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1
A Monte Carlo Approach to Estimation of
G-Renewal Process in Warranty Data Analysis
Mark P. Kaminskiy and Vasiliy V. Krivtsov
1. Introduction
For many years, the most commonly used models for the failure process have been the renewal process
(RP) and the nonhomogeneous Poisson process (NHPP). In the framework of the repairable system applications,
RP is used to model the situations with restoration to "good-as-new" condition (perfect repair assumption),
meanwhile NHPP is applied to the situations with the “same-as-old” restoration (minimal repair assumption). In a
sense, these two assumptions can be considered as extreme ones from both theoretical and practical standpoints.
In order to avoid this “extremism”, several generalizing models have been introduced in recent years. References
include Brown & Proschan (1982), Kijima & Sumita (1986), Filkenstein, (1993), Lindqvist (1999). Among these
models, the G-Renewal Process (GRP) introduced by Kijima & Sumita (1986) is very attractive, since it covers
the intermediate "better-than-old-but-worse-than-new" repair assumption and results in a G-renewal equation,
which is a generalization of the well-known ordinary renewal equation. Unfortunately, a closed form solution of
the equation is unavailable, which makes the respective statistical estimation challenging.
The objective of this paper is limited to statistical estimation of the parameters of G-Renewal Process,
based on typical one-dimensional warranty (grouped failure time) data. The Monte Carlo approach considered
below can be also applied to parameter estimation of Kijima’s Model I and II (Kijima, 1989).
2. G-Renewal Process
Kijima and Sumita (1986) introduced a G-Renewal Process, which can model restoration conditions
ranging from "good-as-new" to "same-as-old". The GRP is introduced using the notion of virtual age.
Let A
n
be the virtual age of a system immediately after the nth repair. If A
n
= y, then the system has the
time to the (n + 1)th failure X
n+1
, which is distributed according to the following cumulative distribution function
(CDF):
)(1
)()(
) ( yF
yFyXF
yAXF
n
−
−
+
==
, (1)
where F(X) is the CDF of the time-to-first-failure (TTFF) distribution of a new system. The sum
∑
=
=
n
i
in
XS
1
,
with S
0
= 0, is called the real age of the system.
In the framework of the GRP it is assumed that the nth repair can remove the damage incurred only
during the time between the (n –1)th and the nth failures, so that the respective virtual age after the nth repair is
A
n
= A
n – 1
+ q X
n
= q S
n
, n =1, 2, . . .
where q is the parameter of rejuvenation (or repair effectiveness parameter) and the virtual age of a new system
A
0
= 0, so that the TTFF is distributed according to F(t|0)
≡
F(t).
The time between the first and second failures is distributed according to (1) with A
1
= qX
1.
Respectively, the time between the second and third failure is distributed according to (1) with A
2
= q ( X
1
+ X
2
),
and so on.
.
2
It is clear that for q = 0, the considered process coincides with an ordinary RP, thus, modeling the
"good-as-new" repair assumption. With q = 1, a system is restored to the "same-as-old" condition, which is
similar to NHPP. The case of 0 < q < 1 corresponds to the intermediate "better-than-old-but-worse-than-new"
repair assumption. Finally, with q > 1, the virtual age A
n
> S
n
, i.e., the repair damages the system to a higher
degree than it was just before the respective failure, which corresponds to the "worse-than-old" repair
assumption.
The expected number of failures in (0, t], which is called a cumulative intensity function, is given by a
solution of G-renewal equation (Kijima, et al., 1988):
,)|()()0|()(
0 0
τττ
τ
ddxxxgxhgtH
t
∫ ∫
−+=
(2)
where
,0,,
)(1
)(
) ( ≥
−
+
= xt
qxF
qxtf
xtg
is such that h(t) = d(H(t))/dt, g(t|0) = f(t), and F(t) and f(t) are the CDF and PDF of the TTFF distribution.
The closed form solution of (2) is not available, and even numerical solutions are difficult to obtain,
since the equation contains a recurrent infinite system (Finkelstein, 1997). A Monte Carlo based solution is,
however, possible and was discussed by Kaminskiy and Krivtsov (1998).
3. Warranty Data
Typical one-dimensional warranty data are collected as results of observations on a large population of
identical repairable units. The population size, N
0
, is known and it can be assumed as constant in time (the
number of lost units, if any, is negligible with respect to N
0
). The following table provides an example of real
warranty data, which are analyzed in the next section.
Table 1. Example of Warranty Data for a Repairable System. Population Size, N
0
= 100000.
Month in Service, t 3 6 9 12 15 18 21 24 27
Cumulative Number of
Failures Per System, H
emp
(t)
0.03
0.09
0.14
0.24
0.38
0.54
0.70
0.90
1.17
4. Estimation Procedure
Based on the warranty data, the empirical cumulative intensity function, H
emp
(t), is calculated as
0
)(
)( N
tN
tH
i
ie mp
=
, t
i
< t
i+1
, i = 1, 2, . . . , n (3)
where N(t) is the cumulative number of failures in (0, t].
Denote a solution of G-renewal equation (2) obtained by Monte Carlo simulations by
Η
mc
(t) = f(F(
τ|α
), q, t)), (4)
where F(
τ|α
) is a given time-to-first-failure CDF with unknown vector of parameters,
α
.
Using (3) and (4), the least squares estimates of GRP parameters
α
and q can be obtained as a solution
of GRP parameters
α
and q can be obtained as a solution of the following optimization problem:
Η−Η
∑
=
n
i
imciemp
q
tqFt
1
2
,
)),),|(()((min
ατ
α
.
3
5. Examples
5.1
Simulated Data
The empirical cumulative intensity function, H
emp
(t), was obtained by simulating a GRP with a Weibull
distributed TTFF (shape parameter,
β
= 1.5 and scale parameter,
Θ
= 1) and the GRP rejuvenation parameter,
q = 0.5 using N
0
= 100 realizations over the observation interval, T = 5
Θ
.
Estimates of
β, Θ
, and q were obtained based on n
0
= 1000 realizations of GRP as follows:
0.49
^
,00.1
^
1.48,
^
=== q
θβ
. Tables 2 - 3 show the sample correlation and covariance matrices for the
obtained estimates of GRP parameters for 30 simulated empirical cumulative intensity functions, H
emp
(t).
Table 2. Sample Correlation Matrix
β
ββ
β
Θ
ΘΘ
Θ
q
β
ββ
β
1.000
Θ
ΘΘ
Θ
0.702 1.000
q 0.079 -0.523 1.000
Table 3. Sample Covariance Matrix
β
ββ
β
Θ
ΘΘ
Θ
q
β
ββ
β
2.6 10
-3
Θ
ΘΘ
Θ
1.9 10
-3
2.8 10
-3
q 2.3 10
-4
-1.6 10
-3
3.2 10
-3
With the empirical cumulative intensity function simulated for N
0
= 100000 realizations (which is a
more typical case for a company concerned with mass production) and using n
0
= 1000000, the estimation
procedure returns the original GRP parameters with close to zero variance.
5.2
Real Data
The warranty data collected on a system during first 18 months (see Table 1) were used for estimation of
GRP parameters. The Weibull distribution with the shape parameter,
β
, and the scale parameter,
Θ
, was assumed
as the underlying TTFF distribution. The solid line in Figure 1 represents the least squares fit from a family of G-
renewal functions simulated in the following parameter domain: {1 <
β
< 2, 10 <
Θ
< 50, 0 < q < 1}.
Figure 1. A Lest Square Fit of G-Renewal Function
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 5 10 15 20 25 30 35
time in months
number of failures per system
Actu al data
GRP, q = 0.7
GRP, q = 1
ORP
.
4
The obtained estimates of GRP parameters are
0.70
^
,42
^
1.8,
^
=== q
θβ
. The estimated G-renewal
function shows a good fit to the data not only in the interval (0, 18] months (used for estimation) but also in the
remaining interval (18, 30] months (obtained by prediction), see Figure 1. The figure also shows the extreme
repair conditions modeled by the RP (q = 0) and the GRP (q = 1).
It is reasonable to conclude that the approach considered above is not only practically applicable for
estimation of the GRP parameters, but also for prediction of the G-renewal function, which is often essential in
warranty data analysis.
References
1. M. Brown, and F. Proshan, "Imperfect Maintenance" - in Crowley J. and Johnson R. (ed) Survival Analysis.
Vol. 2, 1982, pp 179-188
.
2. M. S. Filkenstein, "The Concealed Age of Distribution Function and the Problem of General Repair",
Journal of Statistical Planning and Inference, # 65, 1997, pp. 315-321.
3. M. P. Kaminskiy and V.V. Krivtsov, "A Monte Carlo Approach to Repairable System Reliability Analysis" -
in Probabilistic Safety Assessment and Management, Springer-Verlag London Ltd, 1998, pp. 1063-1068.
4. M. Kijima and N. Sumita, "A Useful Generalization of Renewal Theory: Counting Process Governed by
Non-negative Markovian Increments", Journal of Applied Probability, # 23, 1986, pp. 71-88.
5. M. Kijima, "Some Results for Repairable Systems with General Repair", Journal of Applied Probability, #
26, 1989, pp. 89-102.
6. H. Lindqvist, "Statistical Modeling and Analysis of Repairable Systems" - in Statistical and Probabilistic
Models in Reliability, Birkhauser, Berlin, 1999, pp. 3-25.
