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
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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.
Frequently asked questions
How do I use Markov Chain Simulator step by step without mistakes?
Follow the order given in the usage section: Compute N-step transition probabilities for a Markov chain. Includes stationary distribution and a state diagram. The step-by-step part matters less than the unit check that sits inside it. Confirm every unit matches the thing you are actually trying to measure before you press calculate - that single check catches more wrong answers than any other habit, and it costs about ten seconds.
What are the common mistakes when using Markov Chain Simulator?
Most common mistakes come down to four habits: entering 20 when the field expects 0.2 because the field is a percentage; leaving a field blank and accepting the default without checking what it stands for; changing one input and reading a result that was calculated before the change; and quoting the output as an exact figure when the page describes it as a reference estimate. Press calculate again after any edit, and treat the number as a reference point rather than a final answer.
How do I read the result from Markov Chain Simulator?
Read the final figure first, then come back to this section for the conditions behind it: Compute N-step transition probabilities for a Markov chain. Includes stationary distribution and a state diagram. Keep two things apart: the number itself, which comes from the values you typed, and the assumptions it depends on, which decide whether that number is meaningful. The page sets out what a higher or lower result means for your case, so check that direction as well as the magnitude.
Can I act on the Markov Chain Simulator result directly?
Treat it as one reference sheet before you decide, rather than as the answer itself: 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. P ProbL The output describes a magnitude under the conditions you entered. It does not include what it actually costs to act, what might change while you act, or how the other side behaves. Before you commit, compare the result against reality and note which assumptions do not hold where you are.
What inputs does Markov Chain Simulator need?
Markov Chain Simulator takes 4 inputs: Preset、Transition matrix (rows = from, cols = to)、Initial state index (0-based)、Steps N. They are listed in the order they feed the calculation, so an earlier value can change a later one. Do not skip a field you do not recognise - work out what it means first, because a silently guessed value produces a number that looks reasonable and is not. Before typing anything, make sure every figure uses the same unit and the same time period.
What should I check before I run Markov Chain Simulator?
Three checks pay for themselves. First, confirm the question you need answered is really inside the scope of this tool rather than a neighbouring one. Second, confirm that the figures you have are drawn from the same source and the same period - mixing monthly and annual numbers, or two different definitions of the same quantity, quietly distorts the result. Third, keep the key inputs written down, so that later you can tell which set of numbers produced a figure you are about to quote.