Compound Markov negative binomial distribution

Let {Yi}i≥1{Yi}i≥1 be a sequence of {0,1}{0,1} variables which forms a Markov chain with a given initial probability distribution and one-step transition probability matrix. Define NnNn to be the number of trials until the nnth success (“1”) in {Yi}i≥1{Yi}i≥1. In this paper, we study the distribution of the random variable T=∑i=1NnXi, where {Xi}i≥1{Xi}i≥1 is a sequence of independent and identically distributed random variables having a common phase-type distribution. The distribution of TT is obtained by means of phase-type distributions.