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  2. Stochastic matrix - Wikipedia

    en.wikipedia.org/wiki/Stochastic_matrix

    In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability . [ 1 ] [ 2 ] : 10 It is also called a probability matrix , transition matrix , substitution matrix , or Markov matrix .

  3. Markov chain - Wikipedia

    en.wikipedia.org/wiki/Markov_chain

    If the Markov chain is time-homogeneous, then the transition matrix P is the same after each step, so the k-step transition probability can be computed as the k-th power of the transition matrix, P k. If the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution π. [41]

  4. Markovian arrival process - Wikipedia

    en.wikipedia.org/wiki/Markovian_arrival_process

    A Markov arrival process is defined by two matrices, D 0 and D 1 where elements of D 0 represent hidden transitions and elements of D 1 observable transitions. The block matrix Q below is a transition rate matrix for a continuous-time Markov chain. [5]

  5. Chapman–Kolmogorov equation - Wikipedia

    en.wikipedia.org/wiki/Chapman–Kolmogorov_equation

    where P(t) is the transition matrix of jump t, i.e., P(t) is the matrix such that entry (i,j) contains the probability of the chain moving from state i to state j in t steps. As a corollary, it follows that to calculate the transition matrix of jump t, it is sufficient to raise the transition matrix of jump one to the power of t, that is

  6. Transition matrix - Wikipedia

    en.wikipedia.org/wiki/Transition_matrix

    Change-of-basis matrix, associated with a change of basis for a vector space. Stochastic matrix , a square matrix used to describe the transitions of a Markov chain . State-transition matrix , a matrix whose product with the state vector x {\displaystyle x} at an initial time t 0 {\displaystyle t_{0}} gives x {\displaystyle x} at a later time t ...

  7. Discrete phase-type distribution - Wikipedia

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

    The transition matrix is characterized entirely by its upper-left block . Definition. A distribution on { 0 , 1 , 2 , . . . } {\displaystyle \{0,1,2,...\}} is a discrete phase-type distribution if it is the distribution of the first passage time to the absorbing state of a terminating Markov chain with finitely many states.

  8. Continuous-time Markov chain - Wikipedia

    en.wikipedia.org/wiki/Continuous-time_Markov_chain

    Strictly speaking, the EMC is a regular discrete-time Markov chain. Each element of the one-step transition probability matrix of the EMC, S, is denoted by s ij, and represents the conditional probability of transitioning from state i into state j. These conditional probabilities may be found by

  9. 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 π i q i j = π j q j i {\displaystyle \pi _{i}q_{ij}=\pi _{j}q_{ji}} holds, then by summing over j {\displaystyle j} , the global balance equations are satisfied and π {\displaystyle \pi } is the stationary ...