Calculated Result
Input Error
Calculator Form
Example Data Table
This example uses a three state chain with labels A, B, and C.
| From \ To | A | B | C | Row Sum |
|---|---|---|---|---|
| A | 0.50 | 0.30 | 0.20 | 1.00 |
| B | 0.20 | 0.60 | 0.20 | 1.00 |
| C | 0.10 | 0.40 | 0.50 | 1.00 |
| State | Steady State Probability | Percentage |
|---|---|---|
| A | 0.244898 | 24.4898% |
| B | 0.469388 | 46.9388% |
| C | 0.285714 | 28.5714% |
Formula Used
Let P be the transition matrix. The steady state row vector is π.
πP = π
∑πi = 1
This calculator solves the linear system formed from PT - I and replaces one equation with the normalization condition.
It also checks the iterative relation vk+1 = vkP to show whether the distribution approaches the stationary vector.
If the chain is irreducible and aperiodic, the long run distribution approaches the steady state from any valid starting distribution.
How to Use This Calculator
- Choose the number of states.
- Enter state labels or keep the defaults.
- Fill the transition matrix with row probabilities.
- Make sure every row sums to 1.
- Enter an initial distribution or leave the default values.
- Set tolerance, decimal places, and maximum iterations.
- Press Calculate Steady State.
- Review the solved stationary vector, diagnostics, and convergence graph.
- Use CSV or PDF export when needed.
Frequently Asked Questions
1) What is a steady state probability?
It is the long run probability of being in each state. It does not depend on the current step when the chain has settled into equilibrium.
2) Why must each row sum to 1?
Each row lists all possible next step outcomes from one state. Since one of those outcomes must occur, their total probability must equal 1.
3) Does every Markov chain have a unique steady state?
No. A finite irreducible chain has a unique stationary distribution, but a reducible chain can have several stationary distributions or initial dependent limits.
4) Why can solved and iterative results differ?
The exact solution solves the stationary equations directly. Iteration depends on chain behavior, tolerance, starting distribution, and whether the chain is periodic or reducible.
5) What does tolerance control?
Tolerance sets the stopping rule for the power iteration. A smaller value usually needs more steps but gives a tighter convergence check.
6) Can I enter fractions like 1/3?
This version expects decimal values. Convert fractions to decimals before entry so the matrix can be validated and calculated correctly.
7) What does period mean in this result?
Period describes cycle timing in an irreducible chain. A period of 1 means aperiodic behavior, which supports convergence to the long run distribution.
8) How many states can this calculator handle?
This page supports two to eight states. That range keeps the matrix readable while still covering many textbook and applied examples.