Markov Chain Analysis
Model a system that moves between states with fixed probabilities. A Markov chain uses a transition matrix to predict how a system evolves and where it settles in the long run, from customer states to machine conditions to weather.
Analyze Markov Chain →What is a Markov Chain?
A Markov chain is a mathematical model of a system that moves between a set of states, where the probability of moving to the next state depends only on the current state, not on the history of how the system got there. This defining feature is the memoryless, or Markov, property.
The chain is described by a transition matrix, whose entries give the probability of moving from each state to each other state in one step. From this matrix, multi-step transition probabilities are found by matrix powers, letting you predict the distribution of states after any number of steps from a given starting point.
Many Markov chains converge to a steady-state (stationary) distribution: a long-run distribution of states that no longer changes from step to step, regardless of where the system started. This steady state answers questions like the long-run fraction of time a machine spends broken, or the eventual market share among competitors, and is one of the most useful outputs of Markov analysis.
In plain terms: A Markov chain models something that hops between states, working/broken, loyal/switching, sunny/rainy, where the next step depends only on where you are now, not how you got there. Give it the transition probabilities and it tells you where the system heads, including the long-run balance it settles into.
Key Concepts
Memoryless Property
The next state depends only on the current state, not the past. This is what defines a Markov chain.
Transition Matrix
A matrix of one-step probabilities between states. Its powers give multi-step transition probabilities.
Steady State
The long-run distribution of states that stops changing, independent of the starting state, for many chains.
Key Formulas
Using the Results
Short-term, the chain predicts the state distribution after any number of steps from a starting point. Long-term, the steady-state distribution gives the fraction of time spent in each state once the system settles, which for many chains is independent of where it began.
The steady state is often the practical goal: the long-run availability of equipment, the eventual market share, or the stationary distribution of a process. It exists and is unique when the chain is irreducible and aperiodic, conditions worth checking before relying on a single steady-state answer.
Assumptions & Validation
Markov Property
The next state depends only on the current state.
If violated: If history matters, expand the state definition or use a higher-order model.
Stationary Transitions
Transition probabilities do not change over time.
If violated: For time-varying probabilities, use a non-stationary model.
Well-Defined States
The states are mutually exclusive and collectively exhaustive.
If violated: Define states so every situation maps to exactly one.
⚠️ Check assumptions first
The Markov property, that the future depends only on the present state, is a strong assumption; if the system's behavior genuinely depends on its history, a simple Markov chain will mislead, and the state definition must be enriched to capture the relevant history. A unique steady state exists only when the chain is irreducible and aperiodic, so check those conditions before treating a steady-state distribution as the single long-run answer.
When NOT to Use Markov Chain
History-Dependent Systems
When outcomes depend on the full path, not just the current state, a basic Markov chain is inappropriate.
Waiting-Line Metrics
For queue lengths and waiting times specifically, queuing theory gives direct formulas.
Optimizing Decisions
To choose optimal actions across states over time, use a Markov decision process or dynamic programming.
Industry Applications
Reliability & Maintenance
Model machine states (working, degraded, failed) to find long-run availability.
Customer Behavior
Model transitions between customer states, such as loyalty or churn, and long-run shares.
Market Share
Model brand switching to project eventual market shares.
Process States
Model any system moving among discrete states with fixed transition probabilities.
Frequently Asked Questions
What is a Markov chain?
A Markov chain is a model of a system that moves between states, where the probability of the next state depends only on the current state and not on how the system arrived there. This is called the memoryless or Markov property. The chain is described by a transition matrix of one-step probabilities, from which future state distributions and long-run behavior can be computed.
What is the Markov (memoryless) property?
The Markov property states that the future evolution of the system depends only on its current state, not on the sequence of states that preceded it. In other words, knowing the present makes the past irrelevant for predicting the future. This assumption is what makes Markov chains tractable, but it must genuinely hold, or the state definition must be enriched so that the relevant history is captured in the current state.
What is a transition matrix?
A transition matrix is a square matrix whose entry in row i, column j gives the probability of moving from state i to state j in one step. Each row sums to one, since the system must move to some state. Raising the transition matrix to the nth power gives the n-step transition probabilities, and multiplying an initial distribution by the matrix advances the system one step.
What is a steady-state distribution?
A steady-state, or stationary, distribution is a long-run distribution of states that no longer changes from step to step. For many Markov chains, the system converges to this distribution regardless of its starting state. The steady state answers long-run questions such as the fraction of time a machine is operational or the eventual market share, and it is found by solving the equation that the distribution is unchanged by the transition matrix.
When does a unique steady state exist?
A unique steady-state distribution exists when the Markov chain is irreducible, meaning every state can eventually reach every other state, and aperiodic, meaning it does not cycle through states in a fixed period. Under these conditions the chain converges to a single stationary distribution independent of its start. If these conditions fail, the long-run behavior may depend on the starting state or may not settle to a single distribution.
How is a Markov chain different from a Markov decision process?
A Markov chain models how a system evolves under fixed transition probabilities, with no decisions involved. A Markov decision process adds actions and rewards, so at each state a decision-maker chooses an action that influences the transition and yields a reward, and the goal is to find an optimal policy. Markov decision processes are solved with dynamic programming, extending the descriptive Markov chain into a prescriptive decision model.
Model How a System Evolves and Settles
Compute transitions and the steady-state distribution of a Markov chain. Free during Beta.
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