Back-of-the-envelope Estimation

Learn to estimate system capacity and performance requirements in system design interviews using back-of-the-envelope calculations.

January 3, 2023

Power of Two for Data Volume

Understanding data volume units based on powers of two is crucial for distributed systems. A byte consists of 8 bits. An ASCII character typically uses one byte. The table below illustrates common data volume units:

Power	Approximate Value	Full Name	Short Name
10	1 Thousand	1 Kilobyte	1 KB
20	1 Million	1 Megabyte	1 MB
30	1 Billion	1 Gigabyte	1 GB
40	1 Trillion	1 Terabyte	1 TB
50	1 Quadrillion	1 Petabyte	1 PB

Latency Numbers for Programmers

Familiarity with typical computer operation latencies is beneficial. While some numbers may be updated due to technological advancements, they provide a general understanding of performance differences:

Operation Name	Time
L1 cache reference	0.5 ns
Branch mispredict	5 ns
L2 cache reference	7 ns
Mutex lock/unlock	100 ns
Main memory reference	100 ns
Compress 1KB with Zippy	10,000 ns = 10 µs
Send 2KB over 1 Gbps network	20,000 ns = 20 µs
Read 1MB sequentially from memory	250,000 ns = 250 µs
Round trip within same datacenter	500,000 ns = 500 µs
Disk seek	10,000,000 ns = 10 ms
Read 1MB sequentially from network	10,000,000 ns = 10 ms
Read 1MB sequentially from disk	30,000,000 ns = 30 ms
Send packet CA -> Netherlands -> CA	150,000,000 ns = 150 ms

Notes:

ns = nanosecond (10^-9 seconds)
µs = microsecond (10^-6 seconds = 1,000 ns)
ms = millisecond (10^-3 seconds = 1,000 µs = 1,000,000 ns)

Key takeaways from latency analysis:

Memory operations are fast; disk operations are slow.
Minimize disk seeks.
Simple data compression is efficient.
Compress data before network transmission when possible.
Inter-datacenter communication introduces significant latency.

Availability Metrics

High availability ensures a system remains operational for a desired period. It is measured as a percentage, with 100% indicating zero downtime. Most services aim for 99% or higher.

A Service Level Agreement (SLA) is a formal commitment between a service provider and a customer regarding uptime. Cloud providers typically set SLAs at 99.9% or above. Uptime is often expressed in “nines,” with more nines signifying less downtime. The table below shows the correlation between availability percentage and expected downtime:

Availability %	Downtime per Day	Downtime per Week	Downtime per Month	Downtime per Year
99%	14.40 minutes	1.68 hours	7.31 hours	3.65 days
99.99%	8.64 seconds	1.01 minutes	4.38 minutes	52.60 minutes
99.999%	864.00 ms	6.05 seconds	26.30 seconds	5.26 minutes
99.9999%	86.40 ms	604.80 ms	2.63 seconds	31.56 seconds

Example: Estimating Twitter’s QPS and Storage

(Note: The following figures are for illustrative purposes only and do not represent actual Twitter data.)

Assumptions:

300 million monthly active users.
50% of users are daily active.
Users post an average of 2 tweets per day.
10% of tweets include media.
Data retention: 5 years.

Estimations:

Daily Active Users (DAU): 300 million * 50% = 150 million.
Tweets QPS (Queries Per Second): (150 million users * 2 tweets/user) / (24 hours/day * 3600 seconds/hour) ≈ 3500 tweets/second.
Peak QPS: 2 * Tweets QPS = 7000 tweets/second.

Media Storage Estimation:

Average Tweet Size Components:
- tweet_id: 64 bytes
- text: 140 bytes
- media: 1 MB
Daily Media Storage: 150 million users * 2 tweets/user * 10% (with media) * 1 MB/media = 30 TB/day.
5-Year Media Storage: 30 TB/day * 365 days/year * 5 years ≈ 55 PB.

Tips for Back-of-the-Envelope Estimation

Focus on the process, not just the exact result. Interviewers assess your problem-solving approach:

Rounding and Approximation: Simplify calculations. For example, 99987 / 9.1 can be approximated as 100,000 / 10. Precision is not the primary goal.
Document Assumptions: Clearly state all assumptions you make during the estimation process.
Label Units: Always specify units (e.g., “5 MB” instead of “5”) to avoid ambiguity.
Practice Common Estimations: Familiarize yourself with estimating QPS, peak QPS, storage, cache, and server counts. Practice is key.

Based on your screenshot, you are using Vercel Analytics for your Astro-based personal blog, but you are not seeing any visitors or page views being tracked.

Here’s a checklist to help you troubleshoot and get analytics working:

1. Install the Analytics Package

Make sure you have installed the Vercel Analytics package in your project:

npm install @vercel/analytics

yarn add @vercel/analytics

2. Add the Analytics Component

You need to import and use the <Analytics /> component in your Astro project, typically in your main layout (e.g., src/layouts/BaseLayout.astro or similar):

---
import { Analytics } from '@vercel/analytics/react';
---
<html>
  <body>
    <!-- ... your content ... -->
    <Analytics />
  </body>
</html>

If you are using Astro 3.x or later, you may need to use the Astro integration for Vercel Analytics.

3. Deploy Your Site

Analytics only works on your deployed site (not locally). Make sure you have deployed your site to Vercel and are visiting the production URL (e.g., https://your-site.vercel.app).

4. Visit Your Site

After deployment, visit your site in an incognito/private window to generate some page views. Wait a few minutes and refresh the analytics dashboard.

5. Check for Ad/Content Blockers

Some browser extensions (like uBlock Origin, AdBlock, etc.) can block analytics scripts.
Try disabling these extensions or use a different browser.

6. Check Console for Errors

Open your browser’s developer console on your deployed site and look for any errors related to analytics.

7. Verify Environment

Make sure you are looking at the correct environment in the Vercel dashboard (Production, Preview, etc.).

8. Astro SSR vs. Static

If you are using Astro in static mode, the analytics script should still work, but if you have custom server logic, ensure nothing is interfering with the analytics script.

If You Still Don’t See Data

Double-check the deployment: Make sure the latest code (with <Analytics /> added) is deployed.
Try a hard refresh on your site and the Vercel dashboard.
Check Vercel documentation: Vercel Analytics for Astro

If you want, I can check your layout file to ensure the <Analytics /> component is correctly added. Would you like to share your main layout file (e.g., BaseLayout.astro)?