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One method of finding the stationary probability distribution, π, of an ergodic continuous-time Markov chain, Q, is by first finding its embedded Markov chain (EMC). Strictly speaking, the EMC is a regular discrete-time Markov chain, sometimes referred to as a jump process.
Suppose that one starts with $10, and one wagers $1 on an unending, fair, coin toss indefinitely, or until all of the money is lost. If represents the number of dollars one has after n tosses, with =, then the sequence {:} is a Markov process. If one knows that one has $12 now, then it would be expected that with even odds, one will either have ...
The mixing time of a Markov chain is the number of steps needed for this convergence to happen, to a suitable degree of accuracy. A family of Markov chains is said to be rapidly mixing if the mixing time is a polynomial function of some size parameter of the Markov chain, and slowly mixing otherwise. This book is about finite Markov chains ...
The Markov-modulated Poisson process or MMPP where m Poisson processes are switched between by an underlying continuous-time Markov chain. [8] If each of the m Poisson processes has rate λ i and the modulating continuous-time Markov has m × m transition rate matrix R , then the MAP representation is
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to ...
Markov chain approximation method; Markov chain central limit theorem; Markov chain mixing time; Markov chain tree theorem; Markov Chains and Mixing Times; Markov chains on a measurable state space; Markov decision process; Markov information source; Markov kernel; Markov chain; Markov property; Markov renewal process; Markov reward model ...
Intuitively, a stochastic matrix represents a Markov chain; the application of the stochastic matrix to a probability distribution redistributes the probability mass of the original distribution while preserving its total mass. If this process is applied repeatedly, the distribution converges to a stationary distribution for the Markov chain.
A Markov chain with two states, A and E. In probability, a discrete-time Markov chain (DTMC) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past.