ABSTRACT
In Chapter 5 entitled The Non-Homogeneous Markov System the definition of a NHMS in discrete space and time is provided firstly for an expanding population and secondly for a fluctuating one but without forced wastage. The study evolves with by evaluating the expected and relative expected population structure as functions of what are called the basic parameters of the system. That is, parameters that uniquely determine a NHMS and they are estimated from available data on the phenomenon under study. In the section that follows a synopsis of areas of applications of NHMS are discussed, illustrating the breath of applications and little of the reasons why the theoretical results that will follow are central for the study of the population at hand. These areas include Manpower Planning in Corporate Organizations such as in hierarchical manpower systems, sociology and labor markets, military establishments. Also, Ecological Modelling for various ecological systems Health care systems specializing in various health problems for patients, Fish Populations restricted in fisheries and being interested in problems of optimal harvesting and cost. Then Non-Homogeneous Markov Jump Systems which attracted much attention for their practical applications and are closely related with NHMS. There is also an immense literature on Biological Markov Population Models which are closely related, or potentially related or partially related with the present NHMS theory. There is also a large interesting literature on Open Markov Populations, % which actually are special cases of the process NHMS and hence the results in the present book are directly applicable. The eighth area of application entitled Social Physics and Statistical Mechanics collects Open Markov population models in discrete space and continuous time which are closely related with statistical physics. Social physics, biological physics and physics of particles are related probably through the strong concepts of entropy, stability and energy which exists in all kind of populations. Finally, we conclude the areas of applications with Non-Homogeneous Semi-Markov Systems which has been a very successful extension of the NHMS theory. Sections 5.5 and 5.6 contains the more general definition of a Non homogeneous Markov systems and Change of measure in a NHMS and uses some measure theory with some insight. The not familiar with measure theory reader/lecturer could omit these sections or read them only superficially. Section 5.7 studies the space or random population structures as a Hilbert space. The mathematical level is rather advanced and not usual in applied probability but I believe it could be accessible. We conclude the chapter with a section on the estimation of the transition probabilities of a NHMS.
