For a repairable item the number of units in the system barely changes, so purchasing covers condemnations and growth while the availability you get is set by how many units are stuck in the loop. Palm's 1938 result gives the expected pipeline as the removal rate multiplied by the mean turnaround time, and the variability of repair time does not enter that expression, which makes mean turnaround the lever that moves the most stock. Most of a turnaround is dwell and queue rather than work, so the measurement worth having splits the loop into stages and names an owner for each. Condemnation is the only genuine leak in a closed population, it sets the annual buy, and it drifts upward quietly as piece parts go obsolete.
The pool of hydraulic manifolds has sat at 180 units for four years and nobody has bought one since 2022. Every failure gets swapped in the field, the failed unit comes back, the shop strips and rebuilds it, and it returns to stock as serviceable. Last quarter availability on that manifold fell to 71 percent and the request that came up the chain was to buy thirty more at four thousand pounds each. At the same moment the shop had ninety of them stacked against a wall waiting for a seal kit worth eleven pounds.
Consumables and repairables usually get planned by the same team, in the same system, under the same policy, and the mechanics underneath them have very little in common. A consumable leaves the system when it is used, so replenishment is the whole story. A repairable circulates, and the useful question is what share of a fixed population is available at any given moment.
The population is closed and only the flow moves
Count everything: serviceable units in central stock, units at forward locations, units installed in machines, units in transit in both directions, and units somewhere inside the repair shop. That total is the population, and it changes only when you buy a unit or scrap one. Nothing else in the operation moves it.
Two consequences follow immediately. Purchasing for a repairable is not a replenishment activity in any ordinary sense, because the annual buy quantity equals the units condemned plus whatever the installed base grew by. And availability is a distribution question, since the entire pool already exists and the only thing in play is whether a given unit is sitting on a shelf ready to issue or sitting in a queue somewhere.
An accounting trap sits underneath this and breaks the numbers before anyone reaches the interesting part. A repairable part number exists in at least two conditions, serviceable and unserviceable, and many ERP configurations carry a single on-hand figure for it. When the ninety units waiting for seal kits count as stock, the system believes there are 90 available, the planner reads the shortage as a demand problem, and the purchase request looks reasonable to everyone. Separate condition codes, or separate locations with separate part numbers, are the precondition for every calculation below.
Palm's theorem and the number it hands you
The core result is older than the discipline that uses it. Palm published it in 1938 in the context of telephone traffic, and it says that if failures arrive as a Poisson process and repair capacity is ample, the number of units in resupply at any moment is Poisson distributed with a mean equal to the arrival rate multiplied by the mean repair time.
The property that matters operationally is what the expression leaves out. Only the mean of the repair time appears. The variance of repair time, the shape of its distribution, whether some units take three days and others take four months, none of it changes the expected pipeline. Sherbrooke built METRIC on this foundation in Operations Research in 1968 for exactly this class of item, and the network placement question that grew out of it belongs with the multi-echelon treatment elsewhere rather than here.
Work an example. Across the installed base you remove 40 manifolds a month. Turnaround from removal to return-to-serviceable averages 45 days, so 1.5 months. Expected units in the pipeline is 40 times 1.5, which is 60. To hold availability at a high level you need enough units that the pipeline rarely exceeds the pool, and with a Poisson mean of 60 the ninety-fifth percentile sits near 73. A pool of 73 supports that service level, and the 180 units currently in the system are more than double what the loop requires when it runs at its design speed.
Now change one input. Cut mean turnaround to 20 days. Expected pipeline becomes 40 times 0.667, which is 26.7, and the ninety-fifth percentile lands around 36. The pool requirement drops from 73 to 36. On a four thousand pound unit, halving the loop time released close to 150,000 pounds of capital, and it did so without a purchase order, a supplier negotiation or a forecast.
That is the argument for treating turnaround as the primary lever. It enters the expression multiplicatively, it is under your own control, and buying stock is the expensive way to compensate for a slow loop. One caution on the arithmetic: the normal approximation used above is fine at a mean of 60 and wrong at a mean of 2, so read low-volume rotables straight off the Poisson quantile.
Most of a turnaround is dwell and queue
The loop has more stages than the shop's own cycle time report covers, and the shop only measures the part it owns.
- Dwell at the point of removal. The failed unit sits in a van, a site cupboard or a branch cage until somebody ships it. No engineer has ever been measured on this.
- Transport inbound, usually consolidated to save freight cost, which trades days for pounds without anyone pricing the days.
- Receipt and induction, where the unit joins a queue for a diagnostic bay.
- Waiting for piece parts, which is the seal kit in the opening paragraph and the single most common blocker.
- The repair itself, plus test and certification.
- Transport outbound back to serviceable stock.
Take that 45 day loop and put plausible numbers on it. If diagnosis takes a day and the bench work takes four hours, then roughly 43 of the 45 days are dwell, transport and queue. Every stage on that list has a different owner, and the two longest ones typically sit with people who have never been shown the pool size their delay is funding.
The return leg is worth a specific mention because it is the stage nobody has ever been accountable for. A field engineer is measured on fix rate and on jobs closed, and returning a dead unit to a depot appears in none of that. Measuring return dwell by region and publishing it is the cheapest intervention in this discipline, and it usually moves within a month of anyone looking.
Condemnation sets the annual buy
Since the population only shrinks through scrapping, the condemnation rate at induction determines what you have to purchase. With 480 removals a year and 8 percent condemned, you buy 38 units a year to hold the pool flat. If condemnation drifts to 14 percent, the buy rises to 67.
That drift happens for reasons that have nothing to do with reliability. A repair standard gets tightened after a warranty dispute. A piece part goes obsolete and the only economic disposition becomes scrap. A new technician condemns anything ambiguous. The purchase requests that follow are booked as routine replenishment, spread across a year, approved individually, and no single one of them is large enough to prompt a question.
Track condemnation rate as a monthly series per part number with the reason code attached. A step change in it is a policy change upstream, and finding out whose policy changed is cheaper than funding it.
The related figure worth pulling in the same query is the no fault found rate, meaning units returned from the field that test serviceable on the bench. Those units consume a full turnaround, occupy pool capacity, and never had a failure. They also inflate your measured removal rate, which inflates the pipeline calculation, which sizes the pool larger, which is a self-reinforcing loop paid for in capital. If no fault found sits in double digits, the fix is field diagnostics and engineer training rather than inventory.
Repair capacity is a queue and behaves like one
Palm's result assumes ample repair capacity, and that assumption is the first one to break in a real shop. When the shop is a constrained resource, waiting time grows nonlinearly as it fills up. Kingman's 1961 heavy traffic approximation gives the shape: queue time scales with utilisation divided by one minus utilisation. At 80 percent utilisation that factor is 4, at 90 percent it is 9, and at 95 percent it is 19.
A repair shop run at 95 percent utilisation looks efficient on its own report and quietly converts labour savings into inventory investment elsewhere on the balance sheet. Reading those two numbers together is the point of the exercise, because the shop budget and the inventory budget almost never belong to the same person and are almost never compared.
Batching does the same thing more visibly. If the shop inducts in lots of 20 to make setup economic, the first unit of a lot waits for 19 companions before work starts. That wait sits in every unit's turnaround time and therefore in the pool size, and the setup saving that justified the batch is usually much smaller than the capital it consumes.
The repair or buy line has moved and nobody re-drew it
Most operations carry a threshold rule, something like repair when the quoted repair cost is under 60 percent of a new unit. The rule is missing two terms.
The first is expected condemnation loss. When you induct a unit with a 14 percent chance of scrapping it, you pay diagnostic cost on all of them and buy new on 14 percent regardless, so the effective cost of the repair route is the repair price plus that probability multiplied by the purchase price and the diagnosis already spent.
The second is the pipeline cost of the turnaround itself. A repair route that takes 45 days consumes 45 days of a pool unit, and the pool unit costs the full purchase price of the item in capital. Where a new unit arrives from the supplier in 20 days, the repair route has to beat the purchase on price by enough to cover 25 extra days of tied-up capital across every unit in the loop.
Run those two terms across your top twenty rotables and a few will cross the line. Cheap items with slow, parts-dependent repairs and high condemnation rates should be bought new, and they get repaired because the policy was written when the item was expensive and the piece parts were freely available.
Obsolescence of the piece parts is the quiet end state here. When the seal kit stops being made, the repairable becomes a consumable by default, and the sizing question turns into a final order on the piece part governing the service life of the entire pool. That is the all-time buy problem and it has its own treatment.
Where this stops
Everything above requires serialised traceability, and that is where most operations discover they cannot run the calculation. Computing mean turnaround needs a per-unit event history linking the removal of serial 4471 to its return to serviceable stock, with timestamps at each stage in between. A great many systems record a goods receipt and an issue with no link between them, which leaves the mean turnaround as an opinion held by whoever runs the shop. Establishing serial-level tracking on the top fifty rotables is a prerequisite, and it takes months rather than days.
Palm's theorem also needs its conditions checked. It assumes one-for-one replenishment, which holds for rotables and fails the moment somebody batches removals. It assumes Poisson arrivals, which describes random failure reasonably and describes scheduled overhaul badly. A component removed every 4,000 running hours produces a deterministic, lumpy removal stream where the pipeline is a calendar calculation rather than a stochastic one, and treating it as Poisson will oversize the pool during quiet periods and undersize it when a cohort of machines reaches its interval together.
Small pools deserve their own caution. When the expected pipeline is 2 units, the difference between availability of 90 and 99 percent is one physical unit, the normal approximation is meaningless at that scale, and a single unit lost in transit is a 30 percent capacity event. Those items want exact Poisson quantiles and a named person who knows where each unit is.
Pull the last fifty repair events for your highest value rotable, timestamp each stage from field removal through to return-to-serviceable, and compute both the mean and the share of it that is queue rather than work.
More on this in service parts on the DemandLabs platform.
