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

Eutichia Vaggelatou ()
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Eutichia Vaggelatou: National and Kapodistrian University of Athens

Methodology and Computing in Applied Probability, 2024, vol. 26, issue 4, 1-26

Abstract: Abstract 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 , … , S r and a failure F), $$r\ge 1$$ r ≥ 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 , … , 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.

Keywords: Success runs; Marked point process; Continuous-time Markov chain; Markov chain embedding technique; Waiting time; Longest run; Exact distribution; Laplace transform; Approximating distribution; Exponential distribution; Erlang distribution; Extreme value distribution; Erdos-Renyi type law; Reliability model with shocks; Quality control charts; Independence test; Primary 60J28; 60E10; Secondary 62E15; 60G40 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11009-024-10127-5

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