A study on an optimal replacement policy for a degenerative system under partial product process

In this paper, we study a degenerative reparable system with two types of failure states. Any system after repair can not be as good as new. A general monotone process model for a degenerative system under partial product process is used. We use a replacement policy N based on the failure number of the system and to determine an optimal replacement policy N∗ such that the average cost rate is minimized.


1.Introduction
In a maintenance degenerative reparable system assumes that a failed system after repair will be as good as new, this is the perfect repair model. For a deteriorating repairable system, its successive working times of the system after repair become shorter and shorter while the consecutive repair times of the degenerative system may become longer and longer. Such a system can not work any longer. Neither can it be repaired or replaced new one. Lam [1988] first introduced a geometric process repair model in which he studied two kind of replacement policy, one based on the working age T of the system and the other based on the number of failures N of the system. Cheng and Li (2014) first introduced a α-series process for an optimal replacement policy for a degenerative system with two types of failure states where as both operating times and repair times are assumed to follow partial product process in Babu, Govindaraju and Rizwan (2018) paper.
In this paper, we assume successive operating time and the consecutive repair time for a degenerative system with two types of failure states under a partial product process.

2.Basic Definitions and Model assumptions
The preliminary definitions and results about partial product process are given below.
Definition 2.1 For a given two random variables X and Y, X is said to be stochastically larger than Y (or Y is stochastically less than X) if P (X > α) ≥ P (Y > α) for all real α.

Lemma 2.5 Given a partial product process {X
Definition 2.7 An integer valued random variable N is said to be stopping time for the sequence of independent random variables X 1 , X 2 , ... if the event {N = n} is independent of X n+1 , X n+2 , ... Theorem 2.8 Wald's equation, If X 1 , X 2 , ... are independent and identically distributed random variables having finite expectations and if N is the stopping time for X 1 , X 2 , ... such that E(N ) < ∞, then We consider a monotone process model for a degenerative system with two types of failure state by making the following assumptions.

A1
At the beginning, a new system is installed. Whenever the system fails, it will be repaired or replaced by an identical new one. A2 Assume that there are three states in the system, that is, 0,1 and 2 are respectively denoted as the working state. First type failure state and second type failure state of the system. And assume that the occurrence of a failure state of two types is mutually exclusive and stochastic. If the system fails, then with probability p 1 , the system will be in state i, i = 1, 2 and p 1 + p 2 = 1. A3 Let X 1 be the operating time of the system after installation. Let X n be the operating time of the system after n − 1 th repair. Let Y n be the repair time after the n th failure. Let us denote the time of the n th failure by t n . Let S(t n ) denote the type of the n th failure where S(t n ) ∈ {1, 2}, n = 1, 2, ...X n , Y n are independent. Assume that Where m j ∈ {1, 2}, j = 1, 2, ..., n; β 1 ≤ 0; β 2 ≤ 0. A4 Let c w be the reward rate per unit time of the system when it is operating c r be the repair cost rate per unit time of the system. A5 T n and Y n n = 1, 2, 3, ... are independent sequence. A6 The policy N is applied.

3.The replacement policy N
Definition 3.1 A replacement policy N is a policy in which we replace the system at the N th failure of the system. By the renewal reward theorem, Ross (1983), the lens run average cost per unit time under the replacement policy N is given by The expected cost incurred in a cycle The expected length of a cycle