On The Semi-Markovian Random Walk Process with Reflecting and Delaying Barriers

On The Semi-Markovian Random Walk Process with Reflecting and Delaying Barriers Probability characteristics of random walks with or without barriers are being used to solve a number of very interesting problems in the fields of inventory, queues and reliability theories, mathematical biology etc,. These barrier can be reflecting, delaying or absorbing depending on concrete problems at hand. In this study, the semi-Markovian random walk process X(t) with reflecting barrier on the zero-level and delaying barrier on the P (p > 0) -level and the important boundary functionals of it, y, - the first falling moment of the process into the delaying barrier and y2- the first reflection moment of the process from the reflecting barrier are constructed mathematically, explicit formulae are given for the distribution functions, moment generating functions, expected value and variances of y, and y 2. One dimensional distribution functions of X(t) are expressed by means of the probability characteristics of a renewal process JTn:n > 0} and a random walk (Yn:n > 0}. In special cases in which the duration between two jump instants has exponential, Erlang or Chi-square distributions explicit formulae are obtained for distribution functions of y, and y2 and one dimensional distribution functions of X(t). Furthermore, under some assumptions, the ergodic theorem for the process X(t) is proved and the ergodic distribution function of the process X(t) is given by means of the probability characteristics of a renewal process {Tn:n > 0} and a random walk {Y":n > 0}. Key words : Stochastic process, random walk, renewal process, reflecting barrier, delaying barrier, Laplace transform, semi-Markovian random walk, Chi-square distribution, Erlang distribution, exponential distribution, ergodic.