265. Delays
Time lags between cause and effect. Critical for explaining oscillation and instability in feedback systems.
265.1. Three kinds
Material delays: physical movement / transformation takes time (shipping, manufacturing, aging).
Information delays: perception / measurement lags (reporting, accounting cycles).
Decision delays: time between observing data and acting (administrative).
265.2. Pipeline (pure) delay
Constant time lag :
Output is just input shifted by . Variance preserved.
265.3. First-order exponential delay
A stock holds material in transit; outflow rate is . Mean delay , but each unit’s transit time is random (exponentially distributed). High variance.
265.4. Higher-order Erlang delays
Cascade first-order delays in series, each of length :
Output = .
The impulse response is Erlang-distributed: mean , variance . As , variance and the cascade approaches a pure pipeline delay.
| Behavior | |
|---|---|
| 1 | Pure first-order (high variance, smooth) |
| 3 | Visible peak, moderate spread (typical SD default) |
| 10 | Tight pulse, low variance |
| ∞ | Pipeline delay (no spread) |
In Vensim: DELAY1 (order 1), DELAY3 (order 3), DELAYN (order n).
265.5. Why delays drive oscillation
Imagine a thermostat that responds late. The room overheats; the system finally responds, cooling too much; oscillation. The delay turns a balancing loop into an oscillator.
Generic structure:
When delay, smooth approach. When delay, oscillation. When delay, instability.
265.6. Common SD examples
- Beer game: each echelon has ordering + shipping delays → bullwhip oscillation
- Capacity planning: long construction lead times → boom-bust cycles
- Workforce: hiring + training delay → over/under-staffing oscillation
- Public policy: legislation lag → policy ineffectiveness
265.7. See also
- Feedback Loops
- Smoothing Operators
- Second-Order Systems — formal oscillation analysis
- Beer Game