Circular success and failure runs in a sequence of exchangeable binary trials

Let X1,…,Xn be an exchangeable sequence of binary trials arranged on a circle with possible values “1” (success) or “0” (failure). In an exchangeable sequence, the joint distribution of X1,X2,…,Xn is invariant under the permutation of its arguments. For the circular sequence, general expressions for the joint distributions of run statistics based on the joint distribution of success and failure run lengths are obtained. As a special case, we present our results for Bernoulli trials. The results presented consist of combinatorial terms and therefore provide easier calculations. For illustration purposes, some numerical examples are given and the reliability of the circular combined k-out-of-n:G and consecutive kc-out-of-n:G system under stress–strength setup is evaluated.