Some Results on the Embeddable Problem for Discrete-Time Markov Models in Manpower Planning

Abstract

For discrete-time 2-stated Markov chains the embeddable problem as well as the inverse problem are discussed in detail based on the concept of m-th root probability matrix: necessary and sufficient conditions are formulated under which the considered Markov chain is compatible with a transition matrix regarding time unity . In case of an embeddable Markov chain, the m-th root probability matrices are expressed in analytic form.

A Markov chain with states that are ordered, is expected to be embedded in a discrete-time state-wise monotone Markov chain. For this type of Markov chains with the number of states equal to 2, 3, or 4, the trace of the transition matrix at least equal to 1 is proved to be a necessary condition for the embeddability in a Markov chain regarding time unity , for m an even number.

Keywords:

• Markov chain
• Embeddable problem
• State-wise monotone

Mathematics Subject Classification:

• 15A23
• 15A51
• 60J10

Acknowledgement

The author thanks the reviewers for their remarks and valuable suggestions.