💰 AWS Cost Optimization: Mathematical Models and Formulas

Understanding the mathematical foundations behind AWS cost optimization can help you make better decisions about resource allocation and scaling strategies.

Cost Models

Linear Cost Model

For services with linear pricing (like data transfer), the cost function is:

\[C(x) = a \cdot x + b\]

Where:

  • $C(x)$ = Total cost
  • $x$ = Usage amount
  • $a$ = Cost per unit
  • $b$ = Fixed cost

Reserved Instance Savings

The break-even point for Reserved Instances can be calculated as:

\[T_{breakeven} = \frac{C_{upfront}}{C_{ondemand} - C_{reserved}}\]

Where:

  • $T_{breakeven}$ = Time to break even (in months)
  • $C_{upfront}$ = Upfront payment for reserved instance
  • $C_{ondemand}$ = Monthly cost of on-demand instance
  • $C_{reserved}$ = Monthly cost of reserved instance

Auto Scaling Optimization

For optimal auto-scaling, we want to minimize:

\[\min_{n} \left[ C_{compute}(n) + C_{latency}(n) \right]\]

Subject to:

  • $n \geq n_{min}$ (minimum instances)
  • $n \leq n_{max}$ (maximum instances)
  • $L(n) \leq L_{target}$ (latency constraint)

Storage Cost Analysis

S3 Storage Classes

The optimal storage class selection follows:

\[\text{Class}_{optimal} = \arg\min_{c \in \{S3, IA, Glacier\}} \left[ C_{storage}^c + C_{retrieval}^c \cdot R \right]\]

Where:

  • $C_{storage}^c$ = Storage cost for class $c$
  • $C_{retrieval}^c$ = Retrieval cost for class $c$
  • $R$ = Expected retrieval frequency

Data Transfer Costs

Cross-region data transfer costs follow:

\[C_{transfer} = \sum_{i,j} D_{i,j} \cdot P_{i,j}\]

Where:

  • $D_{i,j}$ = Data transferred from region $i$ to region $j$
  • $P_{i,j}$ = Price per GB for transfer from region $i$ to region $j$

Lambda Cost Optimization

Memory vs Duration Trade-off

For Lambda functions, the optimal memory allocation minimizes:

\[C_{total} = \frac{M}{1024} \cdot D \cdot P_{gb-second} \cdot N\]

Where:

  • $M$ = Memory allocation (MB)
  • $D$ = Duration (seconds)
  • $P_{gb-second}$ = Price per GB-second
  • $N$ = Number of invocations

The relationship between memory and duration often follows:

\[D(M) = \frac{k}{M^{\alpha}}\]

Where $k$ and $\alpha$ are function-specific constants.

Practical Applications

Example: EC2 Right-sizing

Given current utilization $U_{current}$ and target utilization $U_{target}$, the optimal instance size is:

\[\text{Size}_{optimal} = \text{Size}_{current} \cdot \frac{U_{current}}{U_{target}}\]

Example: RDS Storage Optimization

For RDS, the total cost including IOPS is:

\[C_{RDS} = C_{storage} + C_{IOPS} + C_{backup}\] \[C_{RDS} = S \cdot P_{storage} + I \cdot P_{IOPS} + B \cdot P_{backup}\]

Conclusion

These mathematical models provide a foundation for making data-driven decisions about AWS cost optimization. By understanding the underlying formulas, you can:

  1. Build automated cost optimization tools
  2. Make informed decisions about resource allocation
  3. Predict cost impacts of architectural changes
  4. Optimize for specific business constraints

Remember that these models are simplified representations. Real-world scenarios often involve additional factors and constraints that should be considered in your optimization strategies.

Next post: Implementing these formulas in practice with AWS Cost Explorer APIs