Steady-state distribution

Definition

A steady-state distribution is a probability distribution that remains unchanged as time progresses in a stochastic process, particularly in the context of Markov chains. In this state, the system's long-term behavior can be analyzed, revealing the probabilities of being in each state after many transitions. The steady-state distribution is crucial for understanding equilibrium in systems modeled by Markov processes and indicates how the system behaves as it evolves over time.

5 Must Know Facts For Your Next Test

1. The steady-state distribution is often represented as a vector, with each entry corresponding to the probability of being in a particular state after a long time.
2. For an irreducible and aperiodic Markov chain, the steady-state distribution exists and is unique.
3. To find the steady-state distribution, one can solve the equation πP = π, where π is the steady-state vector and P is the transition matrix.
4. In practice, steady-state distributions help model various real-world processes like population dynamics, queuing systems, and financial markets.
5. If a Markov chain has absorbing states, the steady-state distribution may assign positive probabilities only to these absorbing states.

Review Questions

• How does the concept of steady-state distribution relate to the long-term behavior of a Markov chain?
  • The steady-state distribution reflects the long-term probabilities of being in each state of a Markov chain after many transitions. As time progresses, regardless of the initial state, the probabilities stabilize into this distribution. It helps analyze the equilibrium behavior of systems modeled by Markov chains, showing how they evolve over time and ultimately settle into predictable patterns.
• Describe how one would mathematically find the steady-state distribution for a given Markov chain and discuss its significance.
  • To find the steady-state distribution for a Markov chain, one typically sets up and solves the equation πP = π, where π represents the steady-state vector and P is the transition matrix. This involves determining probabilities for each state such that they sum to one and satisfy this equation. The significance lies in its ability to provide insights into long-term behaviors of systems under study, allowing predictions about future states based on current probabilities.
• Evaluate the implications of having multiple steady-state distributions in a Markov chain and how this affects system predictability.
  • When a Markov chain has multiple steady-state distributions, it indicates that different initial conditions or paths can lead to distinct long-term outcomes. This situation complicates predictions because it suggests that even with identical transition probabilities, starting from different states can yield different equilibria. Analyzing such systems requires understanding their structure and identifying conditions under which certain distributions are favored, making predictability more nuanced and dependent on initial conditions.

Related terms

Markov Chain: A stochastic model that describes a sequence of possible events where the probability of each event depends only on the state attained in the previous event.

Transition Matrix: A square matrix used to describe the transitions of a Markov chain, where each element represents the probability of moving from one state to another.

Ergodicity: A property of a Markov chain that ensures it will eventually reach a steady-state distribution regardless of its initial state.