On the Longest Run and the Waiting Time for the First Run in a Continuous Time Multi-State Markov Chain

In this paper, a marked point process with \(r+1\) $r+1$ types of marks (r types of successes \(S_{1},S_{2},\ldots ,S_{r}\) $S_1,S_2,\dots ,S_r$ and a failure F), \(r\ge 1\) $r\ge 1$ , that appear in continuous time according to a continuous-time Markov chain is considered. By constructing an appropriate embedded process using Markov chain embedding technique in continuous time, the exact distribution and its Laplace transform for the waiting time T until the first appearance of an \(S_{i}\) $S_i$ -run of length \(k_{i}\) $k_i$ , for \(i=1,2,\ldots ,r\) $i=1,2,\dots ,r$ (whichever comes first), are provided. The exact distribution of the length \(L_{t}\) $L_t$ of the longest run of successes in the time interval [0, t] is also derived. Further, the asymptotic distributions of T and \(L_{t}\) $L_t$ are obtained under general assumptions. Finally, numerical examples and applications in reliability theory, quality control and hypothesis testing are presented.