Compound Markov negative binomial distribution.

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