Weak Ergodicity in G-NHMS

Abstract In the present we provide the definition of the new concept of the General Non-Homogeneous Markov System (G-NHMS) and establish the expected population structure of a NHMS in the various states. These results will be the basis to build on the new concepts and the basic theorems of what follows.We then establish the set of all possible expected relative distributions of the initial number of memberships at time t and all possible expected relative distributions of the expansion memberships at time t . We call this set the general expected relative population structure in the states of a G-NHMS. We then proceed by providing the new definitions of weak ergodicity in a G-NHMS and weak ergodicity with a geometrical rate of convergence. We then prove the Theorem 4 which is a new building block in the theory of G-NHMS. We also prove a similar theorem under the assumption that relative expansion of the population vanishes at infinity.We then provide a generalization of the coupling theorem for populations. We proceed then to study the asymptotic behavior of a G-NHMS when the input policy consists of independent non-homogeneous Poisson variates for each time interval $$\left( t-1,t\right] $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mi>t</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mfenced> </mml:math> . It is founded in Theorem 7 that it displays a kind of weak ergodicity behavior, that is, it converges at each step to the row of a stable matrix. This row is independent of the initial distribution and of the asymptotic input policy unlike the results in previous works. Hence it generalizes the result in that works. Finally we illustrate our results numerically for a manpower system with three states.