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

In this paper, we consider an optimal maintenance policy for a reparable deteriorating system subject to random shocks. For a reparable deteriorating system, the repair time by a partial product process and the failure mechanism by a generalized δshock process. Develop an explicit expression of the ling run average cost per unit time under N policy is studied.


1.Introduction
In reliability, the study of maintenance problem is always an important topic. The replacement problem for a reparable system has aroused great attention. Barlow and Proschan (1965) used the replacement policy in which the system will have a preventive replacement as soon as the age of the system reaches T and a failure replacement as soon as it fails, which ever occur earlier and the replacement time is assumed to be negligible. Park (1979) studied the replacement policy in which the system will have a replacement as soon as the number of failures of the system reaches N. The problem is to choose an optimal replacement policy N * such that the long run average cost per unit time is minimized. Lam (1988) introduced a geometric process and Cheng and Li (2014) introduced a generalization of the α-series process model. Babu, Govindaraju and Rizwan(2018) introduced a partial product process model.
Shock models were used to model operating time. A system fails due to the shock effect on the system. Li (1984) first introduced the δ shock model to avoid measuring amount of damage. The δ shock model focuses mainly on the frequency of shocks rather than the magnitudes of shocks. The δ shock models if the time interval between two successive shocks is smaller than a thresold value δ, the system fails.
In this paper, we adopt a general δ shock model by the δ be an exponentially distributes a random variable with parameter varying with number of repairs.

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.
Definition 2.7 An integer valued random variable N is said to be stopping time for the sequence of independent random variables X 1 , .. are independent and identically distributed random variables having finite expectations and if N is the stopping time for We consider the maintenance model for a deteriorating system and make the following assumptions.
A1 At the beginning a new system is installed. Whenever the system fails, it may be repaired or replaced by a new and identical one. A2 Shocks arrive according to a Poisson process with rate λ 1 or EX i = 1 λ i , where X i is the i th inter arrival time of two consecutive shocks. Let δ i be another exponentially distributed random variable associated with X i . We assumed that the sequence {δ i , i = 1, 2, ...} form an increasing partial product process with 0 < β 0 ≤ 1. Then δ i has cumulative distribution function Q(β 2 i−1 0 x), where Q(X) is the cumulative distribution function of δ 1 . (X i , δ i ) follows a δ−shock model if the system fails at i th shock which satisfies X i ≤ δ i and the life time or equivalently the operating time is the sum of all X i until the one satisfying the above condition. We assume that X i is independent of δ i . A3 Let T n be the operating time after the (n − 1) th repair {T n , n = 1, 2, ...} is a stochastically decreasing random variable sequence induced by the δ shock model. A4 Let Y n be the repair time after the n th failure and forms an increasing partial product process with 0 ≤ γ 0 ≤ 1. Then Y n has cumulative distribution function A5 T n and Y n n = 1, 2, 3, ... are independent sequence. ISSN: 2456-8686, Volume 4, Issue 1, 2020:48-54 DOI :http://doi.org/10.26524/cm65 A6 Let Z be the replacement time with E(Z) = τ. A7 The repair cost rate is c, the reward rate is r and the replacement cost is R.
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 long 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 Let W be the length of a renewal cycle under N replacement policy. We have We first calculate E(T n ) the expected operating time of the system after the (n − 1) th failure. Let l ni be the inter arrival time between the (i − 1) th and i th shock following the (n − 1) th repair, where i = 1, 2, ... Define Where M n denotes the number of shocks till the first deadly shock occurs and M n has a geometric distribution with P (M n = k) = q k−1 n p n , k = 1, 2, ...
Where p n is the probability of a shock, following the (n−1) th repair and q n = 1−p n . We have EM n = 1 pn . As M n is a stopping time with respect to the random sequence {ln i , i = 1, 2, ...} which are independence identically distributed random variables, using Wald equation, we have Since F (x) and Q(x) are all exponentially distributed, we have As l ni and δ i (β 2 i−1 0 δ 1 ) are independent and have the marginal exponential distribution with means of 1 λ 1 and and We obtain, Since Y n , n = 1, 2, ... is increasing partial product process with ratio 0 < δ 0 ≤ 1, we have The long run average cost C(N ) of the system under the policy N is given by The optimal replacement policy N * can be calculated by minimizing C(N ) 4.The optimal replacement policy N * Equation (1) can be re-written as The difference between B(N + 1) and B(N ).

Conclusions
In this paper, we have studied the optimal replacement policy for a reparable and deteriorating systems using δ−shock model, we derived the optimal replacement policy N * by minimizing the average cost rate (CN).