Markov Chain Steady State Calculator
Find the steady-state (stationary) distribution of a 2×2 or 3×3 Markov chain transition matrix.
Calculate long-run probabilities for any Markov process.
What Is a Markov Chain? A Markov chain is a stochastic process where the next state depends only on the current state (not history). Named after Andrei Markov (Russia, 1906). Described by a transition matrix P where P[i][j] = probability of moving from state i to state j. Each row of P must sum to 1 (stochastic matrix).
Steady-State Distribution The steady-state (stationary) distribution π satisfies: π × P = π (π is unchanged after one step) Sum of all π_i = 1 This represents the long-run proportion of time spent in each state.
For a 2-State Chain Given: P = [[p, 1-p], [q, 1-q]] Steady state: π₁ = q / (q + (1-p)); π₂ = (1-p) / (q + (1-p)) Or: π₁ = q / ((1-p) + q), π₂ = (1-p) / ((1-p) + q)
Solving for 3-State Chains Set up the system: πP = π with constraint Σπ = 1. Replace one equation with Σπ_i = 1 and solve the linear system.
Convergence Any ergodic (irreducible + aperiodic) chain converges to the unique stationary distribution. Convergence speed depends on the second-largest eigenvalue (spectral gap). After many steps, starting state doesn’t matter — all chains reach the same steady state.
Real-World Examples Web page rank (Google PageRank uses Markov chains). Customer churn: loyal/at-risk/churned states. Weather: sunny/cloudy/rainy transitions. Gene regulatory networks, queuing theory, inventory models.
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This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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