In this paper, six new \(({k}_{1},{k}_{2},\dots ,{k}_{m})\)-runs with multi-state trials are proposed creatively, which can satisfy the practical needs in many fields. The exact distributions of proposed runs are obtained by applying finite Markov chain imbedding approach. This paper not only studies the case of independent identical distribution (i.i.d.) multi-state trials, but also independent non-identical distribution (non-i.i.d.) multi-state trials. Numerical examples have served the purpose to illustrate the effectiveness of the proposed approach. This study is of reference value and application significance for similar runs.

在本文中，创造性地提出了六个新的\(({k}_{1},{k}_{2},\dots ,{k}_{m})\) -runs with multi-state trial，它们可以满足很多领域的实际需要。通过应用有限马尔可夫链嵌入方法获得建议运行的精确分布。本文不仅研究了独立同分布（iid）多态试验的案例，也研究了独立非同分布（non-iid）多态试验的案例。数值示例的目的是说明所提出方法的有效性。本研究对类似运行具有参考价值和应用意义。