The exact distribution of overlapping counting patterns associated with a sequence of homogeneous Markov-dependent multi-state trials: An application to SECON.

*(English)*Zbl 0901.60059The distribution of the number of non-overlapping consecutive \(k\) successes \((N_{n,k})\) in a sequence of \(n\) Bernoulli trials has been widely studied and applied to many practical situations. For instance, the probability of \(N_{n,k} =0\) is the reliability of a consecutive-\(k\)-out-of-\(n:F\) system, and the tail probability of \(N_{n,k} <m\) is the reliability of an \(m\)-consecutive-\(k\)-out-of-\(n:F\) system. When the counting method is overlapping permissible and the sequence restrictions relax to the case of homogeneous Markov-dependent multi-state trials, then the resulting distribution problem of \(N_{n,k}\) can be used to model a variety of real-world processes, but such cases have not been investigated in the literature. In this article, we present some results based on the finite Markov chain imbedding technique for the exact distribution of \(N_{n,k}\) with \(k=2\), and focus on its application to an important measurement of continuity in the health-care sector called SECON. Here continuity means that a patient sees the same health-care provider (physicians or nurse) on two consecutive visits. The objective of this article is to present new mathematical formulations for and to link the medical care system to consecutive-\(k\)-out-of-\(n\) systems.

##### MSC:

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

90B25 | Reliability, availability, maintenance, inspection in operations research |

##### Keywords:

stationary Markov chain; transition probability matrix; sequential continuity of care; finite Markov chain imbedding
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\textit{W. Y. W. Lou}, Nonlinear Anal., Theory Methods Appl. 30, No. 6, 3349--3357 (1997; Zbl 0901.60059)

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