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Article Excerpt
1. Introduction
The most important problem in reliability theory is to estimate statistically at what time an operating unit will fail in the near future. Using such a reliability point of view, failure distributions and their parameters have been estimated, some reliability quantities have been well defined and maintenance policies to prevent failures have been practically considered and analytically discussed (Barlow and Proschan, 1965, 1975; Pham, 2003; Nakagawa, 2005).
On the other hand, when we detect the failure of a unit at time t and its failure time is unknown, we often want to know when it failed. For example, suppose that some products are weighed and shipped out, using a scale whose accuracy is checked at periodic times. Then, one problem is how many of the products do we have to reweigh when the scale is uncalibrated and is judged to be inaccurate (Sandoh and Nakagawa, 2003). Another example is the backup policy for a database system (Nakagawa et al., 1998). When a failure occurs in the operation of a database system, we execute the rollback operation until the last checkpoint is reached and recover the data. The problem is where to place the checkpoints in terms of planned suitable times.
First, we define the probability that a unit failed during (t - x, t], given that it is detected to have failed at time t. Using this definition, when a unit is detected to have failed at time t, we consider the optimization problem of how much time we go back to search for its failure time (Nakagawa et al, 2005). Introducing two costs of excess and shortage backward times, we discuss analytically an optimum backward time which minimizes the expected cost. It is theoretically shown that the reversed failure rate plays an important role in analyzing the optimum policy. Furthermore, we apply the backward policy to the inspection model where a unit is checked at periodic times and the inspection is not perfect. We solve the problem of how many checks we need to go back to search for the failure time, given that the failure is detected at time KT.
The reversed hazard rate was first defined in Keilson and Sumita (1982), and some results on its ordering were obtained in Shaked and Shanthikumar (1994) and Gupta and Nanda (2001). The monotonic properties of the reversed hazard rate were investigated in Block et al. (1998), Chandra and Roy (2001) and Finkelstein (2002). However, there is no paper which has discussed the application of reversed failure rates in maintenance models.
Next, as practical applications of backward time, we consider the backup model of a database (Naruse et al., 2005) and the model of reweighing products by a scale (Sandoh and Nakagawa, 2003). For the respective two models, we derive analytically an optimum checkpoint interval for which the rollback operation is made until the last checkpoint at which a failure was detected, and an optimum volume of products reweighed by an adjusted scale when it is uncalibrated. Finally, we try to apply the backward time to the traceability problem in production systems. There exist many actual situations in which we go back to some point and restore a normal condition after maintenance, when a failure has been detected.
2. Backward time
Suppose that a unit begins to operate at time and a random variable X denotes its failure time with a probability distribution F(t) [equivalent to] Pr{X [less than or equal to] t}, a density function f(t) and a finite mean time 1/[lambda] [equivalent to] E{X} = [[integral].sub.0.sup.[infinity]] [bar.F](t)dt < [infinity] where [bar.F](t) [equivalent to] 1 - F(t). Then, the probability that the unit failed during (t - x, t] (0 [less than or equal to] x [less than or equal to] t), given that it is detected to have failed at time t (0 < t < [infinity]) is
H(x|t) [equivalent to] Pr{t - x [less than or equal to] X|X [less than or equal to] t} = F(t) - F(t - x)/F(t) [less than or equal to] 1, (1)
for F(t) > (Fig. 1). In this case, we call t - X the backward time which is called the waiting time in Finkelstein (2002). Since
[FIGURE 1 OMITTED]
[bar.H](x|t) [equivalent to] 1 - H (x|t) = F(t - x)/F(t), (2)
its density function is
h(x/t) = dH(x/t)/dx = f(t - x)/F(t), (3)
and its mean backward time, i.e., the mean time from t to a failure is
[beta](t) [equivalent to] E{T - X|X [less than or equal to] t} = [[integral].sub.0.sup.t] [bar.H](x|t) dx = 1/F(t)...
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