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Continuous-time Markov chain
In probability theory, a continuous-time Markov chain (CTMC or continuous-time Markov process) is a mathematical model which takes values in some finite state space and for which the time spent in each state takes non-negative real values and has an exponential distribution. It is a continuous-time stochastic process with the Markov property which means that future behaviour of the model (both remaining time in current state and next state) depends only on the current state of the model and not on historical behaviour. The model is a continuous-time version of the Markov chain model, named because the output from such a process is a sequence (or chain) of states.
==Definitions==
A continuous-time Markov chain (''X''''t'')''t'' ≥ 0 is defined by a finite or countable state space ''S'', a transition rate matrix ''Q'' with dimensions equal to that of the state space and initial probability distribution defined on the state space. For ''i'' ≠ ''j'', the elements ''q''''ij'' are non-negative and describe the rate of the process transitions from state ''i'' to state ''j''. The elements ''q''''ii'' are chosen such that each row of the transition rate matrix sums to zero.
There are three equivalent definitions of the process.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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