Theoretical Fiber Latency Formula Verification

The worked example uses Tokyo (TYO) to Sydney (SYD) and a fixed 101.7 ms RTT recorded on July 18, 2026. Keeping this fixture unchanged lets the regression test check the equations independently of later measurement rounds. Coordinates are public GeoNames city centres.

Symbols and reference inputsCopy Anchor Link

1. WGS-84 geodesic verificationCopy Anchor Link

The first step is the WGS-84 inverse geodesic:

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

Using the ellipsoid avoids the spherical-Earth approximation. Its surface distance supplies the geographic lower bound for the remaining steps.

2. Vacuum propagation floorCopy Anchor Link

RTT covers the distance twice. The factor $10^3$ converts seconds to milliseconds:

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

Kilometres cancel in the ratio; the result is in seconds before conversion.

3. Silica propagation floorsCopy Anchor Link

Fiber group velocity is given by:

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

Substituting the standard single-mode group index gives:

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

For the 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 resulting propagation times must follow this ordering:

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

4. Engineering path, attenuation, and amplifiersCopy Anchor Link

The engineering reference applies the stated path allowance:

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

Optical power loss is calculated as:

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

Span and inline-amplifier counts are calculated separately:

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

Convert each amplifier's microsecond contribution to milliseconds:

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

Adding the terms gives the engineering floor:

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

Attenuation enters through $N_{\mathrm{span}}$ and $N_{\mathrm{amp}}$. It does not appear in the propagation-speed term, whose units and physical meaning are different.

5. Mapped-fiber research referenceCopy Anchor Link

The separate mapped-fiber research comparison uses 1.33×:

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

The result is close to the fixed Tokyo-to-Sydney fixture. That numerical agreement does not establish the route's cable length.

6. Efficiency and latency inflationCopy Anchor Link

With the fixed $101.7\ \mathrm{ms}$ observation:

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

Converting efficiency from percent to a ratio makes it the reciprocal of inflation:

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

7. Implementation reconciliationCopy Anchor Link

8. Boundary checksCopy Anchor Link

Automated tests check these invariants:

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

A changed constant, coordinate or rounding rule requires a corresponding update to the reconciliation test. The equations and generated results should continue to agree.

See the model inputs, the published Tokyo to Sydney round, or the full matrix.