Higher order moments and conditional asymptotics of the batch markovian arrival process

We consider the number of arrivals in a Batch Markovian Arrival Process (BMAP) and derive matrix analytic expressions for its moments of arbitrary order. These expressions consist of decomposition formulas connected to the semigroup structure of the moments, forward and backward differential equations, and recursive is well as direct, integral formulas. This extends earlier work by Narayana and Neuts on the first two factorial moment matrices. We next turn to the terminating BMAP i.e., a BMAP with an absorbing state in which no arrivals occur. We consider the asymptotic behavior of the moments conditional on the process not yet having terminated. We show that the conditional mean and variance possess affine asymptotics and derive the coefficients explicitly. Finally, we discuss how parts of our work also apply to the more general class of Rational Arrival Processes (RAPs).