Computing survival functions of the sum of two independent Markov processes: an application to bladder carcinoma treatment

The study of the sum of two independent phase-type (PH)-distributed variables, each of them being associated with a Markovian process with one absorbing state, is considered in this paper. The distribution function of the variable sum is computed, obtaining a new PH-distributed function of higher order. As the order increases in the new function, the exponential function of a block upper triangular matrix is calculated in terms of its respective blocks to reduce the dimension of the problem. The obtained results are applied to bladder carcinoma data.