Markov Chain Steady State Calculator

Welcome to the Markov Chain Steady State Calculator, a powerful mathematical tool for computing the long-run stationary distribution of any finite Markov chain. Enter your transition matrix and instantly see the steady-state probabilities, an interactive state transition diagram, convergence visualization, and detailed step-by-step solution. Ideal for students, researchers, and professionals working with stochastic processes.

What is a Steady-State Distribution?

A steady-state distribution (also called a stationary distribution) of a Markov chain is a probability vector \(\pi\) such that:

This means that if the system starts in distribution \(\pi\), it remains in \(\pi\) after any number of transitions. Intuitively, \(\pi_i\) represents the long-run proportion of time the system spends in state \(i\).

Key Concepts

How to Solve for the Steady State

The steady-state vector \(\pi\) can be found by solving the system of linear equations derived from \(\pi P = \pi\):

1. Rewrite the equation: \(\pi P = \pi\) becomes \(\pi(P - I) = 0\), or equivalently \((P^T - I)\pi^T = 0\).
2. Add normalization: Replace one redundant equation with \(\pi_1 + \pi_2 + \cdots + \pi_n = 1\).
3. Solve the system: Use Gaussian elimination or matrix methods to find \(\pi\).

For ergodic chains, repeated multiplication converges to the unique steady state regardless of the starting distribution.

How to Use This Calculator

1. Enter the transition matrix: Input your matrix with each row on a new line. Values can be separated by commas or spaces. Each row must sum to 1.
2. Add state labels (optional): Provide descriptive names for your states (e.g., Sunny, Rainy) separated by commas.
3. Set decimal precision: Choose the number of decimal places (2-15) for results.
4. Calculate: Click "Calculate Steady State" to see the full analysis including the stationary distribution, convergence chart, state diagram, and step-by-step solution.

Understanding Your Results

Steady-State Vector

The main output is the vector \(\pi = (\pi_1, \pi_2, \ldots, \pi_n)\), where each \(\pi_i\) represents the long-run probability of being in state \(i\). The state with the highest probability is the dominant state.

Convergence Chart

This shows how the probability distribution evolves from a uniform start through successive multiplications by P. Faster convergence indicates a more strongly mixing chain.

State Transition Diagram

An interactive visual representation where:

• Node size reflects the steady-state probability
• Edge thickness represents transition probability
• Curved arrows show the direction of transitions
• Self-loops indicate probability of remaining in the same state

Real-World Applications

When Does a Unique Steady State Exist?

A Markov chain has a unique steady-state distribution when it is ergodic (both irreducible and aperiodic):

• Irreducible: Every state can be reached from every other state (no disconnected components)
• Aperiodic: The GCD of all cycle lengths through any state is 1 (no fixed periodicity)

If the chain is reducible or periodic, it may still have a stationary distribution, but it may not be unique, and convergence is not guaranteed from all starting distributions.

Frequently Asked Questions

What is a steady-state distribution of a Markov chain?

A steady-state (or stationary) distribution is a probability vector π such that πP = π, where P is the transition matrix. It represents the long-run proportion of time the system spends in each state, regardless of the initial state. For an irreducible and aperiodic Markov chain, the steady-state distribution is unique.

How do you calculate the steady-state probabilities?

To find the steady-state vector π, solve the system πP = π subject to the constraint that all probabilities sum to 1 (Σπᵢ = 1). This is equivalent to solving (Pᵀ - I)π = 0 with the normalization constraint. You can also use power iteration: repeatedly multiply an initial distribution by P until convergence.

When does a Markov chain have a unique steady-state distribution?

A Markov chain has a unique steady-state distribution when it is both irreducible (every state can be reached from every other state) and aperiodic (the chain does not cycle with a fixed period). Together, these properties make the chain ergodic, guaranteeing convergence to a unique stationary distribution.

What is the mean return time in a Markov chain?

The mean return time for state i is the expected number of steps to return to state i starting from state i. For an ergodic Markov chain, the mean return time equals 1/πᵢ, where πᵢ is the steady-state probability of state i. States with higher steady-state probability have shorter mean return times.

What is the difference between a transition matrix and a steady-state vector?

A transition matrix P is an n×n matrix where P(i,j) gives the probability of moving from state i to state j in one step. Each row sums to 1. The steady-state vector π is a 1×n probability vector representing the long-run distribution across states. While P describes single-step dynamics, π describes the equilibrium behavior.

Additional Resources

• Markov Chain - Wikipedia
• Stationary Distribution - Wikipedia
• Stochastic Matrix - Wikipedia