Linking Equation

352. Linking Equation

A simple but powerful approximation for how variability propagates through tandem queues. The departure-process variability from one station is the arrival variability for the next.

352.1. Formula

For a single station with utilization $𝜌$ , arrival CoV $𝑐_{𝑎}$ , and service CoV $𝑐_{𝑠}$ , the departure CoV $𝑐_{𝑑}$ is:

𝑐_{𝑑}^{2} = (1 - 𝜌^{2}) 𝑐_{𝑎}^{2} + 𝜌^{2} 𝑐_{𝑠}^{2}

— a weighted average of arrival and service variability, weighted by $𝜌^{2}$ and $1 - 𝜌^{2}$ .

352.2. Two extreme regimes

Low utilization ( $𝜌 \to 0$ ): $𝑐_{𝑑} \approx 𝑐_{𝑎}$ . The station is mostly idle; departures inherit the arrival pattern.

High utilization ( $𝜌 \to 1$ ): $𝑐_{𝑑} \approx 𝑐_{𝑠}$ . The station is always busy; departures inherit the service-time pattern.

In between: blend.

352.3. Tandem queues: variance flows downstream

Tandem layout:

Station 1 → Station 2 → Station 3 → ... → Station N

Arrival CoV at station 2 = departure CoV from station 1 = $𝑐_{𝑑}^{(1)}$ .

Applying the linking equation repeatedly:

𝑐_{𝑎}^{(𝑘 + 1)} = 𝑐_{𝑑}^{(𝑘)}

So variance propagates downstream. Stations near the bottleneck (high $𝜌$ ) propagate their service variability strongly; idle stations pass through arrival variability.

352.4. Why it matters: the Hopp-Spearman corollary

In a balanced line (all stations have similar $𝜌$ and $𝑐_{𝑠}$ ), variability builds up but stays bounded. In an unbalanced line, one bad station injects variance that downstream stations can’t fully absorb — leading to queue blowups far from the original problem.

This is part of why line balancing (here) and variance reduction (eliminating setups, breakdowns) are foundational manufacturing improvements.

352.5. Bottleneck analysis

The linking equation lets you predict where queues will build before operations start:

Compute $𝜌$ at each station
Apply the linking equation from upstream to downstream
Combine with VUT / Sakasegawa to estimate queue lengths

Identifies bottleneck location and severity quantitatively.

352.6. Limitations

Approximation — exact only for special distributions
Independence assumed — auto-correlation in service or arrivals breaks the formula
Open queues — doesn’t handle blocking / starvation directly (need finite-buffer extensions)

For finite buffers: see two-machine finite-buffer models (Buzacott-Shanthikumar) — more complex but capture blocking effects.

352.7. See also

Kingman VUT
Sakasegawa
Best/Worst/PWC — Hopp-Spearman line performance
Line Balancing