Periodic strong ergodicity in non-homogeneous Markov systems

Abstract

This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov systems under different sets of assumptions. First the periodic case is studied and the limiting evolution of the individual cyclically moving subclasses of the state space of the associated Markov replacement chain is completely determined. A special case of the above result is the aperiodic or strongly ergodic convergence. Two numerical examples from the literature on manpower planning highlight the practical aspect of the theoretical results.

A generalized formulation for the performability indicator

“…and in the case of an irreducible and aperiodic chain, the transition matrix from time 0 to time n tends to an ergodic matrix, therefore if 7r is the stationary distribution probability vector, In this special case of nonhomogeneous Markov chains [22,23], the transition probabilities have a periodical behavior, that is they may vary but remain constant from one period to another: this special feature is called cyclicity. The transition function (p~(i,j); n E N, i,j E E) is called cyclic of period d (d > 1), if d is the smallest integer verifying Prnd+r : Pr for m,r E N. The major benefit of these chains is that an asymptotic analysis is possible due to their eventual weak ergodicity.…”

A continuous time markov‐renewal replacement model for manpower systems

“…This fact was exploited in a series of recent articles by Gerontidis 18,23i43,44 and Gerontidis and Vassiliou. 45 In cases where we postulate that the system has given input instead of given total size, the imbedded Markov chain is absorbing and under certain conditions easily met in practice it can be shown that the limiting distribution of the grade sizes is multivariate Poisson (see bar tho lo me^^^ and V a~s i l i o u~~) .…”

On Existence of Limiting Distribution for Time-Nonhomogeneous Countable Markov Process

“…This type of analysis is similar to an investigation of stability for Markov chain and is associated with a verification of stationarity (quasi-stationarity) (see [1,4,5,[7][8][9][10]21,29] and many others.…”