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  2. Examples of Markov chains - Wikipedia

    en.wikipedia.org/wiki/Examples_of_Markov_chains

    A finite-state machine can be used as a representation of a Markov chain. Assuming a sequence of independent and identically distributed input signals (for example, symbols from a binary alphabet chosen by coin tosses), if the machine is in state y at time n , then the probability that it moves to state x at time n + 1 depends only on the ...

  3. Markov chain - Wikipedia

    en.wikipedia.org/wiki/Markov_chain

    Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term. While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space. [15]

  4. 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 ...

  5. Absorbing Markov chain - Wikipedia

    en.wikipedia.org/wiki/Absorbing_Markov_chain

    A basic property about an absorbing Markov chain is the expected number of visits to a transient state j starting from a transient state i (before being absorbed). This can be established to be given by the (i, j) entry of so-called fundamental matrix N, obtained by summing Q k for all k (from 0 to ∞).

  6. Discrete phase-type distribution - Wikipedia

    en.wikipedia.org/wiki/Discrete_phase-type...

    A terminating Markov chain is a Markov chain where all states are transient, except one which is absorbing. Reordering the states, the transition probability matrix of a terminating Markov chain with m {\displaystyle m} transient states is

  7. Balance equation - Wikipedia

    en.wikipedia.org/wiki/Balance_equation

    For a continuous time Markov chain (CTMC) with transition rate matrix, if can be found such that for every pair of states and = holds, then by summing over , the global balance equations are satisfied and is the stationary distribution of the process. [5]

  8. Stochastic matrix - Wikipedia

    en.wikipedia.org/wiki/Stochastic_matrix

    The stochastic matrix was developed alongside the Markov chain by Andrey Markov, a Russian mathematician and professor at St. Petersburg University who first published on the topic in 1906. [3] His initial intended uses were for linguistic analysis and other mathematical subjects like card shuffling , but both Markov chains and matrices rapidly ...

  9. Uniformization (probability theory) - Wikipedia

    en.wikipedia.org/wiki/Uniformization...

    The original chain is scaled by the fastest transition rate γ, so that transitions occur at the same rate in every state, hence the name. The method is simple to program and efficiently calculates an approximation to the transient distribution at a single point in time (near zero). [1] The method was first introduced by Winfried Grassmann in 1977.