Calculator Input
Example Data Table
This example uses three market states. Each row shows the next-period transition probabilities.
| Current State | Low Demand | Medium Demand | High Demand |
|---|---|---|---|
| Low Demand | 0.70 | 0.20 | 0.10 |
| Medium Demand | 0.30 | 0.40 | 0.30 |
| High Demand | 0.20 | 0.30 | 0.50 |
Formula Used
For a row stochastic transition matrix, the steady state vector satisfies:
πP = π
The probability values must also satisfy:
π1 + π2 + ... + πn = 1
For a column stochastic transition matrix, the steady state vector satisfies:
Pπ = π
The iterative method repeatedly applies the transition rule until the largest change is below the selected tolerance.
How to Use This Calculator
- Select the number of states in the Markov chain.
- Choose row stochastic or column stochastic format.
- Enter the transition matrix with one matrix row per line.
- Enter the initial distribution, or leave it blank.
- Choose tolerance, iteration limit, and decimal places.
- Use normalization when small input totals need correction.
- Press the calculate button to view the result above the form.
- Download the report as CSV or PDF when needed.
Steady State Markov Chain Guide
Why Steady State Probabilities Matter
A Markov chain describes movement between states. Each move follows fixed transition probabilities. After many moves, some chains settle into a stable pattern. That pattern is called the steady state distribution. It shows the long run chance of being in each state.
This idea supports many statistical decisions. It can model customer status, website navigation, machine condition, inventory levels, queue positions, weather classes, credit ratings, and game states. The first state may matter early. Yet the steady state often explains the future mix better.
What the Calculator Evaluates
This calculator accepts a transition matrix and an optional starting distribution. It checks whether rows or columns behave like probability groups. It can normalize small entry mistakes when requested. It then estimates the steady state by repeated multiplication. It also solves the stationary equations with a linear system.
The two methods help verify the answer. Iteration is useful because it shows convergence behavior. The equation method is useful because it directly enforces the balance rule. When both answers match, the model is usually stable and well formed.
Interpreting the Output
A larger steady probability means that the process spends more time in that state. It does not mean the chain will remain there forever. It means that, over a long horizon, visits to that state should form that share of all visits.
The convergence error shows the largest change between two final iterations. A small value supports a reliable estimate. A high value may mean the matrix needs more iterations. It may also suggest periodic behavior or a chain with slow mixing.
Good Modeling Practices
Use states that are clear and mutually exclusive. Every transition value should be nonnegative. Each probability group should sum to one. Use enough decimal places when transitions are small. Test several starting distributions when the chain has special structure.
Absorbing states need careful reading. Their steady state may place all mass on final classes. Reducible chains can have more than one limiting answer. In those cases, the starting distribution may affect the long run result. Always connect the numeric answer to the story behind the states.
Document assumptions with the export buttons. Keep records for review, audits, and later model comparisons too.
FAQs
What is a steady state probability?
It is the long run share of time spent in each state after many transitions.
Does every Markov chain have one steady state?
Many finite chains have stationary distributions. A unique limiting distribution needs stronger conditions, such as irreducibility and aperiodicity.
What is a row stochastic matrix?
It is a transition matrix where every row contains nonnegative probabilities that sum to one.
What is a column stochastic matrix?
It is a transition matrix where every column contains nonnegative probabilities that sum to one.
Why use the equation method?
It solves the stationary balance equations directly and gives a useful comparison against iteration.
Why use the iterative method?
It shows how the starting distribution changes over repeated transitions and whether it converges.
What does normalization do?
It rescales each selected row or column so its total becomes one, when the total is positive.
Can absorbing states affect the answer?
Yes. Absorbing states can pull all long run probability into final states, depending on the chain structure.