Application of renewal reward processes in homogeneous discrete markov chain

TLDR: In this paper , a renewal process which is a special type of a counting process, which counts the number of events that occur up to (and including) time has been investigated, in order to provide some insight into the performance measures in renewal process and sequence.

Abstract: A renewal process which is a special type of a counting process, which counts the number of events that occur up to (and including) time  has been investigated, in order to provide some insight into the performance measures in renewal process and sequence such as, the mean time between successive renewals, ; Laplace-Stiltjes transform (LST) of the mean time, ;  the Laplace-Stieltjes transform (LST) of the mean time distribution function,  ;  Laplace-Stiltjes transform (LST) of  fold convolution of distribution function, ;  the time at which the  renewal occurs,  the average number of renewals per unit time over the interval (0, t],   and expected reward, . Our quest is to analyse the distribution function of the renewal process and sequence   using the concept of discrete time Markov chain to obtain the aforementioned performance measures. Some properties of the Erlang , exponential and geometric distributions are used with the help of some existing laws, theorems and formulas of Markov chain. We concluded our study through illustrative examples that, it is not possible for an infinite number of renewals to occur in a finite period of time; Also, the expected number of renewals increases linearly with time; and from the uniqueness property, we affirmed that, the Poisson process is the only renewal process with a linear mean-value  function; and lastly, we obtained the optimal replacement policy for a manufacturing machine which showed that, the exponential property of the lifetime