Consider three basic operations arising in reliability theory: finite mixtures, finite convolutions, formation of coherent systems of independent components. In this paper it is shown that: (i) The class of all phase type distributions is closed under all three operations; (ii) the class of all phase type distributions having a representation in which the matrix is upper triangular is the smallest class which is closed under all three operations and contains all exponential distributions. The paper includes a section of preliminaries in which the relevant material regarding phase type distributions and reliability theory is summarized.
"Closure of Phase Type Distributions Under Operations Arising in Reliability Theory." Ann. Probab. 10 (1) 265 - 269, February, 1982. https://doi.org/10.1214/aop/1176993932