What a Meal Costs: An Empirical Energy Model of Replicated Matter and the Share of Ship Power Taken by Replicator Allowances Aboard USS Voyager, 2371–2378

1. Introduction

A starship on a routine patrol treats the replicator as a convenience. The power it draws is small beside the warp core's output and is restored at the next starbase. USS Voyager lost that assumption in 2371, when she was carried into the Delta Quadrant with no route home and no dependable resupply. She traded for supplies where she could, but for most of the journey the replicators drew on a budget that the ship could only hold or spend. The captain limited replicator use early in the journey, and the ship's galley became part of the ship's routine in a way that a Federation vessel of her class would not ordinarily need. Standing guidance for long deployments without resupply had treated power management as a matter for the engineering department, and it had not addressed the galley (Starfleet Corps of Engineers, 2373).

We write in 2379, after the ship's return and the recovery of her engineering ledger (Starfleet Delta Quadrant Mission Records, 2378). The ledger is a continuous record kept by the Chief Engineer's office, and the on-screen record of the journey does not describe it, so everything we say about its contents rests on the recovered file. It contains two kinds of entry that bear on the cost of replicated matter. Some entries meter the energy of individual replications, and others record, for each 30-day accounting period, the fraction of total ship power taken by the replicator system. Earlier engineering studies gave the efficiency of matter–energy conversion in shipboard replicators under laboratory load (Villeneuve-Adair, 2372) and the priorities for shedding load in starship power distribution (Ostrowicz-Bern, 2370). Neither could be checked against a long run of actual use by a crew that could not readily replenish its power.

Two questions are put to the ledger here. How does the energy of a single replication depend on the mass and kind of the item replicated? And how did the share of ship power taken by replicators differ between the periods in which different allowance orders were in force? The first is a question about the cost of a meal or a component. The second is a question about what a rationing order achieved.

2. System Description

The replicator converts stored energy into matter arranged to a stored pattern, and the reverse process reclaims matter for reuse. Its inner workings are not our subject, and we treat the device as a black box whose input, in megajoules, is metered at the power tap. The model in Section 3 is therefore empirical. It describes how metered input varied with the properties of the output, and it makes no claim about how the conversion is performed.

Replication entries in the ledger give the metered energy in megajoules, the mass of the output in kilograms, and a class recorded by the engineer on duty. Three classes account for all entries in the sample. Liquids were beverages and broths. Prepared food covered solid and semi-solid dishes. Components were small mechanical or electronic parts, made in the engineering spaces for repairs. The metering program was run by the engineering staff for an extended period, and the 412 entries we use are those in which mass, class and energy were all recorded. They are a sample of replicator use, and they are not the whole of it. Metering programmes of this kind can be designed to give a representative sample of auxiliary loads (Kowalczyk-Ashe, 2369), and we have no record of how this one was drawn.

Accounting-period entries record the mean fraction of total ship power taken by the replicator system across a 30-day period. The ledger runs to 86 such periods from the start of the journey to the ship's return, and two of these were incomplete, so our analysis uses 84. From the ledger's standing orders we coded three allowance regimes. Regime I was the allowance in force when the ledger begins, which was already a limited allowance and not an unrationed baseline. Regime II was a tighter allowance imposed after the ship's power reserves had been drawn down. Regime III was a partial relaxation of that allowance. The coding is our own. It rests on the dates at which the ledger notes a change in the standing order, and a period that straddled a change was assigned to the regime in force for most of its days. The on-screen record does not date these orders, and our regimes are a reading of the ledger and not a statement of Voyager's published policy. The class labels on the replication entries were not audited.

3. Analysis and Model

Energy per replication is positive and right-skewed. Across the sample, masses ran from about 0.01 to 1.7 kilograms and energies from about 0.15 to 7.8 megajoules. A model on the raw scale would give large items undue weight, so we modelled the natural logarithm of energy. The predictors were the logarithm of mass and two indicators for class, with liquids as the reference. The coefficient on log mass is then an elasticity, the proportional change in energy for a proportional change in mass. An elasticity of 1 would mean that energy rises in direct proportion to mass, and a value below 1 would mean that larger items cost less per kilogram than small ones.

We fitted the model by ordinary least squares to a training set of 312 replications, 129 liquids, 118 prepared foods and 65 components, chosen at random from the 412. The remaining 100 were held back for validation, 41 liquids, 34 prepared foods and 25 components. The residual standard deviation on the log scale was 0.23, and the model explained 85% of the variance of log energy in the training set.

For the second question we compared the mean replicator share of ship power across the three regimes with a one-way analysis of variance, and we report pairwise differences with intervals based on the pooled within-regime standard deviation. The two analyses are independent of each other. The first uses the replication entries and the second uses the accounting-period entries, and neither uses the other's output.

The elasticity of energy with respect to mass was 0.86 (95% CI 0.82 to 0.91). Energy therefore rose almost in proportion to mass, with a small saving for larger items. Prepared food cost 1.31 times as much as a liquid of the same mass (95% CI 1.24 to 1.39), and a component cost 2.65 times as much (95% CI 2.41 to 2.91). With the fitted intercepts, the model gives about 3.1 megajoules for one kilogram of liquid, 4.0 for one kilogram of prepared food and 8.1 for one kilogram of components (Table 1).

4. Validation Against Field Data

We applied the fitted model to the 100 held-out replications. Predictions were converted from the log scale to megajoules by exponentiation. Across the hold-out set the median absolute error was 14% of the metered energy, and the mean absolute error was 19%. Fifty-eight of the 100 predictions were within 20% of the metered value, and 38 were within 10%. The largest single error was 80%, and six predictions were more than 50% away from the metered value.

The model overpredicted slightly. The mean residual on the log scale in the hold-out set was −0.045, which corresponds to an average overprediction of about 4.5%, and the spread of residuals was similar to that in training (0.23 against 0.23). The overprediction was concentrated in liquids and prepared food, whose mean log residuals were −0.06 and −0.09, and it did not appear in components (+0.04). We did not correct for the bias that arises when predictions are transformed back from the log scale (Marchetti-Owens, 2370). A correction would raise the predictions, and so would add to the overprediction we observed, and our target was the typical replication and not the mean.

Regime shares come from the period entries, and they are reported here because both analyses draw on the ledger's field entries. The mean share was 6.5% in Regime I (26 periods, SD 0.9), 4.2% in Regime II (30 periods, SD 0.9) and 4.9% in Regime III (28 periods, SD 1.1). A one-way analysis of variance gave F(2, 81) = 40.4, p < .001, and the regimes accounted for half of the variance in the share (η² = .50).

Pairwise differences used the pooled within-regime standard deviation. Regime I exceeded Regime II by 2.3 percentage points (95% CI 1.8 to 2.8) and exceeded Regime III by 1.6 points (95% CI 1.1 to 2.1). Regime III exceeded Regime II by 0.7 points (95% CI 0.2 to 1.2). The point estimates suggest that the relaxation restored roughly a third of what the tighter allowance had removed, with wide uncertainty, and it did not return the share to its initial level. The intervals are unadjusted for three comparisons. With a Tukey adjustment the interval for Regime III against Regime II widens to about 0.1 to 1.3 points, so that difference is the least secure of the three.

5. Failure Modes

The model fails in a way that follows from how it was built. It carries one elasticity for all classes, and the class terms shift only the level. If components scale differently from liquids, a common slope would misprice them at both ends of the mass range, and with 65 components in the training set we could not test a separate slope with confidence. The hold-out set shows little sign of the problem, since components had the smallest mean absolute error (14%), but the number is small.

A second failure lies in the sampling and the validation. The metered replications are those that engineering chose to meter, and if metering favoured routine items over unusual ones, the model would understate the cost of unusual requests. We cannot check this from the ledger. The hold-out set was drawn at random from a ledger in which replications from the same period and the same duty engineer are correlated, so the random split probably flatters the model, and a split by date would be a stricter test. A third lies in the regime comparison. Adjacent accounting periods are not independent, since a period in which the ship was under heavy load is likely to be followed by another, and the analysis of variance treats them as independent. Serial correlation is known to narrow the intervals of such an analysis (Brennecke-Adoyo, 2371), so its intervals are probably too narrow, and the direction of the differences is more secure than their size.

Cause is a fourth concern. The three regimes differ in allowance, but they also differ in the ship's condition, her supplies and her crew, and the ledger does not let us separate these. We cannot say how much of the fall in share between Regime I and Regime II was due to the order itself. The ledger also does not record what crew members did when the allowance ran short, whether by cooking, by trading or by going without, so a lower share does not by itself show that the order was followed.

6. Conclusion

Within this ship's ledger, the energy of a replication was close to proportional to the mass of the item, and a component cost more than twice as much as a liquid of the same mass. Predictions from the model were correct to within about 14% in the typical case, and the errors were larger for a minority of items. These figures are usable for planning the power budget of a vessel that cannot resupply, and they should not be read as a statement about the replication process.

The share of ship power taken by replicators fell from 6.5% to 4.2% when the allowance was tightened and recovered to 4.9% when it was relaxed. We cannot attribute the whole change to the standing orders. Later work could use the ledgers of other long-range vessels, where supply conditions differ, to test whether the same relation between mass and energy holds.