Abstract
Continuous-time Markov chainsContinuous-time Markov chain (CTMCs)CTMC seealso Continuous-time Markov chain Markov chain seealso Continuous-time Markov chain are a class of stochastic processes with a discrete state space in which the time between transitions follows an exponential distribution. In this section, we first provide the basic definitions for CTMCs and notations associated with this model. We then proceed with an explanation of the basic concepts for phase-type distributions (PHDs) and the analysis of such models. For theoretical details about CTMCs and related stochastic processes we refer to the literature [151].
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Notes
- 1.
As mentioned in Sect. 2.1.2 we assume that the point mass at zero, i.e., the probability of starting in the absorbing state is 0. If the absorbing state may have an initial probability greater than zero the number of independent parameters increases to 2n and the matrix representation has n 2 + n parameters.
- 2.
If the case that \(\boldsymbol{{\pi }}^{(A)}(n + 1) = 0\), i.e., there is no start in the absorbing state, the random variable X (A) is strictly positive. Then the initial probability vector is given by \(\boldsymbol{{\pi }}^{(C)} = {[\boldsymbol{\pi }}^{(A)},\boldsymbol{ 0}]\) where \(\boldsymbol{0}\) is the row m-vector of 0’s.
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Buchholz, P., Kriege, J., Felko, I. (2014). Phase-Type Distributions. In: Input Modeling with Phase-Type Distributions and Markov Models. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-06674-5_2
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