# Availability Arithmetic — Scalability, Availability & Reliability

Source: https://www.skillbyai.com/en/scalability/a-math

> Calculate composite availability for components in series and in parallel.

## Multiply for chains, add redundancy for parallel

When a request needs **every** component in a chain (load balancer → app → database), the components are **in series** and availabilities **multiply**: three components at 99.9% each give about 99.7%, lower than any single part. Every synchronous dependency you add lowers the ceiling. When components are **in parallel** (redundant copies, any one of which can serve), the system fails only if **all** fail: availability = 1 − (1 − a)^n, assuming independent failures. Two copies at 99% give 99.99%. That independence assumption is the catch: copies in the same zone, with the same bug or the same bad config push, fail together, so real gains are smaller. The arithmetic explains common advice: keep critical paths short, make non-critical dependencies optional (degrade gracefully instead of failing), and put redundancy where failures are truly independent.

## Composite availability calculations

Series multiplies; parallel combines failure probabilities.

```python
def series(*avail):
    total = 1.0
    for a in avail:
        total *= a
    return total

def parallel(a, n):
    return 1 - (1 - a) ** n

chain = series(0.9999, 0.999, 0.999)         # LB, app tier, database
# about 0.9979 -> roughly 99.8%

two_db = parallel(0.99, 2)                   # two independent replicas
# 0.9999 -> 99.99%

improved = series(0.9999, parallel(0.999, 3), parallel(0.999, 2))
# redundancy at each tier lifts the whole chain
```

## Optional dependencies should not count

If recommendations fail, show the page without them. A dependency that can fail without failing the request drops out of the series calculation entirely.

**Quiz:** Three components are each 99.9% available and all are required for a request. What is the approximate overall availability?

- [ ] 99.9%
- [ ] 99.97%
- [x] About 99.7%
- [ ] 99.999%

*Answer:* About 99.7%. 0.999³ ≈ 0.997, so about 99.7%.