267. Delays

Time lags between cause and effect. Critical for explaining oscillation and instability in feedback systems.

267.1. 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).

267.2. Pipeline (pure) delay

Constant time lag $𝐷$:

Output is just input shifted by $𝐷$. Variance preserved.

267.3. First-order exponential delay

A stock $𝑆$ holds material in transit; outflow rate is $\frac{𝑆}{𝐷}$. Mean delay $𝐷$, but each unit’s transit time is random (exponentially distributed). High variance.

267.4. Higher-order Erlang delays

Cascade $𝑛$ first-order delays in series, each of length $\frac{𝐷}{𝑛}$:

Output = ${𝑆}_{𝑛}\cdot \left(\frac{𝑛}{𝐷}\right)$.

The impulse response is Erlang-distributed: mean $𝐷$, variance $\frac{{𝐷}^2}{𝑛}$. As $𝑛\to \infty$, variance $\to 0$ and the cascade approaches a pure pipeline delay.

In Vensim: DELAY1 (order 1), DELAY3 (order 3), DELAYN (order n).

267.5. 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 $𝜏\gg$ delay, smooth approach. When $𝜏\approx$ delay, oscillation. When $𝜏\ll$ delay, instability.

267.6. 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

267.7. See also

• Feedback Loops
• Smoothing Operators
• Second-Order Systems — formal oscillation analysis
• Beer Game