An Extended Birth-Death Processes with Catastrophes——Stochastically Monotone, Feller and Symmetric Properties

A new structure with the special property that catastrophes is imposed to ordinary Birth_Death processes is considered. The necessary and sufficient conditions of stochastically monotone, Feller and symmetric properties for the extended birth_death processes with catastrophes are obtained.

Inspection-replacement policy of system under predictive maintenance

In general, every system is in one of the three states： normal, abnormal, or failure state. When the system is diagnosed as abnormal state, it needs predictive maintenance, lfthe system fails, an identical new one will replace it. The predictive maintenance cannot make the system ＂as good as new＂. Under these assumptions, the reliability index and the inspection-replacement policy of a system were studied. The explicit expression of the reliability index and the average income rate （i.e., the long-run average income per unit time） are derived by using probability analysis and vector Markov process method. The criterion of feasibility for the optimal inspection-replacement policy under the maximum average income rate is obtained. The numerical example shows the optimal inspection-replacement policy can raise the average income rate when the optimal inspection-replacement policy is feasible.

Explicit criteria on separation cutoff for birth and death chains

The criteria on separation cutoff for birth and death chains were obtained by Diaconis and Saloff-Coste in 2006. These criteria are involving all eigenvalues. In this paper, we obtain the explicit criterion, which depends only on the birth and death rates. Furthermore, we present two ways to estimate moments of the fastest strong stationary time and then give another but equivalent criterion explicitly.