Advanced Steady State Probability Calculator

Calculator Inputs

Use a transition matrix with rows representing the current state and columns representing the next state. The calculator supports 2 to 6 states.

Number of States

Displayed Step Count

Maximum Iterations

Convergence Tolerance

Normalize Rows Automatically

Normalize Initial Distribution Automatically

Transition Matrix

Rows = current state. Columns = next state.

Initial Distribution

Enter the starting probability for each state.

Example Data Table

This sample shows a 3-state Markov chain. The stationary distribution solves πP = π with probabilities summing to 1.

Current State	To S1	To S2	To S3	Row Total	Stationary Probability
S1	0.70	0.20	0.10	1.00	0.405405
S2	0.25	0.50	0.25	1.00	0.324324
S3	0.15	0.30	0.55	1.00	0.270270

Formula Used

The stationary or steady state probability vector is the long-run distribution of a Markov chain. It satisfies two conditions.

1. πP = π

2. π₁ + π₂ + ... + π_n = 1

3. P is the transition matrix, and π contains the state probabilities.

This calculator uses two ideas:

It solves the stationary system directly using linear equations.
It also multiplies the initial distribution by the transition matrix across many steps to show convergence behavior.

How to Use This Calculator

Select the number of states in your Markov chain.
Enter the transition probabilities for each row of the matrix.
Provide the initial distribution across all states.
Choose the displayed step, tolerance, and iteration limit.
Keep automatic normalization enabled if you want the tool to correct small total mismatches.
Click calculate to view steady state probabilities, step probabilities, row checks, and the Plotly chart.
Use the CSV and PDF buttons to export the result section.

Frequently Asked Questions

1. What is a steady state probability?

It is the long-run probability of being in each state after many transitions. When a chain stabilizes, these probabilities stop changing even after more matrix multiplications.

2. Do my transition rows need to sum to 1?

Yes. Each row represents all possible next-state outcomes from one current state, so the probabilities must total 1. This calculator can normalize rows automatically when enabled.

3. Why does the chart matter?

The chart shows how state probabilities move over iterations from the initial distribution. It helps you see whether the chain settles smoothly, slowly, or with visible oscillation.

4. What does tolerance control?

Tolerance sets how close the step distribution must get to the stationary distribution before the calculator marks the chain as effectively close. Smaller values require tighter agreement.

5. What if convergence is not reached?

The stationary solution may still exist, but the displayed iteration limit may be too low, or the chain may converge very slowly. Increase the maximum iterations and review the transition structure.

6. Can this handle absorbing states?

Yes. Absorbing states can be entered directly in the transition matrix. The results will reflect their long-run effect, including cases where one state eventually dominates.

7. Why normalize the initial distribution?

The starting probabilities should total 1. Automatic normalization is useful when values are proportion-like weights rather than exact probabilities, preventing minor input mistakes from blocking calculation.

8. Should I trust the stationary solution for every chain?

Interpret it with care. Some chains have multiple stationary patterns or unusual dynamics. The residual error, row checks, and convergence plot help you judge whether the result is practically meaningful.