BUS505 · Quantitative Business Analysis ESLSCA · MBA · Dr. Lobna S. Ahmed
Group Project · April 2026

Queue-Based
Capacity Analysis &
Decision Tree
Optimization

An EdTech video-authorization API under exam-night traffic — and which of four scaling plans wins on expected value, risk, and stability.

Two models · Sixteen scenarios · One recommendation PRESS → TO BEGIN
§1 · The Setting 02 / 32
Production reality

2,600 students. One play button. Seven sequential checks before a video starts.

Every "Watch Lesson" tap fires a Laravel/PHP-FPM API call: token validation, subscription, device session, lesson access, log write, signed-URL generation, response. Nine PM on exam-prep night, all 2,600 hit play within a fifteen-minute window.

The Stack
VPS8 vCPU · 32 GB RAM
OSUbuntu 24.04
WebNginx · HLS
AppLaravel · PHP-FPM
DB / CacheMySQL 8 · Redis
Workersk = 40
Single-VPS architecture · shared resources EdTech · Egypt · 9 PM exam window
§1.2 · The Bottleneck 03 / 32
Capacity overshoot
Capacity
72.8 req/s

k · μ = 40 × 1.82

Arrival rate
120 req/s

λ during exam spike

Overshoot
+65 %

Queue grows without bound

When λ exceeds kμ, mobile clients time out, students retry, and arrivals climb to ~180 req/s. The system enters a retry-storm failure mode.

Anderson §11.3 · stability requires kμ > λ Source: production telemetry, Q1 2026
§1.3 · Decision Problem 04 / 32
The question

Engineering offers four scaling plans. Which is right?

A
Keep current
No change.
Hope for the best.
k = 40 · μ = 1.82
+0 EGP/mo
B
Optimize only
Redis cache, indexes, async logs, OPcache.
k = 45 · μ = 4.00
+3,000 EGP/mo
C
Add 2nd server
Horizontal scale-out behind a load balancer.
k = 80 · μ = 1.82
+6,500 EGP/mo
Optimal
D
Hybrid
Optimize and add 2nd server.
k = 90 · μ = 4.00
+9,500 EGP/mo
Four alternatives · 0 → 9,500 EGP/month spread Defend the choice quantitatively
Methodology 05 / 32
Two models, one answer

Feasibility first, then cost.

01
Anderson Ch. 11
M/M/k Waiting Line

Computes seven operating characteristics — P₀, Lq, L, Wq, W, Pw, Pn — for each of 4 decisions × 4 traffic states.

Output: which architectures are technically feasible.

02
Anderson Ch. 13
Decision Tree + EV

Backward-pass expected value across 4 decisions and 4 states; risk profile and EVPI; sensitivity in 3 dimensions.

Output: which feasible option minimizes expected cost.

Both classroom models · applied to one production decision Models converge → Decision D
Part 1 of 2 06 / 32
Model 01

Waiting
Lines

Anderson §11 · M/M/k multi-channel · the math that decides which servers stay up.

Seven operating characteristics · sixteen scenarios P₀ · Lq · L · Wq · W · Pw · Pn