On the joint distribution of success runs of several lengths in the sequence of MBT and its applications

Let $$X_1 ,X_2 ,\ldots ,X_n $$ X 1 , X 2 , … , X n be a sequence of Markov Bernoulli trials (MBT) and $$\underline{X}_n =( {X_{n,k_1 } ,X_{n,k_2 } ,\ldots ,X_{n,k_r } })$$ X ̲ n = ( X n , k 1 , X n , k 2 , … , X n , k r ) be a random vector where $$X_{n,k_i } $$ X n , k i represents the number of occurrences of success runs of length $$k_i \,( {i=1,2,\ldots ,r})$$ k i ( i = 1 , 2 , … , r ) . In this paper the joint distribution of $$\underline{X}_n $$ X ̲ n in the sequence of $$n$$ n MBT is studied using method of conditional probability generating functions. Five different counting schemes of runs namely non-overlapping runs, runs of length at least $$k$$ k , overlapping runs, runs of exact length $$k$$ k and $$\ell $$ ℓ -overlapping runs (i.e. $$\ell $$ ℓ -overlapping counting scheme), $$0\le \ell >k$$ 0 ≤ ℓ > k are considered. The pgf of joint distribution of $$\underline{X}_n $$ X ̲ n is obtained in terms of matrix polynomial and an algorithm is developed to get exact probability distribution. Numerical results are included to demonstrate the computational flexibility of the developed results. Various applications of the joint distribution of $$\underline{X}_n $$ X ̲ n such as in evaluation of the reliability of $$( {n,f,k})\!\!:\!\!G$$ ( n , f , k ) : G and $$>n,f,k>\!:\!\!G$$ > n , f , k > : G system, in evaluation of quantities related to start-up demonstration tests, acceptance sampling plans are also discussed. Copyright Springer-Verlag Berlin Heidelberg 2014