ModelRisk example · General techniques
Sales in a Limited Market
How do sales behave when the market is limited? A pool of buyers that runs down, and growth towards a ceiling.
From the model’s saved run
Alarm system: a pool that runs down
Households buying each year, and over ten years.
| Measure | Unit | 10th pct | Median | 90th pct | Mean |
|---|---|---|---|---|---|
| Year 1 | households | 8,119 | 11,401 | 15,021 | 11,507 |
| Year 5 | households | 6,831 | 7,759 | 8,328 | 7,662 |
| Year 10 | households | 5,825 | 6,150 | 6,476 | 6,149 |
| Ten years | households | 68,688 | 80,539 | 89,267 | 79,678 |
| Measure | Unit | Value |
|---|---|---|
| Long-run sales: new households a year | households | 5,502 |
Line chart: 90th percentile, Median, Mean, 10th percentile.
The business question
A home-security company sells an alarm system that a household buys once. About 55,000 households in its region do not have one yet. Each year a fifth of them might buy (anything from 10% to 35%), and new households arrive at about 10% of today's pool each year.
Every sale shrinks the pool, so sales fall as the market saturates, down towards the flow of new households. The company's new monitoring service sells differently: its annual sales grow towards a ceiling of about 2,300 contracts a year, reaching half of it in about four years. The model forecasts both over ten years.
- Who it is for
- Anyone forecasting sales in a market that can be saturated: durable goods, one-off services, new products.
- Main outputs
- Each product's sales by year at P10 / median / P90 and over ten years · the years the monitoring service needs to reach 90% of its ceiling.
How the model works
Four sheets after this one, read from left to right. Each one feeds the next.
Inputs sheet
For the alarm system, the households that might buy now (a PERT rounded to whole households), the share of that number added each year by new households, and the yearly chance that a potential buyer buys. For the monitoring service, the ceiling on annual sales and the years it takes to reach half of it.
Pool sheet
Each year VoseBinomial(pool, chance) households buy. Next year's pool is this year's, minus the buyers, plus the new households.
Ceiling sheet
Annual sales are (1 − 0.5^(year / half-life)) × ceiling: half of the ceiling after one half-life, three quarters after two.
Results sheet
Each product's sales by year and over ten years.
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The results in detail
Monitoring service: growth to a ceiling
Contracts sold each year, and over ten years.
Line chart: 90th percentile, Median, Mean, 10th percentile.
| Measure | Unit | 10th pct | Median | 90th pct | Mean |
|---|---|---|---|---|---|
| Year 1 | contracts | 313 | 376 | 460 | 382 |
| Year 5 | contracts | 1,150 | 1,376 | 1,670 | 1,397 |
| Year 10 | contracts | 1,649 | 1,965 | 2,371 | 1,992 |
| Ten years | contracts | 11,248 | 13,435 | 16,280 | 13,633 |
| Measure | Unit | Value |
|---|---|---|
| Mean years to reach 90% of the ceiling | years | 13.5 |
How to read these results
- The alarm system's first year is very uncertain, because the chance of buying is: anything from about 8,000 to 15,000 households. By year ten the uncertainty has mostly gone: whatever the chance of buying, the pool has been worn down and sales settle near the 5,500 new households a year.
- So in a finite market a faster take-up brings sales forward rather than adding to them: it empties the pool sooner.
- The monitoring service's sales are still growing after ten years: with a half-life of about four years it takes over thirteen years to reach 90% of the ceiling. Its uncertainty comes from the ceiling and the half-life, not from chance.
- Use a pool when each customer buys once; use a ceiling when sales repeat but the market caps them.
See it move. Open the workbook, change an input on the Inputs sheet and run the simulation: every table and chart on this page updates in Excel.
Inputs and assumptions
Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.
Alarm system: the pool of buyers 3 inputs
Households that might buy, the refill, and the yearly chance of buying.
| Assumption | Unit | Uncertainty | Low | Most likely | High |
|---|---|---|---|---|---|
| Households that might buy, todayRounded to whole households. | households | PERT, rounded | 50,000 | 55,000 | 60,000 |
| New households each year, as a share of today's poolAbout 5,500 a year. | share | Fixed | 10% | ||
| Chance a potential buyer buys in a yearThe same every year: one draw per iteration. | probability | PERT | 10% | 20% | 35% |
Monitoring service: growth to a ceiling 2 inputs
The ceiling on annual sales and how fast sales approach it.
| Assumption | Unit | Uncertainty | Low | Most likely | High |
|---|---|---|---|---|---|
| Ceiling on annual sales | contracts | PERT | 1,800 | 2,300 | 3,600 |
| Years to reach half of the ceiling | years | PERT | 3.5 | 4.0 | 5.0 |
Units and conventions
- Units
- Households (alarm systems) and contracts (monitoring service) per year.
- Half-life
- The years until annual sales reach half of the ceiling.
Key assumptions and limitations
- One chance per year
- The chance of buying is uncertain but the same every year and for every household.
- Refill
- New households arrive at a fixed share of today's pool each year; nobody leaves the pool unless they buy.
- Ceiling
- The service's sales follow the curve exactly; the uncertainty is in its ceiling and half-life.
- Data
- The pool, refill and chance are those of ModelRisk's earlier Finite Market example; the ceiling and half-life those of its Sales With Maximum example.
How to use it
- Set the pool, its refill, the chance of buying, the ceiling and the half-life on the Inputs sheet.
- Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set).
- Use the pool when each customer buys once, and the ceiling when sales repeat but the market caps them.
ModelRisk functions used
- VosePERT
- The pool, the chance of buying, the ceiling and the half-life.
- VoseBinomial
- How many of the pool's households buy in a year.
- VoseOutput
- Each product's annual sales as an output range, and the ten-year totals.
- VoseSimMean, VoseSimPercentile
- Statistics of the simulation and the bands.
Further reading
- Monte Carlo simulation explained (Vose guide)
- Vose, D. (2008). Risk Analysis: A Quantitative Guide, 3rd edition. Wiley: forecasting sales in a finite market.
Download the workbook and run it yourself. It opens in Excel with ModelRisk. The free trial runs every example model.
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An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.