cheat sheets / formulas · 3-min read
Performance Formula Sheet
Every formula on perfday on one printable page.
Concurrency and queueing
| Little's Law | L = λ · W |
|---|---|
| Utilisation | ρ = λ · S |
| M/M/1 response time | W = S / (1 − ρ) |
| M/M/1 queue wait | Wq = ρ · S / (1 − ρ) |
| One lock's max rate | ≈ 1 / hold time |
Scaling
| Amdahl's Law | S = 1 / ((1 − p) + p / N) |
|---|---|
| Amdahl ceiling | 1 / (1 − p) |
| p from a measurement | p = (1 − 1/S) / (1 − 1/N) |
| Gustafson's Law | S = N − s · (N − 1) |
| USL throughput | X(N) = X(1) · N / (1 + σ(N − 1) + κN(N − 1)) |
| USL peak | N* = √((1 − σ) / κ) |
Latency
| Nearest-rank percentile | value at position ⌈q · n⌉ of sorted samples |
|---|---|
| Samples for pq | n ≈ 10 / (1 − q) |
| Fan-out tail | P = 1 − (1 − q)^N |
| Per-backend target | q = 1 − (1 − P)^(1/N) |
| Effective latency | t = h · t_hit + (1 − h) · t_miss |
Capacity and reliability
| Time to a limit | months = ln(limit / load) / ln(1 + g) |
|---|---|
| Doubling time | ≈ 70 / growth % |
| Error budget | (1 − SLO) × window |
| Burn rate | actual spend rate ÷ allowed rate |
| Throughput | ops/s × size per op · 1 Gbit/s = 125 MB/s |
| Queue bound | max queue = drain rate × latency budget |
Sources
Each formula links to its calculator or reference page, where the primary sources are listed.