Back of the Envelope Calculation

1️⃣ What is Back-of-the-Envelope Calculation?

Back-of-the-envelope (BOE) calculations are quick, rough estimates used to evaluate the feasibility of a system before designing it in detail. These help in estimating storage, bandwidth, memory, and processing needs efficiently.

Abstract

🌐 📋 Rapid sizing techniques for HLD interviews: approximate scale, throughput, storage, memory, and compute before diving into detailed architecture.

2️⃣ Understanding Data Size Conversions

Before diving into calculations, let’s define unit conversions to avoid confusion:

Data Size Units

Unit 📏	Equivalent Value
1 KB (Kilobyte)	1,000 bytes = 10³ bytes
1 MB (Megabyte)	1,000 KB = 10⁶ bytes
1 GB (Gigabyte)	1,000 MB = 10⁹ bytes
1 TB (Terabyte)	1,000 GB = 10¹² bytes
1 PB (Petabyte)	1,000 TB = 10¹⁵ bytes
1 EB (Exabyte)	1,000 PB = 10¹⁸ bytes

Large Number Notations

Name	Value	Scientific Notation	Zeros
Thousand	1,000	10³	1,000
Million	1,000,000	10⁶	1,000,000
Billion	1,000,000,000	10⁹	1,000,000,000
Trillion	1,000,000,000,000	10¹²	1,000,000,000,000
Quadrillion	1,000,000,000,000,000	10¹⁵	1,000,000,000,000,000
Quintillion	1,000,000,000,000,000,000	10¹⁸	1,000,000,000,000,000,000

Quick Reference for Calculations

• 1 Million = 1M = 10⁶
• 1 Billion = 1B = 10⁹
• 1 Trillion = 1T = 10¹²

📌 Example:
A 5 MB file = 5 × 1,000 × 1,000 bytes = 5,000,000 bytes (5 million bytes).

3️⃣ Traffic Estimation (Requests Per Second - RPS) ⚡

Estimating the number of requests a system handles helps in designing load balancers, databases, and caching systems.

Formula:

Total RPS = Total Users * Active Users Ratio * Requests per Active User per Second

Example Calculation:

🔹 Given:

• A system has 100 million registered users.
• 10% are active daily → 10 million active users.
• Each user makes 10 requests per minute.

🔹 Calculation:

• Requests per second per user = (10/60 = 0.167) requests per second.
• Total RPS:

  10,000,000 * 0.167 = 1,670,000 { RPS (1.67 million requests/sec)}

4️⃣ Storage Estimation 🏦

Estimating storage helps in planning databases, replication, and backup strategies.

Formula:

Total Storage per Year = Data per Request * Total Requests per Year

Example Calculation:

🔹 Given:

• Each request generates 2 KB of data.
• System handles 1.67 million RPS.
• Requests per year:

  1.67 * 10^6 * 60 * 60 * 24 * 365

  = 52.6 * 10^{12} { (52.6 trillion requests per year)}

🔹 Storage Required:

52.6 * 10^{12} * 2 { KB} = 105.2 { PB per year}

🔹 Considering Replication & Backups:

• If 3x replication, total storage = 315.6 PB per year.
• If backups are kept for one year, total storage needed = 631.2 PB.

📌 Scale Reference:

• 1 PB = 1,000 TB
• 1 PB ≈ Storage of 223 million HD movies (4.5 GB each)

5️⃣ Memory Estimation (Cache & DB Indexing) 🛠️

Determining cache size ensures efficient system performance.

Cache Size Estimation Formula:

Cache Size = Data per Object * Objects Stored in Cache

Example Calculation:**

🔹 Given:

• 20% of requests are cached.
• Each cached response = 2 KB.
• System handles 1.67M RPS.

🔹 Calculation:

• Data in cache per second =

  1.67 * 10^6 * 0.2 * 2 = 670 { MB per second}

• Data stored in cache for 1 hour:

  670 * 3,600 = 2.41 { TB}

• Total RAM required ≈ 2.5 TB.

6️⃣ Quick Cheat Sheet for BOE Calculations 📌

Metric 🛠️	Formula
Requests Per Second (RPS)	Users × Active Ratio × Requests per User
Storage Per Year	RPS × Data per Request × Seconds in a Year
Cache Size	Cache % × RPS × Data per Object × Cache Retention Time
Bandwidth	RPS × Data per Response
CPU Servers Needed	(RPS × CPU Time per Request) / (Cores per Server × Utilization)

7️⃣ CPU Server Estimation 🖥️

Calculating required servers is crucial for capacity planning and cost estimation.

Formula:

Number of Servers = (RPS × CPU time per request) / (Cores per server × Core utilization × Seconds)

Example Calculation:

🔹 Given:

• System handles 1.67M RPS
• Each request takes 50ms (0.05s) of CPU time
• Each server has 32 cores
• Target CPU utilization: 80% (0.8)

🔹 Calculation:

Servers = (1,670,000 × 0.05) / (32 × 0.8 × 1)
        = 83,500 / 25.6
        ≈ 3,262 servers

Server Capacity Planning Matrix

Request Complexity	CPU Time	Max RPS per Core
Light (cache hit)	1ms	1,000
Medium (DB read)	10ms	100
Heavy (DB write/compute)	50ms	20
Very Heavy (complex computation)	100ms	10

Example with Buffers:

Base servers needed = 3,262
Peak load buffer (2x) = 6,524
Growth buffer (50%) = 9,786
Redundancy (30%) = 12,722 servers