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  2. Examples of Markov chains - Wikipedia

    en.wikipedia.org/wiki/Examples_of_Markov_chains

    A game of snakes and ladders or any other game whose moves are determined entirely by dice is a Markov chain, indeed, an absorbing Markov chain. This is in contrast to card games such as blackjack, where the cards represent a 'memory' of the past moves. To see the difference, consider the probability for a certain event in the game.

  3. Markov chain - Wikipedia

    en.wikipedia.org/wiki/Markov_chain

    A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set (often representing time), but the precise definition of a Markov chain varies. [6]

  4. Markov chains on a measurable state space - Wikipedia

    en.wikipedia.org/wiki/Markov_chains_on_a...

    In 1953 the term Markov chain was used for stochastic processes with discrete or continuous index set, living on a countable or finite state space, see Doob. [1] or Chung. [2] Since the late 20th century it became more popular to consider a Markov chain as a stochastic process with discrete index set, living on a measurable state space. [3] [4] [5]

  5. Continuous-time Markov chain - Wikipedia

    en.wikipedia.org/wiki/Continuous-time_Markov_chain

    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 ...

  6. Markov renewal process - Wikipedia

    en.wikipedia.org/wiki/Markov_renewal_process

    A semi-Markov process (defined in the above bullet point) in which all the holding times are exponentially distributed is called a continuous-time Markov chain. In other words, if the inter-arrival times are exponentially distributed and if the waiting time in a state and the next state reached are independent, we have a continuous-time Markov ...

  7. Kolmogorov's criterion - Wikipedia

    en.wikipedia.org/wiki/Kolmogorov's_criterion

    Consider this figure depicting a section of a Markov chain with states i, j, k and l and the corresponding transition probabilities. Here Kolmogorov's criterion implies that the product of probabilities when traversing through any closed loop must be equal, so the product around the loop i to j to l to k returning to i must be equal to the loop the other way round,

  8. Markovian arrival process - Wikipedia

    en.wikipedia.org/wiki/Markovian_arrival_process

    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

  9. Additive Markov chain - Wikipedia

    en.wikipedia.org/wiki/Additive_Markov_chain

    An additive Markov chain of order m is a sequence of random variables X 1, X 2, X 3, ..., possessing the following property: the probability that a random variable X n has a certain value x n under the condition that the values of all previous variables are fixed depends on the values of m previous variables only (Markov chain of order m), and the influence of previous variables on a generated ...