Computation of instant system availability and its applications
In this article we let (R_j) and (L_j) indicate the dimension of jth down and up period, where (L_j) denote the given lifetime in such that after ((j-1)th) period the system is down and (R_j) be the distance required to facilitate the repairing with regarding the replacement job.
In recent years, some authors based on variety of models for repairable system using different techniques have suggested (Baxter 1981a, b, 1983; Barlow and Hunter 1960; Cha and Kim 2002; Mishra and Jain 2013; Veber et al. 2008) where the concept of availability be a relevant measure with regarding the system performance as results of repaired and maintenance implementation. Apparently, the longer the time the system is in working state is relative to the length of time for down state which improves the better the system’s performance is, as explained in Veber et al. (2008).
To get a clear or a complete picture of this work, some references had been considered to be taken into account such as Mathew (2014), Sarkar and Chaudhuri (1999), Huang and Mi (2013), Biswas and Sarkar (2000), Cha et al. (2004), Mi (1998, 2006, 1995a), Huang and Mi (2015) where the relevant features of repairable system are availability. It is known as point availability at times t .
To measure the likelihood that a system is an availability at some particular time t, a quantity denoted as (S_tau (t)) is referred to as instant availability. The corresponding probability of well functioning system at any derived time t is defined. On the other hand (tau) represents a small change with respect to the system availability.
To the best authors studies have been attempted, many theoretical and meaningful results to provide analysis for individual cases of system availability (S_tau (t)) such as Huang and Mi (2013), Huang and Mi (2015), Sarkar and Li (2006) but it remains unclear with regard to the behaviour of the instant system availability as a function of time t, and it is obtained through perfect repair, imperfect repairs or replacement after each failure.
In this paper, many discussions of ({(L_j,R_j),jge 1}) here are supposed to be (i.i.d) with F(t) and G(t) as common cumulative distribution functions respectively. We also consider that ((L_j,jge 1)), ((R_j,jge 1)) being independent of each other.
The article is arranged in the following manner: “Starting point monotonicity analysis” section presents the behaviour of the instant system availability (S_tau (t)) in the given positive interval if it is decreasing or increasing. In “Bounded for availability system” section, we consider the bound for availability system taking into consideration to the upper bound of (S_tau (t)) is derived from this section. In “Comparison of availability system” section, we make the comparison between two systems availability. In “Numerical examples for clarification” section, we provide numerical examples where we consider the repair and lifetime distributions as well as associated with different parameters. “Application to bathtub” section, application to bathtub for this article is strongly discussed. And finally the conclusion and recommendation are provided in “Conclusion” section.