GEOMETRIC PROCESSES AND REPLACEMENT PROBLEM

In this paper, we introduce and study the geometric process which is a sequence of independent non-negative random variables X,X,…such that the distribution function of Xis F(ax),where a is a positive consent. If a>1, then it is a decreasing geometric process, if a<1,it is an increasing geometric process.Then, we consider a replacement model as follows:the successive survival times of the system after repair form a decreasing geometric process or a renewal process while the consecutive repair times of the system constitute an increasing geometric process or a renewal process.Besides the replacement policy based on the working age of the system,a new kind of replacement policy which is determined by the number of failures is considered.The explicit expressions of the long-run average costs per unit time under each replacement policy are then calculated, and therefore the corresponding optimal replacement policies can be found analytically or numerically.