One Transit in Thirty-Five Thousand: A Failure-Rate Model for Short-Range Displacers from 126 Incidents in 4.48 Million Logged Transits, 1900–1949 CE

1. Introduction

A displacer moves a person or a load a short distance through hyperspace, and the Culture uses it routinely for ordinary movement inside vehicles, between neighbouring habitats and across the surfaces of Orbitals. Its risk is small but not zero, and every user knows it (Aldenmoor, 1741 CE). The risk is accepted by choice, as other small risks are, and engineers who maintain displacer terminals have long kept records of the occasions on which a transit does not go as specified.

Those records have rarely been analysed together. The earliest tabulation listed failures from a handful of installations and quoted a single overall rate (Sorrelmaine, 1868 CE). Later work established that the number of transits through a terminal must be counted as carefully as the failures if any rate is to mean something (Hennaquist, 1894 CE), and a theoretical treatment proposed a rise of failure rate with transit distance without testing it against incident data (Taldrisse & Veyth, 1926 CE). None of these fitted a model to incident records and then asked it to predict records it had not seen.

We ask three questions. How does the rate of failed transits vary with distance and with the kind of load? Does a model fitted to earlier records predict the number and pattern of later incidents? And what forms does a failed transit take?

This paper is written in 1950 CE. Our records run from 1900 to 1949 CE and cannot say anything about later years. The series we analyse, which we designate DT-1 (Hyperspatial Physics Group, 1900–1949 CE), is our own compilation from terminal logs, made for this study. It has not been deposited as a standing record series, and the rate model is ours. We offer no physical theory of displacement and make no claim about why a transit fails.

2. System Description

A displacer installation consists of one or more terminals. In the installations we studied, a load enters a terminal, is moved through hyperspace and arrives at a receiving terminal or at a specified arrival point (Aldenmoor, 1741 CE). Each terminal in the installations we studied keeps a transit counter and an arrival check. The counter increments at every dispatch. The arrival check compares the position and time of arrival with the specification and flags a transit as an incident if the position differs by more than the installation's tolerance, if the arrival is later than the installation's time tolerance, or if there is no arrival. Tolerances are set locally and are published in the maintenance notes of each installation (Venkarrow, 1823 CE).

DT-1 draws on all 63 terminals in 14 installations, nine aboard General Systems Vehicles and five on Orbitals, that kept both counter and arrival check throughout the years they appear in the series. Terminals entered the series at different dates, so the number of transits per terminal varies widely. The third author reconciled the counters against dispatch records at 12 terminals chosen at random and found agreement within 0.4% at each. DT-1 holds only the small instrumented minority of terminals that kept both counter and arrival check, and so says nothing of the total use of displacers in the Culture. We treat transit counts as exact and incident counts as complete for events that the arrival check can detect.

Every transit in the series carries a distance, taken from the terminal specification, and a load class. Services in these installations are certified to 60 km, so that is the upper limit of the data. We grouped distance into four bands set from the installations' own service classes before fitting: up to 100 m, 100 m to 1 km, 1 to 10 km and 10 to 60 km. In modelling we used the geometric centre of each band, 50 m, 400 m, 3 km and 25 km. The load class is individual, meaning a single person or a single drone with personal effects, or laden, meaning a group, equipment or cargo. The series records the load and its arrival status, not the condition or welfare of what was moved, and nothing here bears on that.

Two of us, Ralliand and Pryoth, read every incident report independently and coded its failure mode. Disagreements were settled by discussion. The third author, Olk-Vaskett, prepared the exposure counts and did not take part in coding.

3. Analysis / Model

We modelled the number of incidents in each stratum, a combination of distance band and load class, as Poisson with a log link and the log of the stratum's transit count as the offset. The linear predictor was an intercept, the base-10 logarithm of the distance in units of 100 m, and an indicator for laden loads. The incident rate for a transit of distance d is therefore r(100 m) × (d/100 m)^β × L, where r(100 m) is the rate for an individual transit of 100 m and L is 1 for individual loads and a constant factor for laden ones. We fitted this to the 1900–1939 CE records by iteratively reweighted least squares and report Wald intervals.

Over the fitting period there were 3,408,144 transits and 94 incidents, an overall rate of 27.58 per million transits (95% exact Poisson CI 22.29–33.75). Table 1 gives the eight strata. Rates rise steeply with distance in both load classes and are higher for laden loads in all bands except 100 m to 1 km.

Fit was adequate. The Pearson statistic was 5.37 on 5 degrees of freedom, a dispersion of 1.07, so there is no sign of extra-Poisson variation at the level of the strata. For each tenfold increase in distance the incident rate rose by a factor of 3.65 (95% CI 2.90–4.58, p < .001). That corresponds to an incident rate proportional to distance raised to the power 0.56 (0.46–0.66). Laden loads had a rate 2.06 times that of individual loads (1.37–3.10, p < .001). The fitted rate for an individual transit of 100 m was 6.28 per million (4.01–9.82). Scaling the standard errors by the square root of the dispersion widened the intervals slightly, to 2.88–4.62 for distance and 1.35–3.14 for load class, and changed no conclusion.

Incidents are clustered within terminals and installations, and the model treats them as independent. Terminals with more transits will also differ in age and maintenance, which we do not model. The intervals above are therefore a little narrower than they should be, by an amount we cannot estimate from stratum-level counts. Distance enters as a power law because it is the simplest form considered by Taldrisse & Veyth (1926 CE), not because the data single it out over other monotone forms.

4. Validation Against Field Data

Records for 1940–1949 CE were set aside before fitting and used only for validation. They contain 1,072,684 transits and 32 incidents, a rate of 29.83 per million (95% exact CI 20.40–42.11). The overall rate did not differ detectably from that of the fitting period (conditional exact test, p = .68).

Applied to the validation transits with the fitted coefficients, the model predicted 31.65 incidents in total (95% CI 25.85–38.75, from the uncertainty in the coefficients alone). The observed 32 lies close to the centre. A constant rate taken from the fitting period would have predicted 29.59, so for the total the model has no advantage over the simplest alternative.

The distance pattern is where the model failed. Table 2 pools the validation strata into the four bands. The model under-predicted incidents in the two shortest bands and over-predicted them in the longest, where it expected 10.74 and 5 were seen. It also over-predicted the 1 to 10 km band (10.77 expected, 8 seen). The p-values in Table 2 are twice the smaller tail of the exact Poisson distribution at the predicted count, with no correction for four comparisons. The two shortest bands and the longest lie close to the .05 line (p = .09 each) and expected counts are small, but the direction of the departures is systematic. Table 3 gives the eight strata. The deviance across them was 14.89 for the model against 17.80 for the constant rate, a modest gain that we report as descriptive only. Pooled to the four distance bands, the deviances are about equal, 11.14 for the model and 10.75 for the constant rate, so the gain comes from the load-class term and not from distance.

Refitting the same model to the 32 validation incidents gave a rise of 1.90 per tenfold distance (95% CI 1.28–2.81), against 3.65 in the fitting period, and a laden factor of 1.91 (0.95–3.82), close to the earlier 2.06. The difference between the two distance coefficients is large (z = 2.81, p = .005). We tested it after seeing Table 2, so the p-value is conditional on our having looked, and with 32 events we regard the later coefficient as imprecise. We take the result as evidence that the gradient with distance was weaker in 1940–1949 CE, and we have no tested explanation for it.

5. Failure Modes

We coded each of the 126 incidents in DT-1, from both periods, into one of four modes. In an arrival offset, the load arrives intact but outside the position tolerance. In a delayed arrival, the load arrives in the right place but after the time tolerance. In a recovered non-arrival, nothing arrives and the load is later located or returned within 30 days. In an unrecovered non-arrival, it is not. The two coders agreed on 118 of the 126 reports (93.65%, Cohen's kappa 0.91) before discussion.

Arrival offsets were the most common mode, 51 incidents (40.48%, 95% exact CI 31.83–49.58), followed by delayed arrivals, 34 (26.98%, 19.47–35.62), recovered non-arrivals, 27 (21.43%, 14.62–29.62), and unrecovered non-arrivals, 14 (11.11%, 6.21–17.94). The fourteen unrecovered cases correspond to 3.12 incidents per million transits (1.71–5.24). Because the fitted model concerns incident rates, not modes, we cannot say from it how the unrecovered rate itself varies with distance or load.

Mode did not vary detectably with distance. Of 52 incidents at 1 km or less, 22 were offsets, 14 delays, 11 recovered and 5 unrecovered. Of 74 beyond 1 km, 29 were offsets, 20 delays, 16 recovered and 9 unrecovered (χ² = 0.25, 3 df, p = .97). This test has little power with cells this small, and it does not show that the mix is the same.

For operators the practical consequence concerns recovery. Recovery practice after a non-arrival differs between large vehicles, and our recovered cases were handled as Obrevane (1911 CE) describes. We did not study how recovery depends on practice. Terminal arrival checks could not detect any failure that left position and time within tolerance, so the figures describe incidents as they were detected and do not bound the rate of failures that were not.

6. Conclusion

In 4,480,828 logged transits through 63 terminals between 1900 and 1949 CE, 126 incidents occurred, a rate of 28.12 per million (95% exact CI 23.42–33.48) or roughly one in 35,600. Within the fitting period the rate rose with distance and was higher for laden loads. A model fitted to the records of 1900–1939 CE reproduced the total number of incidents in 1940–1949 CE (31.65 predicted, 32 observed), though a constant rate from the fitting period would have predicted the total as well (29.59). It did not reproduce the gradient with distance, which was weaker in those years.

Several limits apply. Counts are small, and the validation set holds 32 incidents. Terminals are clustered and differ in tolerance and maintenance, which the model ignores. Distance enters as an assumed power law over four bands. The series covers 14 installations and cannot be taken to represent every displacer in use, and it records only failures that an arrival check could detect. We cannot say why the gradient changed.

The next step is a model with terminal-level effects, fitted to per-terminal logs and tested on a fresh decade as it closes, with a stated criterion for what would count as a failed validation. Until then, we recommend that the incident rate for a short transit be quoted as a range that depends on distance and load, and not as one figure.