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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 ...
The continuous time Markov chain is characterized by the transition rates, the derivatives with respect to time of the transition probabilities between states i and j. Let X t {\displaystyle X_{t}} be the random variable describing the state of the process at time t , and assume the process is in a state i at time t .
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 ...
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
According to the figure, a bull week is followed by another bull week 90% of the time, a bear week 7.5% of the time, and a stagnant week the other 2.5% of the time. Labeling the state space {1 = bull, 2 = bear, 3 = stagnant} the transition matrix for this example is
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,
So look, I think this is a real testament to the value of business and the importance of business being an example of how continuous learning and renewal plays out.
However, for continuous-time Markov decision processes, decisions can be made at any time the decision maker chooses. In comparison to discrete-time Markov decision processes, continuous-time Markov decision processes can better model the decision-making process for a system that has continuous dynamics , i.e., the system dynamics is defined by ...