Systems and Delays (2026)

I’d be interested in learning what, mathematically, makes these oscillations inevitable

Say you model your inventory as

df/dt = −f(t − τ) / r

Where τ is the delay, and r is the response delay divisor, like your post.

A standard result is that the first-order delay differential equation dx/dt = −ax(t − τ) is stable when aτ < π/2. (e.g. see corollary 3.3 in Stépán's Retarded Dynamical Systems, https://www.mm.bme.hu/~stepan/mm/book/retarded_dynamical_systems.pdf)

With a = 1/r that gives τ/r < π/2, i.e., r > 2τ/π

Applying that to your examples, if τ = 5, you need r > 10/π ~ 3.18

So the system ends up stable when r = 6, but not with r = 2 or r = 1.

Nice writeup, I love little things with emergent behavior like this.

I can't explain it with math, but it seems intuitive to me that if you're trying to work faster than the system can respond to you, you're going to oscillate (i.e. the system responds to your orders by delivering cars in n days but you make orders every day)

I'm also curious how it'd work if cars-ordered-but-not-delivered counted towards behavior. Or if orders only happened every few days. There's a lot of parameters here.