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In manpower planning there is an interesting, practically important and open challenging issue arising from the instances where a transition matrix over a certain short time interval is required but only a transition matrix over a longer time interval may be available. For example, the problem of finding a meaningfulth root of an observable stochastic matrix within the context of Markov chains. The more general problem is divided into three phases, viz. embeddability, inverse, and identification problems. By exploiting the Cauchy’s integral formula with the integrand defined on the Runnenberg’s heart-shaped region, a new representation for a stochastic matrix that is embeddable is derived. New conditions for embeddability and regularization of stochastic matrices are provided. Examples are presented to illustrate the utility of the Cauchy’s representation formula and its consequences.

On the use of Cauchy integral formula for the embedding problem of discrete-time Markov chains