Time-step bias is a modeling choice
A coarse step allows at most one state change per component per interval and can miss rapid fail-repair cycles. Compare delta t with both MTBF and MTTR before trusting the result.
Reliability
System Availability Simulation Calculator
Simulate finite-horizon repairable-system availability with seeded state trajectories.
FINITE-HORIZON AVAILABILITY
Simulate uptime and repair variation across complete system histories
For service planners who need a seeded distribution of experienced k-out-of-n availability rather than only a steady-state average.
Mean availability-
P05 availability-
P95 availability-
Mean downtime hours-
Meet-target probability-
CURRENT DECISION RECORD
Trajectory availability bands
Every row is regenerated from the active inputs and carried into Copy, TXT, and the page-specific PDF payload.
| Availability band | Trajectories | Share (%) |
|---|
CURRENT CALCULATION PROCESS
Formula, substitution, intermediate values, and reconciliation
Pfail=1-e^(-deltat/MTBF); Prepair=1-e^(-deltat/MTTR)
Waiting for valid inputs.
HOW TO USE
Five steps to a reproducible trajectory study
- Define n and required k.
- Enter MTBF, MTTR, and horizon.
- Choose time steps fine enough for transitions.
- Set trajectories, target, and seed.
- Compare seeds and the analytic distribution page.
SIMULATION FUNDAMENTALS
Five concepts behind each system history
- Trajectory
- One seeded finite-horizon history of all component up and down states.
- Time step
- The discrete interval delta t over which failure and repair transition probabilities are applied.
- Failure transition
- 1-exp(-delta t/MTBF), the probability an available component moves down during one step under the exponential assumption.
- Repair transition
- 1-exp(-delta t/MTTR), the probability a down component returns during one step.
- Trajectory availability
- The fraction of simulated steps with at least k of n components available.
DEEP SIMULATION ANALYSIS
Separate discretization, horizon variation, and target claims
Finite horizons create operational variation
Two systems with the same long-run rates can experience different availability over one year because failure timing and repair timing differ. The trajectory percentiles expose that variation.
Target attainment is scenario-dependent
The share meeting a target changes with horizon, initial state, seed, steps, and trajectory count. It is a simulation estimate, not a contractual probability.
WORKED DECISION CASES
A finite-year service case and a grid boundary
Four-unit data service
With four units, three required, MTBF 3000 hours, and MTTR 10 hours, most trajectories remain highly available but a few poorly timed outages create meaningful annual downtime variation.
Coarse-step boundary
If delta t approaches or exceeds MTTR, a repair can begin and finish inside one unobserved interval. The page flags this as a discretization limit even when many trajectories are run.
EVIDENCE RECORD
Make every run reproducible
Retain the component-state rule, horizon, step count, trajectory count, seed, initial-state assumption, MTBF and MTTR data periods, outage definition, target rationale, software revision, and alternate-seed sensitivity. A screenshot without these values is not a reproducible simulation record.
MODEL LIMITS
What the trajectory engine excludes
- Discrete-time transitions approximate continuous failure and repair events.
- Every trajectory starts with all components available.
- Components are identical and independent with exponential up/down transitions.
- The model excludes common cause, repair queues, spare shortages, scheduled maintenance, degraded capacity, and load sharing.
SIMULATION GLOSSARY
Six distinct trajectory terms
- Seed
- The integer that initializes the pseudo-random sequence and makes the same simulation inputs reproducible.
- Trajectory
- A complete simulated sequence of system states over the entered horizon.
- Transition probability
- The chance of moving between up and down states during one discrete time step.
- P05 availability
- A lower simulated percentile: about 5% of trajectories are at or below this value, subject to finite-run error.
- P95 availability
- An upper simulated percentile: about 95% of trajectories are at or below this value.
- Target attainment
- The fraction of trajectories whose availability equals or exceeds the entered target.
FREQUENTLY ASKED QUESTIONS
Questions specific to finite-horizon availability
Why does the simulation differ from the analytic availability page?
This page uses a finite horizon, discrete steps, seeded random transitions, and all-up initial states. The analytic page reports a steady-state probability without finite-run sampling error.
How many time steps should I use?
Choose delta t small relative to both MTTR and MTBF, then compare a finer grid. If the decision metric moves materially, the original grid was too coarse.
How many trajectories are enough?
Increase trajectories until mean, percentiles, and target attainment are stable across multiple seeds at the precision required for the decision.
Is P05 availability a guaranteed minimum?
No. It is a percentile of the simulated scenario and can change with assumptions, seed, iteration count, and model error.
Can the simulation start with components already down?
Not in this page; every trajectory starts all-up. A commissioning, degraded-start, or warm-spare study needs explicit initial-state inputs.
Does this model common-cause outages or repair queues?
No. Component transitions are independent and repairs have no crew or spare constraints. Those dependencies require an expanded event model.
RELIABLE SOURCES
Repairable-system and simulation references
IMPORTANT SIMULATION NOTE
More trajectories cannot repair a wrong model
A large run reduces Monte Carlo noise only. It does not remove time-step bias, repair-resource constraints, common-cause exposure, or bad MTBF and MTTR estimates. Validate those assumptions before using target attainment in a service commitment.