Dependent sequences of MPH* distributed random vectors

Phase-type (PH) renewal processes are renewal processes where the inter-arrival times are PH distributed. A new Markov jump process can be defined from the concatenation of the sequence of Markov jump processes underlying each inter-arrival interval. From this construction one can define the Markovian arrival process as a process where some transitions lead to new arrivals. The inter-arrival intervals are PH distributed but typically now with some dependence structure.

The class of multivariate PH distributions of the MPH* type can be constructed from linear rewards on the states of sub-intensity matrices of PH distributions, so one can define a sequence of MPH* variables just like the PH renewal process. A sequence of MPH* random variables with dependence structure can thus be defined by having the same underlying Markov jump process, where the reward accumulation is reset, at certain transitions. We present formulae for the joint distribution of consecutive intervals and moments. Finally an application of the theory is presented on a bivariate dataset containing times between failures and distances driven by locomotives.