Restoration Factors in BlockSim

Restoration Factors
In repairable system maintainability and availability analysis using ReliaSoft's BlockSim software, the user is given the option to specify a restoration factor that describes the percentage to which a component will be restored after the maintenance action has been performed (repair effectiveness). This provides the ability to model maintenance using "used parts" or imperfect maintenance. The restoration factor in BlockSim is defined as a number between 0 and 1 and has the following effect:

• A restoration factor of 1 (100%) implies that the component is as good as new after repair, which in effect implies that the starting age of the component is 0.

• A restoration factor of 0 implies that the component is the same as it was prior to repair, which in effect implies that the starting age of the component is the same as the age of the component at failure.

• A restoration factor of 0.25 (25%) implies that the starting age of the component is equal to 75% of the age of the component at failure.

Type I and Type II Restoration Factors
BlockSim 7 offers two kinds of restoration factors, as shown in Figure 1.

Figure 1: Restoration Factors in BlockSim

The Type I restoration factor [1, 2] assumes that the repairs can fix only the wear-out and damage incurred during the period of operation since the last repair. Thus, the nth repair can remove only the damage incurred during the time between the (n-1)th and nth failures. The Type II restoration factor [1, 2] assumes that the repairs fix the wear-out and damage accumulated up to the current time. As a result, the nth repair not only removes the damage incurred during the time between the (n-1)th and nth failures, but can also fix the cumulative damage incurred during the time from the first failure to the (n-1)th failure.

To illustrate this, consider a repairable system, observed from time t = 0, as shown in Figure 2.

Figure 2: A repairable system structure

Let the successive failure times be denoted by t₁, t₂ … and let the times between failures be denoted by x₁, x₂ … Let RF denote the restoration factor, then the age of the system v_n at time t_n using the two types of restoration factors is:

Type I Restoration Factor:

v_n= v_n-1 + (1-RF)x_n

Type II Restoration Factor:

v_n= (1-RF)(v_n-1 + x_n)

Example of Type I and Type II Restoration Factors
As an example, consider an automotive engine that fails after six years of operation. The engine is rebuilt. The rebuild has the effect of rejuvenating the engine to a condition as if it were three years old (i.e. a 50% RF). Assume that the rebuild affects all of the damage on the engine (i.e. a Type II restoration). The age of the engine after the first rebuild is three years ((1 − 0.5) × (0 + 6) = 3 years). The engine fails again after three years (when it again reaches an age of six years) and another rebuild is required. This rebuild will also rejuvenate the engine by 50%, thus making it three years old again ((1 − 0.5) × (3 + 3) = 3 years). Now consider a similar engine subjected to a similar rebuild, but suppose that the rebuild affects only the damage since the last repair (i.e. a Type I restoration of 50%). The first rebuild will rejuvenate the engine to a three-year-old condition (0 + (1 − 0.5) × 6 = 3 years). The engine will fail again after three years, but the rebuild this time will affect only the damage accumulated after the first rebuild (i.e. three years' worth). Thus the engine will have an age of four and a half years after the second rebuild (3 + (1 − 0.5) × 3 = 4.5 years). After the second rebuild the engine will fail again after a period of one and a half years and a third rebuild will be required. The age of the engine after the third rebuild will be five years and three months (4.5 + 1.5 × (1 − 0.5) = 5.25 years).

It should be pointed out that when dealing with constant failure rates (i.e. with a distribution such as the exponential distribution), the restoration factor has no effect.

Calculations to Obtain Restoration Factors
The two types of restoration factors discussed above can be calculated using the parametric RDA (Recurrent Data Analysis) tool in Weibull++ 7. This tool uses the General Renewal Process (GRP) model to analyze failure data of a repairable item. More information on the Parametric RDA tool and the GRP model can be found in ReliaSoft's Life Data Analysis Reference [3], and can be accessed on-line at http://www.weibull.com/LifeDataWeb/parametric_recurrence_data_analysis.htm. As an example, consider the times to failure for an air-conditioning unit of an aircraft, recorded in Table 1.

Table 1: Times to failure for an aircraft air-conditioning unit

Assume that each time the unit is repaired the repair can remove only the damage incurred during the last period of operation. This assumption implies a Type I RF, which is specified as an analysis setting in the Weibull++ folio. The Type I RF for the air-conditioning unit can be calculated using the results from Weibull++ shown in Figure 3.

Figure 3: Using the Parametric RDA tool in Weibull++ to calculate restoration factors.

The value of the action effectiveness factor q obtained from Weibull++ is:

q=0.1344

The Type I RF is calculated using q as:

RF=1-q=0.8656

In BlockSim, the parameters of the GRP model can be entered by selecting the Weibull distribution in the Failure Distribution tab and entering the calculated RF value on the Corrective tab as shown in Figure 4. Notice that even though the Weibull distribution is selected in BlockSim, it does not imply that the GRP is the same as the Weibull distribution, since the underlying assumptions and subsequent equations are different between the two (except in the case of q=0). However, the GRP can be modeled in BlockSim by selecting the Weibull distribution and setting the RF to a value other than zero, as explained here. The parameters of the Weibull distribution for the air-conditioning unit to be entered into BlockSim can be calculated as follows. β is obtained from Weibull++ as 1.1976 (same as in the GRP model). η can be calculated using the β and λ values from Weibull++ as:

The values of the Type I RF, β and η calculated above can now be used to model the air-conditioning unit as a component in BlockSim.

Figure 4: Modeling the air-conditioning unit as a component in BlockSim

References
1. Kijima, M. and Sumita, N. "A useful generalization of renewal theory: counting process governed by non-negative Markovian increments," Journal of Applied Probability, 23, 71-88, 1986.
2. Kijima, M. "Some results for repairable systems with general repair," Journal of Applied Probability, 20, 851-859, 1989.
3. ReliaSoft Corporation, Life Data Analysis Reference, ReliaSoft Publishing, Tucson, Arizona, 2007.