Generated with the help of Opus 4.8 [xhigh].

A toy model, not a benchmark. It exists to build intuition for statistical multiplexing: when does pooling many bursty workloads let you run the pool near 100% utilization, and when does it not. Everything below is per simulated day of T = 240 ticks.

One workload's demand

Each workload i draws a demand curve that mixes its own independent bursts with a shared common-mode signal. The --corr knob is the shared-signal weight ρ:

dᵢ(t) = (1−ρ)·indepᵢ(t) + ρ·common(t)

indepᵢ(t) is fresh per workload; common(t) is shared by everyone — so a high ρ means workloads spike together. Both are Pareto draws normalized to mean 1. Because the tails can have infinite variance, ρ is a mixing weight, not a literal Pearson correlation coefficient.

How bursty (--spikiness)

Bursts are heavy-tailed Pareto with tail index set by spikiness s:

α = 1.3 + 1.7·(1−s) + 70·(1−s)^8 (s=1 → α=1.3 heavy / rare tall spikes; s=0 → α=73 nearly flat) xₘ = (α−1)/α chosen so each draw has expected mean 1

Workload sizes (--size-skew)

Each workload also has a size wᵢ (its share of total load). With skew k = 0 all sizes are equal; with k > 0 they're Pareto:

αₛ = 1.06 + 2.6·(1−k) (smaller αₛ = a few elephants dominate)

When 1 < αₛ < 2 the mean exists but variance is infinite. The largest tenant's share still trends toward zero, but only as N^(1/αₛ − 1), so near αₛ = 1 it can dominate practical fleet sizes.

The fleet aggregate

For a fleet of N workloads the total demand and the headline metric are:

D(t) = Σᵢ wᵢ·dᵢ(t) peak/mean = maxₜ D(t) / meanₜ D(t) (1× = perfectly flat = run at 100%)

One floor and one finite-size estimate

• Correlation floor = maxₜ E[d(t)] / meanₜ E[d(t)] — the shape of the expected demand. Independent noise averages out; the shared common-mode does not.
• Indivisible estimate = 1 + topShare(N)·(burst − 1), where topShare(N) is the largest single workload's share of the total and burst is one workload's own peak/mean over the simulated day. It assumes the rest of the fleet is near its mean when the largest workload peaks, so it is a heuristic pressure line rather than a hard lower bound.

The extrapolated curve is max(correlation-decay, indivisible estimate).

Amortization with scale

Independent variation cancels as workloads stack, so the excess over the correlation floor decays as a power law fit from the simulation:

peak/mean(N) ≈ correlationFloor + C·N^(−p) p ≈ 0.5 for light tails, smaller for heavy / size-skewed fleets

How it's actually computed

• Simulate a representative sample of N_SAMPLE = 4096 workloads with a fixed seed.
• Measure peak/mean at growing counts, fit C and p, extrapolate to 100M. When the selected fleet is within the 4096-workload sample, the status line and marker use the exact simulated aggregate instead of the fit.
• topShare(N) past the sample is extrapolated from the size tail index (~ N^(1/αₛ − 1)) — directional, a model, not a measurement.

Caveats

• Sizes are independent of shape; ticks are independent rather than long-lived sessions; one synthetic day; no placement/bin-packing — just the statistical limit.
• Presets are illustrative parameter choices, not measured from real systems.

Further reading

• Warfield, "Building and operating a pretty big storage system (S3)" (2023) — the "managing heat" argument this sandbox is built around.
• Statistical multiplexing (Wikipedia) — the pooling-gain idea: sharing capacity across bursty users beats static allocation.
• Pareto distribution (Wikipedia) — the heavy-tailed law behind both the bursts and the workload sizes here.
• Central limit theorem (Wikipedia) — why independent variation averages out as ~1/√N when you pool many workloads.
• Heavy-tailed distribution (Wikipedia) — when variance is infinite (α<2) the largest terms can dominate practical fleet sizes and convergence is slower than 1/√N.
• Leland, Taqqu, Willinger & Wilson, "On the Self-Similar Nature of Ethernet Traffic" (1994) — empirical evidence that real traffic really is bursty and heavy-tailed.