Theoretical Fiber Latency Formula Verification

This page independently reconciles the equations used by the latency-page generator with a worked Tokyo (TYO) → Sydney (SYD) example. It starts with GeoNames city-centre coordinates, carries units through every calculation, and uses a fixed 101.7ms RTT fixture captured on July 18, 2026. Keeping the fixture fixed separates mathematical regression checks from routine latency-matrix refreshes.

Symbols and reference inputs

1. WGS-84 geodesic verification

The implementation solves the inverse geodesic on the WGS-84 ellipsoid:

$$d_g = \operatorname{InverseGeodesic}_{\mathrm{WGS84}}(A,B) = 7\,792.8\ \mathrm{km}$$

This avoids the distance error introduced by treating Earth as a perfect sphere. The returned surface distance is the geographic lower-bound input for every later step.

2. Vacuum propagation floor

Round-trip propagation traverses the distance twice. Multiplying by $10^3$ converts seconds to milliseconds:

$$\begin{aligned} RTT_{\mathrm{vac}} &= \frac{2d_g}{c} \times 10^3 \\ &= \frac{2(7\,792.8\ \mathrm{km})} {299\,792.458\ \mathrm{km\,s^{-1}}} \times 10^3\ \mathrm{ms\,s^{-1}} \\ &= 51.99\ \mathrm{ms} \end{aligned}$$

The kilometer units cancel, leaving seconds before the final conversion to milliseconds.

3. Silica propagation floors

The group velocity of light in fiber is:

$$v_g = \frac{c}{n_g}$$

For standard single-mode fiber:

$$\begin{aligned} RTT_{\mathrm{fiber}} &= \frac{2d_g n_s}{c} \times 10^3 \\ &= \frac{2(7\,792.8)(1.4679)} {299\,792.458} \times 10^3 \\ &= 76.31\ \mathrm{ms} \end{aligned}$$

For the optimistic low-latency material comparison:

$$\begin{aligned} RTT_{\mathrm{ULL}} &= \frac{2d_g n_u}{c} \times 10^3 \\ &= \frac{2(7\,792.8)(1.4620)} {299\,792.458} \times 10^3 \\ &= 76.01\ \mathrm{ms} \end{aligned}$$

The required physical ordering is therefore:

$$RTT_{\mathrm{vac}} < RTT_{\mathrm{ULL}} < RTT_{\mathrm{fiber}}$$

4. Engineering path, attenuation, and amplifiers

The transparent engineering reference applies the explicit path multiplier:

$$\begin{aligned} d_e &= \lambda d_g \\ &= 1.05(7\,792.8) \\ &= 8\,182.5\ \mathrm{km} \end{aligned}$$

Attenuation determines optical power loss:

$$\begin{aligned} L_{\mathrm{dB}} &= \alpha d_e \\ &= (0.20\ \mathrm{dB\,km^{-1}}) (8\,182.5\ \mathrm{km}) \\ &= 1\,636.5\ \mathrm{dB} \end{aligned}$$

The span and inline-amplifier counts are:

$$\begin{aligned} N_{\mathrm{span}} &= \left\lceil \frac{d_e}{S} \right\rceil = \left\lceil \frac{8\,182.5}{80} \right\rceil = 103 \\ N_{\mathrm{amp}} &= \max(0,N_{\mathrm{span}}-1) = 102 \end{aligned}$$

The amplifier contribution must be converted from microseconds to milliseconds:

$$\begin{aligned} RTT_{\mathrm{amp}} &= 2N_{\mathrm{amp}}\tau_a \\ &= 2(102)(0.05\ \mathrm{\mu s}) \\ &= 10.2\ \mathrm{\mu s} = 0.0102\ \mathrm{ms} \end{aligned}$$

The complete engineering floor is:

$$\begin{aligned} RTT_{\mathrm{eng}} &= \frac{2d_e n_s}{c} \times 10^3 + RTT_{\mathrm{amp}} \\ &= \frac{2(8\,182.5)(1.4679)} {299\,792.458} \times 10^3 + 0.0102 \\ &= 80.14\ \mathrm{ms} \end{aligned}$$

Attenuation is absent from the propagation-speed term. It affects the model through $N_{\mathrm{span}}$ and $N_{\mathrm{amp}}$, which is dimensionally and physically distinct from group velocity.

5. Mapped-fiber research reference

The separate 1.33× research comparison is:

$$\begin{aligned} RTT_{\mathrm{mapped}} &= 1.33RTT_{\mathrm{fiber}} \\ &= 1.33(76.31\ \mathrm{ms}) \\ &= 101.50\ \mathrm{ms} \end{aligned}$$

This reference is close to the fixed Tokyo-to-Sydney RTT fixture, but it remains a comparison factor rather than a claimed cable length.

6. Efficiency and latency inflation

Using the fixed $101.7\ \mathrm{ms}$ verification RTT:

$$\begin{aligned} \eta &= 100\frac{RTT_{\mathrm{fiber}}}{RTT_p} \\ &= 100\frac{76.31}{101.7} \\ &= 75.03\% \approx 75.0\% \end{aligned}$$ $$\begin{aligned} I &= \frac{RTT_p}{RTT_{\mathrm{fiber}}} \\ &= \frac{101.7}{76.31} \\ &= 1.3327 \approx 1.33 \end{aligned}$$

The two comparison metrics are reciprocal after converting efficiency from percent to a ratio:

$$\frac{\eta}{100} = \frac{1}{I}$$

7. Implementation reconciliation

8. Boundary checks

The implementation and its automated tests enforce these invariants:

$$d_g \ge 0$$ $$RTT_{\mathrm{vac}} < RTT_{\mathrm{ULL}} < RTT_{\mathrm{fiber}} < RTT_{\mathrm{eng}}$$ $$N_{\mathrm{span}} \ge 1, \qquad N_{\mathrm{amp}} = N_{\mathrm{span}} - 1$$ $$RTT_{\mathrm{amp}} \ge 0, \qquad \eta > 0, \qquad I > 0$$

If a future constant, coordinate, or rounding rule changes, the reconciliation test must change with it. This prevents the rendered explanation from silently diverging from the generator implementation.

Continue to Latency Coordinates & Fiber Model Inputs, compare the live Tokyo (TYO) → Sydney (SYD) route, or return to the global latency matrix.