  Phase-type distribution has been an important probabilistic tool in the
analysis of complex stochastic system evolution. It was introduced by Neuts
\cite{Neuts1975} in 1975. The model describes the lifetime distribution of a
finite-state absorbing Markov chains, and has found many applications in wide
range of areas. It was brought to survival analysis by Aalen \cite{Aalen1995}
in 1995. However, the model has lacks of ability in modeling heterogeneity and
inclusion of past information which is due to the Markov property of the
underlying process that forms the model. We attempt to generalize the
distribution by replacing the underlying by Markov mixtures process. Markov
mixtures process was used to model jobs mobility by Blumen \cite{Blumen} et al.
in 1955. It was known as the mover-stayer model describing low-productivity
workers tendency to move out of their jobs by a Markov chains, while those with
high-productivity tend to stay in the job. Frydman \cite{Frydman2005} later
extended the model to a mixtures of finite-state Markov chains moving at
different speeds on the same state space. In general the mixtures process does
not have Markov property. We revisit the mixtures model \cite{Frydman2005} for
mixtures of multi absorbing states Markov chains, and propose generalization of
the phase-type distribution under competing risks. The new distribution has two
main appealing features: it has the ability to model heterogeneity and to
include past information of the underlying process, and it comes in a closed
form. Built upon the new distribution, we propose conditional forward intensity
which can be used to determine rate of occurrence of future events (caused by
certain type) based on available information. Numerical study suggests that the
new distribution and its forward intensity offer significant improvements over
the existing model.
