Investigating replacement policy for systems with linear degradation using stochastic processes

In this study, a mechanical system with imperfect maintenance and gradual degradation is considered. It is assumed that at initial time the system has 𝑧 > 0 available resource and when system is in operating time resource is decreasing consistently and gradually (𝑐𝜉𝑛 ). The resource of the system will reach 𝑧 − 𝑐𝜉0 level in the end of the first period. Also, at the end of the first period a certain maintenance policy is applied and general resource of the system is increasing by a random amount (𝜁1). The subsequent periods will proceed similarly and eventually when the total available resource reaches zero, the process will restart repeat in a similar manner. The total level of the mechanical system describe, is expressed by a stochastic process 𝑋(𝑡). The investigated 𝑋(𝑡) process is a stochastic process with dependent components. The main purpose of this study is to determine when the system expressed by 𝑋(𝑡) will replace a new and similar system for the first time by using method of stochastic processes. For this aim, the process 𝑋(𝑡) and certain boundary functional 𝐾0(𝑧), 𝑁0(𝑧), 𝑁(𝑧), 𝑆𝑁0(𝑧)+1, 𝑆𝑁(𝑧)+1, 𝑋𝑁0(𝑧)+1, 𝑋𝑁(𝑧)+1, 𝑇𝑁0(𝑧) of the process are defined mathematically. On the purpose of investigating the significant boundary functional 𝑁0(𝑧), the supplementary boundary functional 𝐾0(𝑧) has been constructed mathematically and obtained both exact and asimptotic results for the expected value and variance by using mthods of renewal theory. Next, asymptotic expressions for the expected value, variance of the boundary functionals 𝑁0(𝑧) and 𝑁(𝑧) have been obtained. Then boundary functionals 𝑆𝑁0(𝑧)+1 and 𝑆𝑁(𝑧)+1 which are neccessary for investigating 𝑋𝑁0(𝑧)+1 and 𝑋𝑁(𝑧)+1 have been analyzed. Thereafter 𝑋𝑁0(𝑧)+1 and 𝑋𝑁(𝑧)+1 which are important for determining maintenance policy to the system on the previous period in which the process reaches to level zero was studed. In addition to them both the exact and approximate formulas have been derived for the boundary functional 𝑇𝑁0(𝑧) which is the first time in which the process 𝑋(𝑡) reached to level zero. One of the important result obtained in this study is to suggest the maintenance policy for the previous period in which 𝑋(𝑡) process drops below zero. Obtained asymptotic results provide convenience in terms of implementation.