What is a steady-state distribution?

The steady-state (or stationary) distribution is a probability vector π such that πP = π, where P is the transition matrix. It represents the long-run fraction of time spent in each state for an ergodic chain.

Why must each row sum to 1?

A row-stochastic matrix models probabilities of moving from one state to all possible next states. Because the chain must go somewhere, those probabilities must total 1 for every row.

What is the k-step distribution?

Starting from an initial probability distribution v, the distribution after k steps is v × P^k. This tells you the probability of being in each state after exactly k transitions.

What method is used to find the steady state?

The tool uses power iteration: it repeatedly multiplies the current distribution vector by the transition matrix until successive results differ by less than 1e-10, indicating convergence to the stationary distribution.

Does every Markov chain have a unique steady state?

Only ergodic chains (irreducible and aperiodic) are guaranteed a unique stationary distribution. Chains with absorbing states or multiple communicating classes may have multiple stationary distributions or none that the power method converges to.

How many states can I model?

This tool supports 2 to 6 states, which covers most textbook and introductory examples. Larger matrices are better handled with dedicated numerical software.

Markov Chain Calculator

Enter a stochastic transition matrix and an initial distribution to get the steady-state vector and the k-step distribution instantly.

Number of States (n)

Steps (k)

Transition Matrix P

Row i = probabilities of moving FROM state i. Each row must sum to 1.

Initial Distribution v₀

Probability of starting in each state. Must sum to 1.

Summary

This Markov Chain Calculator accepts an n×n row-stochastic transition matrix and computes two key results: the steady-state (stationary) probability vector that the chain converges to regardless of starting conditions, and the state distribution after exactly k steps from a user-supplied initial distribution. All computation runs locally in your browser using power-iteration and basic linear algebra — no data is sent to any server.

How it works

Select the number of states n (2 to 6) to generate the n×n matrix grid.
Fill in each cell with the transition probability from row-state i to column-state j.
Each row must sum to 1; a live row-sum indicator confirms validity.
Enter the initial probability distribution (must also sum to 1).
Set the number of steps k and click Calculate.
The tool runs power iteration to find the steady-state vector and matrix-multiplies the initial vector k times for the k-step distribution.

Use cases

Model customer retention and churn across subscription tiers.
Analyze page-rank style link graphs for small networks.
Simulate weather-pattern Markov models in statistics coursework.
Verify hand-calculated steady-state answers on homework problems.
Prototype board-game or inventory-level Markov models quickly.
Understand convergence behavior by comparing k-step vs. steady-state.

Frequently Asked Questions

Last updated: 2026-05-28 · Reviewed by Nham Vu

Markov Chain Calculator

Transition Matrix P

Initial Distribution v₀

Steady-State Distribution (π)

Distribution after 5 Steps

P^k (Transition Matrix after 5 Steps)

Summary

How it works

Use cases

Frequently Asked Questions

Markov Chain Calculator

Use the Markov Chain Calculator

Transition Matrix P

Initial Distribution v0

Steady-State Distribution (π)

Distribution after 5 Steps

Pk (Transition Matrix after 5 Steps)

Summary

How it works

Use cases

Frequently Asked Questions

Initial Distribution v₀

P^k (Transition Matrix after 5 Steps)