Dual and Feller-Reuter-Riley transition functions
Duality and Feller-Reuter-Riley (FRR) property of continuous-time Markov chains (CTMC) are studied. CTMC is described by a transition function \(P(t)=(p_{ij}(t); i,j \in E)\) where \(E=Z_+={0,1,2,...}\). A transition function \(P(t)\) is dual if \(\sum_{k=0}^j p_{ik}(t) \downarrow 0\) as \(i \to \infty\) for \(j \in E\) and \(t \geq 0\). It is FRR transition function if \(\lim_{i \to \infty} p_{ij}(t)=0\) for \(j \in E\) and \(t \geq 0\). A transition function \(P(t)\) is called to be a \(Q\)-function if \( P'(0)=Q\) (componentwise), where \(Q=(q_{ij}\); \(i,j \in E)\) is an infinite stable \(q\)-matrix s.t. \(0 \leq q_{ij} < +\infty\), \(\sum_{i \neq j} q_{ij} \leq -q_{ii}<+\infty\). The following questions are considered. A criterion of dual \(q\)-functions is given in terms of their \(q\)-matrices. For a dual \(q\)-matrix \(Q\), a necessary and sufficient conditions for the minimal \(Q\)-function to be a FRR transition function are also given. Finally, by using dual technique, a criterion of FRR \(Q\)-functions is given when \(Q\) is monotone. Earlier these problems have been considered by \textit{H.~Zhang} and \textit{A.~Chen} [J. Math. Anal. Appl. 234, No.~2, 482--499 (1999; Zbl 0936.60065)].
scientific article; zbMATH DE number 5016113