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Markov Chain Steady State Calculator

Work out markov chain steady state instantly with clear inputs, formula shown and shareable results.

About Markov Chain Steady State Calculator

A two-state Markov chain with transition probabilities b (leaving state 1) and c (leaving state 2) has steady state π₁ = c/(b + c). The chain forgets its starting point at a rate governed by the second eigenvalue a + d − 1.

Formula

Two-state steady state

π₁ = c / (b + c), π₂ = b / (b + c)

Worked example

With the default values below, Markov Chain Steady State Calculator returns Long-run probability of state 1 of 0.57142857. Change any input above and every figure here is recalculated in your browser.

Inputs used

Probability of staying in state 1accepts 0–100 %
70 %
Probability of staying in state 2accepts 0–100 %
60 %
Starting probability in state 1accepts 0–100 %
100 %
Steps to simulateaccepts 1–10,000
5

Results

Long-run probability of state 1
0.57142857
Long-run probability of state 2
0.42857143
Probability in state 1 after your steps
0.57247000
Remaining gap to the steady state
0.0010414286

Step by step

  1. 1Values used
  2. 2Two-state steady state
  3. 3Long-run probability of state 1= 0.57142857
  4. 4Long-run probability of state 2= 0.42857143
  5. 5Probability in state 1 after your steps= 0.57247000
  6. 6Remaining gap to the steady state= 0.0010414286

Assumptions and limits

  • The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

With 70% staying in state 1 and 60% in state 2?

b = 0.3 and c = 0.4, so the steady state is π₁ = 4/7 ≈ 0.5714 and π₂ = 3/7 ≈ 0.4286.

Does the starting state matter?

Only in the short run. For an irreducible aperiodic chain the long-run distribution is independent of where you start.

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