The Semi Markov random wlak process with two reflecting barriers

The Semi-Markov Random Walk Process With Two Reflecting Barriers In particular, a number of very interesting problems of stock control, queuing and relia bility theories can be expressed by means of random walk processes with two barriers. Nume rous studies have been done about these processes because of their theoretical and practical importance. But most of these studies belong to the boundary-value problems for the random walk processes which has a finite state space. The boundary-value problems are important, so are the investigation of proper characteristics of processes at hand. For this reason although there are some studies on proper characteristics of random walk processes with two barriers, more detailed studies in this field have to carried out. In particular, random walk processes with two reflecting barriers are not studied well. Therefore, it is necessary to construct and investigate this process since it has theoretical and practical importance. Moreover, it is more interesting to look at semi-Markov random walk processes that is a general class instead of random walk processes. In this study, the semi-Markov random walk process X(t) that has a denumerable space with two reflecting barriers on the O(zero)-level and on the ?(?> 0) -level and the important boundary functional of it, t, -the first reflection moment of the process from the lower reflec ting barrier are constructed mathematically, explicit formulae are given for the moment gene rating functions of x,. One dimensional non-stationary distribution functions of X(t) are exp ressed by means of the probability characteristics of a renewal process (Tn }n?o and a random walk process {Yn )n?o. In the special cases in which the duration between two jump instants has exponential or Erlang distributions explicit formulae are obtained for one dimensional distri bution functions of X(t). Furthermore, under the most general conditions, the ergodic theorem for the process X(t) is proved and the most general ergodic distribution function of the process X(t) is given by means of the probability characteristics of the processes {Tn }n?o and { Yn }n?o Finally, the limit theorem is proved for the sequence of the process mentioned earlier. Key words : Stochastic Process, Random Walk Process, Renewal Process, Semi-Markov Random Walk Process, Reflecting Barrier, Delaying Barrier, Laplace Transform, Exponential Distribution, Erlang Distribution, Ergodic, Limit Process. Y