations¶

Read/Write Ratios (Typical)¶

Database Capacity¶

Query Performance¶

Caching Estimations¶

Cache Hit Ratio¶

Memory Sizing¶

Cache Performance¶

Estimation Framework¶

Step-by-Step Approach¶

Quick Reference Formulas¶

QPS Calculation¶

Storage Calculation¶

Bandwidth Calculation¶

Servers Needed¶

Example: Design Twitter¶

Requirements¶

Calculations¶

Example: Design URL Shortener¶

Requirements¶

Calculations¶

Common Mistakes to Avoid¶

Cheat Sheet¶