Live model
Turn an observed event history into a future exposure and budget plan
Fit a simple events-per-exposure rate, scale it to a planned period, and translate the expected count into a linear impact budget. The model keeps expected count, count uncertainty, chance of any event, variable impact, and fixed readiness cost distinct.
| Stage | Exposure / quantity | Events or unit value | Derived value | Role |
|---|
Current calculation process
Formula, substitution, intermediate quantities, and check
r = c/T; lambda_future = rE; E[C] = lambda_future; E[impact] = lambda_future v + F; P(C>=1)=1-e^(-lambda_future)
This page estimates one constant rate from observed count divided by observed exposure. It then scales, rather than re-fits, that rate to future exposure. Variable impact is linear in expected count; fixed readiness cost is added once.
Use the calculator
Five planning steps
- Audit the historical count. Apply one incident definition and remove duplicates before entering events.
- Match its exposure denominator. Enter only the time, assets, transactions, or distance during which those events could occur.
- Translate the future plan into the same unit. Convert years to months or fleets to machine-hours before entry.
- Separate variable and fixed impacts. Unit impact scales with events; readiness cost is incurred once in this model.
- Read expectation with uncertainty. Use expected count for average load, SD for natural count spread, and P(>=1) for event-free planning.
Five foundations
Expected value for an exposure-scaled count
1. A rate needs a denominator
Eighteen events alone do not define risk. Eighteen in twelve machine-months produces 1.5 events per machine-month.
2. Scaling assumes comparability
Multiplying by future exposure assumes comparable operating conditions, event definitions, and observation quality.
3. Expected count is not a forecasted integer
lambda=13.5 is a valid average across many comparable plans even though one period cannot realize half an event.
4. Linear impact is an assumption
The page assumes each additional event contributes the same average impact and does not model caps, queues, or economies of scale.
5. Fixed cost changes the reverse check
Subtract fixed readiness cost before dividing total impact by unit impact to recover expected event count.
Calculation anatomy
Symbols and the default bridge
| Symbol | Meaning | Unit |
|---|---|---|
| c | Observed event count | whole events |
| T | Observed exposure | chosen exposure units |
| r | Fitted event rate c/T | events per exposure |
| E | Future exposure | same units as T |
| v | Average impact per event | currency/event |
| F | Fixed readiness cost | currency/plan |
Default substitution: r=18/12=1.5, lambda=1.5x9=13.5, variable impact 13.5x$275, then add $900 once. No annualization occurs unless the entered exposures are already annual.
Deep analysis
Three decisions around the same expected count
Workload staffing
Expected count supports average throughput. Add the count SD and a chosen service quantile before turning that mean into a capacity commitment.
Event-free probability
When readiness cost depends on any event occurring, P(C>=1) is more decision-relevant than a fractional expected count alone.
Budget decomposition
Keep variable and fixed components visible. Otherwise a policy change in readiness cost can be mistaken for a change in event frequency.
Decision cases
Routine and boundary planning
Maintenance callouts
Eighteen callouts across twelve machine-months are scaled to nine future machine-months. The manager uses expected variable impact for budget and retains the readiness contract as a separate fixed line.
Zero future exposure
If a line is shut down and future exposure is zero, expected events and variable impact are zero, while an entered fixed readiness cost remains. That boundary confirms the cost definitions are separated correctly.
Terms
Expected-value vocabulary
- Observed exposure
- Opportunity base that produced the historical count.
- Fitted rate
- Observed count divided by observed exposure.
- Future lambda
- Rate multiplied by planned exposure.
- Expected count
- Long-run average realized count across comparable periods.
- Variable impact
- Expected count multiplied by average per-event impact.
- Readiness cost
- Fixed amount added once regardless of realized count.
FAQ
Questions for event-rate budgeting
Why can expected events be fractional?
Expectation is an average across repeated periods; realized counts remain whole.
How is future exposure handled?
The fitted rate is multiplied by future exposure in the same unit.
Is rate-estimation uncertainty included?
No. Use the confidence calculator or a predictive model when sparse-data uncertainty matters.
Is fixed cost charged with zero events?
Yes. Enter zero if that does not match the contract.
Can impacts vary?
This model uses one average linear impact; variable severity needs a richer model.
Why also show P(>=1)?
It answers whether the plan remains event-free, a different question from average load.
Limits and evidence
Planning boundaries
- The historical rate is treated as fixed; no confidence or trend adjustment is included.
- Observed and future exposures must be comparable and use the same units.
- Impact per event is linear and constant; no frequency-severity dependence is modeled.
- Fixed cost is incurred once regardless of event realization.
- The Poisson process assumes independent occurrences and constant intensity within exposure.
Sources and related tools