Restoration Factors in
In repairable system maintainability and availability analysis using
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
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
To illustrate this, consider
a repairable system, observed from time t = 0, as shown in Figure
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
vn at time tn using the two types of
restoration factors is:
Type I Restoration Factor:
Type II Restoration Factor:
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
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
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
, and can be accessed online at
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
The value of the action
effectiveness factor q obtained from Weibull++ is:
The Type I RF is calculated
using q as:
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
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.