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Stationary Distribution Calculator

Solve balance equations to find the long-term equilibrium probabilities of Markov chains

Equilibrium Analysis
Balance Equations
Long-term Behavior
Transition Matrix Setup
Enter your Markov chain's one-step transition matrix
P0:
P1:
About Stationary Distributions

Definition

A stationary distribution π is a probability distribution that satisfies:

π=πP\boldsymbol{\pi} = \boldsymbol{\pi} \mathbf{P}

where ∑jπj=1\sum_{j} \pi_j = 1 and πj≥0\pi_j \geq 0

Balance Equations

Each component satisfies:

πj=∑iπipij\pi_j = \sum_{i} \pi_i p_{ij}

This means the "flow into" state j equals the probability of being in state j.

Interpretation

  • Long-run proportion of time spent in each state
  • Limiting distribution as n → ∞
  • Equilibrium probabilities of the system

Existence & Uniqueness

  • Irreducible + Aperiodic + Positive Recurrent: Unique solution exists
  • Reducible: Multiple solutions possible
  • Transient/Null Recurrent: No non-trivial solution
Learn More About Stationary Distributions