The Application Of Markov Skeleton Processes In A Mathematical Model

Markov skeleton process is a new type of stochastic process and containing many classical processes as special cases. In 1997,prof Hou Zhenting and his colleagues raised this kind of processes and applied it to fields in queue theory and reliability.In this thesis, applied with the theory of Markov skeleton process , we analyze GI/G/1 queue with set-up period and of the reliability of repairable system in series. As for the GI/G/1 queue with set-up period, compared with the former researches, the distribution of every parameter in this thesis is more general. As for the model of the reliability of repairable system in series, differing from the former researches , the distribution of the lifetime and the repairing time of each component is general. they can get some reliability index, or Laplace-transform, or Laplace-Stieltjes transform of reliability index.Be aimed at the two models which mentioned above, we apply the Markov skeleton process theory, which was firstly put forward by professor Hou. Zhenting etc, to study the transient and limit distribution of its state. In this thesis, we draw the following conclusions:Firstly, applying the Markov skeleton process approach, we present the equations which satisfy the transient distribution of the length of GI/G/1 queue with set-up period {L(t),θ₁(t),θ₂(t),θ₃(t)}, and prove that its probability distribution is the minimal nonnegative solution of some equations. And we study the limit distribution of {L(t),θ₁(t),θ₂(t),θ₃(t)}. Furthermore we find out a Doob skeleton process of GI/G/1 queue with set-up period, and give the limit distribution,generalized limit distribution and the existence of invariant probability measure and its expression by use of the theory of Doob skeleton process and limit theory.Secondly, applying the density evolution approach and the Markov skeleton process approach, we present the equations which satisfy the transient distribution of the state of {L(t),X₁(t),X₂(t),Y₁(t),Y₂(t)} for the model of the reliability of repairable system in series. And give out the minute proof of the method of density evolution. At last, we give a simple comparison to the different equations, which are got for two different methods.