Tools/Markov Chain Simulator

Markov Chain Simulator

Enter a transition matrix (rows sum to 1) and get N-step transition probabilities, the stationary distribution, and a state diagram. Includes a weather preset for intuition.

Setup

Preset Transition matrix (rows = from, cols = to) Initial state index (0-based) Steps N

N-step distribution

Stationary distribution

If the chain is irreducible and aperiodic, this is the long-run fraction of time spent in each state.

Why this exists

A Markov chain has "no memory" — the next state depends only on the current state, not on the history of how you got there. This makes them tractable to compute exactly (linear algebra) and they're used everywhere: Google's PageRank, weather modeling, customer churn, board game state machines.

Who uses it

Data scientists modeling customer journeys and retention.
Game designers tuning AI behavior trees.
Educators teaching the basics of stochastic processes.

Limitations

If the chain has absorbing states (rows of zeros except for one column = 1), the stationary distribution isn't defined as we report it. Try a different matrix.
Period 2 chains (e.g. alternating states) don't converge — we still report the limit of P^N.