A Markov jump process associated with the matrix-exponential distribution

Abstract Let f be the density function associated to a matrix-exponential distribution of parameters $(\boldsymbol{\alpha}, T,\boldsymbol{{s}})$ . By exponentially tilting , we find probabilistic interpretation which generalizes one phase-type distributions. More specifically, show that for any sufficiently large $\lambda\ge 0$ $x\mapsto \left(\int_0^\infty e^{-\lambda s}f(s)\textrm{d} s\right)^{-1}e^{-\lambda x}f(x)$ can described in terms finite-state Markov jump process whose generator is tied T Finally, how revert exponential order assign itself.