LAUSR

In this paper, we study the distribution of $$X_{n,k}^c $$Xn,kc, the number of occurrences of success runs of length k in the sequence of n Markov Binary Trials arranged on circle under four popular schemes of counting runs. Such distribution is referred as circular Markov binomial distribution of order k and is studied for the first time in this paper. The pgf of $$X_{n,k}^c $$Xn,kc is expressed in the form of matrix polynomial and an algorithm is developed to obtain the exact probability distribution. We include some numerical results in order to demonstrate feasibility and simplicity of the theoretical results developed. Further we discuss applications of the distribution of circular binomial run statistic $$X_{n,k}^c $$Xn,kc in studying the distribution of length of longest success run on circle and also in evaluation of circular reliability systems.