Calculator Inputs
Enter state names, a valid initial distribution, and a square transition matrix. Large screens use three columns, smaller screens use two, and mobile uses one.
Example Data Table
| State | Initial Probability | Reward | Transition to Sunny | Transition to Cloudy | Transition to Rainy |
|---|---|---|---|---|---|
| Sunny | 0.50 | 8 | 0.70 | 0.20 | 0.10 |
| Cloudy | 0.30 | 5 | 0.30 | 0.40 | 0.30 |
| Rainy | 0.20 | 1 | 0.20 | 0.30 | 0.50 |
This example models weather as a three-state stochastic process. You can replace the labels and probabilities with customer churn stages, credit ratings, machine states, or queue conditions.
Formula Used
1. State forecast: π(n) = π(0)Pⁿ, where π(0) is the initial distribution and P is the transition matrix.
2. Expected reward at step n: E[Rₙ] = π(n) · r, where r is the reward vector.
3. Cumulative expected reward: ∑ from t=0 to n of π(t) · r.
4. Expected visits over the horizon: ∑ from t=0 to n of π(t) for each state.
5. Approximate steady-state distribution: repeatedly multiply the current distribution by P until changes become negligible.
6. Absorbing analysis: when absorbing states exist, the calculator uses N = (I − Q)⁻¹ and B = NR to estimate absorption probabilities.
How to Use This Calculator
- Enter the state names in the order you want to model them.
- Provide the initial distribution with one probability per state.
- Paste a square transition matrix using one row per line.
- Add a reward vector to score each state numerically.
- Choose forecast steps and the number of simulation runs.
- Optionally set a target state, forced start state, and random seed.
- Press the submit button to place results above the form.
- Use the CSV or PDF buttons to export the generated output.
Frequently Asked Questions
1. What does this simulator calculate?
It forecasts future state probabilities, simulates sample paths, estimates expected visits, computes rewards, approximates steady-state behavior, and checks absorbing-state outcomes when applicable.
2. Do my matrix rows need to total exactly 1?
Yes. Each row represents all possible next-state outcomes from one current state, so the total probability for that row must equal 1.
3. What is the difference between forecast and simulation?
The forecast uses exact matrix multiplication. The simulation uses random sampling many times, which helps you compare theoretical and empirical results.
4. Why include a reward vector?
Rewards let you convert state occupancy into business value, cost, utility, or score. This is useful for finance, operations, reliability, and customer lifecycle analysis.
5. What is a steady-state distribution?
It is the long-run probability pattern that remains stable after repeated transitions, assuming the chain converges under the structure of your matrix.
6. When does absorbing analysis appear?
It appears when the matrix contains at least one absorbing state, meaning a state that transitions to itself with probability 1.
7. Can I model non-weather examples here?
Yes. You can model churn stages, website journeys, disease progression, machine conditions, inventory states, credit ratings, and many other discrete processes.
8. Why use a random seed?
A fixed seed makes simulation output reproducible. This helps when auditing results, comparing assumptions, or sharing the same scenario with teammates.