Markov Skeleton Process Theory In Two Mathematical Models

In this thesis, we discuss the solding of perishable goods model and the reliability of repairable hot reserve model with two-part applying the Markov skeleton process theory. Markov skeleton process is a kind of comprehensive stochastic process, which is firstly put forward by Prof. Hou zhenting and his colleagues in 1997. The process contains many extant classical stochastic process models,such as Markov process,semi-Markov process, piecewise deterministic Markov process etc. They have important value in theory and application. Prof. Hou zhenting and his colleagues have successfully solved a series of classical difficult problems of transient distribution, steady-state distribution, ergodicity ergodicity in queueing system, meanwhile posed many new problems and new thoughts.Recently, the theory has got complemented and perfected.As for the solding of perishable goods model, we introduce the payoff to the inventory model, and get this model. In the inventory model, the predecessors mainly studied the property of the stocks. In the solding of perishable goods model ,we use Markov skeleton process theory not only study the property of the stocks ,but also study the property of the payoff. We also get the limit distribution and instantaneous distribution of the state of the model.As for the repairable hot reserve system with two-part model, the predecessors mainly studied this model when both the distribution of the age of the work part and hot reserve part and the distribution of the repair time of the fault part are exponential distribution. We use Markovskeleton process theory study the model when both the distribution of the age of the work part and the distribution of the repair time of the fault part are general distribution. Only the distribution of the the age of the hot reserve part be exponential distribution.In this dissertation, we drew the following conclusions:Firstly, in the solding of perishable goods model, applying the Markov skeleton process method and the density evolution method respectively, we present the equations which satisfy the transient distribution of the payoff level {X(t), S(t),θ(t),(?)(t); t≥0} for the solding of perishable goods model. To Markov skeleton process method, we prove that the probability distribution of the payoff level is the minimal nonnegative solution of some equation. And we study the limit distribution of payoff level. Furthermore,we find out a Doob skeleton process of the model and give out its limit distribution, generalized limit distribution, the existence condition and its expression of invariant probability measure. To the density evolution method, we list the partial differentio-integral equations system ,boundary conditions and initial conditions that the state probability density P_i(t…) satisfies,and give out the detailed proof. Secondly, in the repairable hot reserve system with two-part model, applying the density evolution method, we list the partial differentio -integral equations system ,boundary conditions,the initial conditions and regular conditions that the state probability density P_i (t…) satisfies,and give out the detailed proof. Furthermore, we use Markov skeleton process theory list the partial integral equations system that the state {L(t);X(t),Y(t);t≥0} satisfies, and give out the detailed proof. In the end, we discuss its limit distribution.