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Boundedness plus hysteresis

How can two identical systems under identical load be in different states?

Feedback loops gave you gain. Gain alone does not explain outages that outlive their cause. An unbounded loop with R > 1 runs to destruction and ends there: out-of-memory, crash, exhaustion. What this article adds is boundedness plus .

Hysteresis means that the state of a system depends on where it has been, not only on its present inputs (a light switch that stays where you put it is the simplest example).

When the loop's gain is above 1 but the amplification is capped, for example by a finite number of clients, a finite number of retries, or a , the system does not destroy itself. It settles. It settles into a second equilibrium that is stable, self-sustaining and useless.

The consequence: Two identical systems under identical load can be in different states, permanently, depending only on what happened to them an hour ago. Every debugging instinct you have assumes that the input determines the output.

The hidden-state principle. A system whose per-unit cost depends on its own recent state has memory, and memory means identical inputs produce non-identical outputs. Cache warmth, depth, heap occupancy, connection warmth, JIT state, debt, buffer-pool contents: each is a hidden state variable that no input-output test can reveal.

  • Operationally: entry and exit thresholds differ, so any control loop over such a system needs two thresholds, not one. A with a single threshold will chatter (switch on and off rapidly).
  • Experimentally: a measurement can be reproduced only if the hidden state is specified. That is why "cold versus warm" must be a declared condition of any benchmark, not an accident.
  • Predictive form: wherever per-request cost varies with recent history, expect hysteresis, and expect the system to be untestable by replaying inputs alone.

A system has two answers to "what does it do at this load?", and which one you get depends on where it has been.