Computing Waiting Time Probabilities Related To (<i>k</I><sub>1< <i>k</I><sub>2< ..., <i>k<sub>l</Sub>< Pattern

Abstract

For a sequence of multi-state trials with l possible outcomes denoted by {1, 2, ..., l}, let E be the event that at least k(1) consecutive is followed by at least k(2) consecutive 2s,..., followed by at least k(l) consecutive ls. Denote by T-r the number of trials for the rth occurrence of the event E in a sequence of multi-state trials. This paper studies the distribution of the waiting time random variable T-r when the sequence consists of independent and identically distributed multi-state trials. In particular, distributional properties of T-r are examined via matrix-geometric distributions.