Presented in Montreal, Quebec, July 7, 2001

Properties of the Duane Plot for Repairable Systems

Steven E. Rigdon Department of Mathematics and Statistics Southern Illinois University Edwardsville Edwardsville, IL 62026-1653 srigdon@siue.edu www.siue.edu/~srigdon/ Phone: 618-650-2193, FAX: 618-650-3771

Abstract The Duane plot is a plot of cumulative operating time against the cumulative failure rate on double logarithmic axes. It is often claimed that the power law process, a nonhomogeneous Poisson process with intensity function λ(t) = (β/θ) (t/θ)^{β−1} , implies a linear Duane plot. We show that this implication and its converse are false.

Keywords: Repairable system, power law process, reliability growth, deterioration.

1 Introduction

Consider a repairable system whose failures occur according to some random process. The repairs are either instantaneous or the repair times are not counted in the operating time. Let t_{1} < t_{2} < ⋯ < t_{n} < ⋯ denote the failure times, measured from the initial start-up of the system and let N(t) denote the number of failures through time t. Duane (1964) suggested plotting the failure time t_{i} (on the horizontal axis) against the cumulative failure rate, which he defined to be N(t_{i})/t_{i}, on double logarithmic axes. Equivalently, the Duane plot can be seen as a plot of log t_{i} against log [N(t_{i})/t_{i}] . Since N(t_{i}) = i, the quantity being plotted on the vertical axis is actually equal to log (i/t_{i}) . Although Duane (1964) suggested that such a plot would show the extent of reliability growth or deterioration his procedure cannot be considered a statistical model because it does not provide a probabilistic mechanism that governs the failure times.

To illustrate the use of the Duane plot, suppose that a repairable system experienced failures at the times shown in Table 1. The increased length of the spacings, that is, the times between failures, indicates reliability growth. The Duane plot, shown in Figure 1, shows a downward trend in the cumulative failure rate.

Table 1: Failure times of a repairable system.

10  55  166  205  341  488  567
731  1308  2050  2453  3115  4017  4569

The nonhomogeneous Poisson process with intensity function

λ(t) = \frac{β}{θ} (\frac{t}{θ})^{β−1} , t > 0

is often used to model the failures of a repairable system. This model is called the power law process because the intensity function is proportional to a power of t. See Crow (1974), Bain and Engelhardt (1991, Chapter 9), Ascher and Feingold (1984), and Rigdon and Basu (1989, 2000). For the power law process, the expected number of failures through time t is

E [N (t)] = ∫_{0}^{t} λ (x) dx = (\frac{t}{θ})^{β} , t > 0.  (1)

In the next section we study the connection between the power law process and the Duane plot. We show that the power law process does not imply a linear Duane plot, and a linear Duane plot does not imply the power law process.

2 Linearity and the Power Law Process

Dividing both sides of (1) by t gives

E [\frac{N (t)}{t}] = \frac{t^{β−1}}{θ^{β}} , t > 0.

If we then take the logarithm of both sides we obtain

log E [\frac{N (t)}{t}] = (β − 1) log t − β log θ,

which shows that log E [N (t) /t] and log t are linearly related. But these are not the quantities plotted on a Duane plot; a Duane plot is a graph of (log t_{i}, log [N(t_{i})/t_{i}]) , not a graph of (log t, log [N(t)/t]) .

It is often suggested (MIL-HDBK-189 p. 115 and p. 130, and MIL-HDBK-781A, pp. 19-20) that the Duane plot be used either as a graphical goodness-of-fit test for the power law process, or as a way to get rough estimates of the parameters β and θ of the power law process. Such suggestions are based on the erroneous assumption that the power law process implies a linear Duane plot and vice-versa. Ascher and Feingold (1984, p. 77) say “If the process generating failures can be modeled as an NHPP with [intensity function λ(t) = (β/θ) (t/θ)^{β−1}], then it can be shown that, subject to sampling variability, a linear plot will be obtained on log-log paper.” Crowder et al. (1991, p. 175) say “… if [λ(t) = (β/θ) (t/θ)^{β−1}] is appropriate, then the [Duane] plot should be roughly linear.” Tobias and Trindade (1995, p. 374) say “… the Duane relationship is equivalent to M(t) = at^{b} …. This expression is just the power relationship model … The intensity function, or instantaneous repair rate, for this process is the derivative m(t) = M′(t) = abt^{b−1}.” The confusion between the linearity of the Duane plot and the power law process is widespread.

2.1 A Power Law Process Does Not Imply a Linear Duane Plot

When viewed before testing begins the failure times are random variables which we will denote by T_{i}, i = 1, 2, …. In this case N (T_{i}) /T_{i} = i/T_{i} is also a random variable, so (log T_{i}, log N(T_{i})/T_{i}) , i = 1, 2, …, n, are random points in the plane. (Note that when T_{i} is random, N (T_{i}) is still equal to i.)

The expectations of these points are

E [ (log T_{i}, log [N(T_{i})/T_{i}]) ] = ( E [log T_{i}] , E [log N (T_{i})] − E [log T_{i}] )
= ( E [log T_{i}] , log i − E [log T_{i}] ) .

Computation of these expectations, therefore, hinges on the computation of E [log T_{i}] . Since the distribution of the time to the ith failure has a generalized gamma distribution with pdf

f_{i}(t) = \frac{1}{Γ(i)} \frac{β}{θ} (\frac{t}{θ})^{iβ−1} exp [− (\frac{t}{θ})^{β}] , t > 0

(Rigdon and Basu (2000), pp. 56-57) we can write

E [log T_{i}] = ∫_{0}^{∞} (log t) \frac{1}{Γ(i)} \frac{β}{θ} (\frac{t}{θ})^{iβ−1} exp [− (\frac{t}{θ})^{β}] dt

= log θ + \frac{1}{βΓ(i)} ∫_{0}^{∞} (log y) y^{i−1} e^{−y} dy.

This last line is obtained by making the transformation y = (t/θ)^{β} . If we apply integration by parts to the integral

A_{i} = ∫_{0}^{∞} (log y) y^{i−1} e^{−y} dy

we see that for i > 1

A_{i} = [− (log y) y^{i−1}e^{−y}]_{0}^{∞} + ∫_{0}^{∞} e^{−y} [(log y) (i − 1) y^{i−2} + y^{i−2}] dy
= (i − 1) A_{i−1} + Γ(i − 1).

When i = 1, we have

A_{1} = ∫_{0}^{∞} (log y) e^{−y}dy.

The substitution u = log y transforms this integral into

A_{1} = ∫_{−∞}^{∞} u exp (u − e^{u}) du

This integral gives the mean of the standard extreme value distribution, which can be shown to be −γ ≈ 0.577216, where γ is Euler’s constant. (See Gumbel (1958).)

The sequence {A_{i}} therefore satisfies the recurrence relation

A_{1} = −γ
A_{i} = (i − 1) A_{i−1} + Γ(i − 1), i ≥ 2  (2)

Theorem 1, which is proved in the Appendix, gives the solution of this recurrence relation.

Theorem 1 The solution of the recurrence relation given in (2) is A_{i} = Γ(i)ψ(i), where ψ(z) = Γ′(z)/Γ(z) is the digamma function.

The expectations of the points on the Duane plot are therefore

( E [log T_{i}] , i − E [log T_{i}] ) = (log θ + \frac{1}{βΓ(i)}A_{i}, log i − log θ − \frac{1}{βΓ(i)}A_{i})

= (log θ + \frac{ψ(i)}{β}, log i − log θ − \frac{ψ(i)}{β}) .

Thus the points on the Duane plot are linear if and only if

log i − log θ − \frac{ψ(i)}{β} = m (log θ + \frac{ψ(i)}{β}) + b

for some m and b. This is equivalent to

log i − \frac{m + 1}{β}ψ (i) = (m + 1) log θ + b.  (3)

Thus, the points on a Duane plot are linearly related if and only if log i and ψ(i) are linearly related, which is, of course, false. Figures 2 and 3 show the extent of the nonlinearity for particular values of n and β. This tells us that nonlinearity in a Duane plot does not indicate a departure from a power law process.

Figures 2 and 3 show that for large i the Duane plot is nearly linear and that the greatest curvature occurs for small values of i. The explanation for this phenomenon is as follows. Equation (6.3.22) in Abramowitz and Stegun (1964) implies that

log z − ψ(z) = \frac{1}{2z} + 2 ∫_{0}^{∞} \frac{x}{(x^{2} + z^{2}) (e^{2πx} − 1)} dx.  (4)

Since the right side of (4) tends to zero as z → ∞ we conclude that for large values of z, log z ≈ ψ(z). The choice of m = β − 1 and b = −β log θ in (3) leads to

log i − ψ(i) = 0.  (5)

Although (5) is never true for finite values of i, the left side approaches zero as i → ∞. Thus, for large i, the Duane plot should be nearly linear with slope m = β − 1 and intercept b = −β log θ.

2.2 A Linear Duane Plot Does Not Imply a Power Law Process

It is easier to demonstrate that the converse (i.e., a linear Duane plot implies a power law process) is false. Consider a system that fails deterministically at every multiple of four hours. The failure times are therefore t_{i} = 4i so the points on the Duane plot are

(log 4i, log [i/(4i)]) = (log 4i, log \frac{1}{4}) .

These points are linearly related (the line is perfectly flat) but the process is not a power law process.

3 Conclusion

It is a common misconception that the linearity of a Duane plot is synonymous with the power law process, a nonhomogeneous Poisson process with intensity function λ(t) = (β/θ) (t/θ)^{β−1} .

We have shown that the power law process does not imply a linear Duane plot and that the linearity of a Duane plot does not imply the power law process.

References

1. Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, National Bureau of Standards: Washington, 1964.

2. Ascher, H. and Feingold, H. Repairable Systems Reliability: Modeling, Inference, Misconceptions and their Causes, Marcel Dekker: New York, 1984.

3. Bain, L. J. and Engelhardt, M. Statistical Analysis of Reliability and Life Testing Models: Theory and Methods, Second Edition, Marcel-Dekker: New York, 1991.

4. Crow, L. R. “Reliability Analysis for Complex Systems,” Reliability and Biometry, Eds. F. Proschan and R. J. Serfling, 1974, pp. 379–410.

5. Crowder, M. J., Kimber, A. C., Smith, R. L. and Sweeting, T. J. Statistical Analysis of Reliability Data, Chapman and Hall: New York, 1991.

6. Duane, J. T. “Learning Curve Approach to Reliability Monitoring,” IEEE Transactions on Aerospace 1964 2, pp. 563–566.

7. Gumbel, E. J. Statistics of Extremes, Columbia University: New York, 1958.

8. MIL-HDBK-189 Reliability Growth Management, U.S. Department of Defense, 1981.

9. MIL-HDBK-781A Handbook for Reliability Test Methods, Plans, and Environments for Engineering, Development, Qualification, and Production, U. S. Department of Defense, 1996.

10. Rigdon, S. E. and Basu, A. P. “The Power Law Process: A Model for the Reliability of Repairable Systems,” Journal of Quality Technology 1989 10, pp. 251–260.

11. Rigdon, S. E. and Basu, A. P. Statistical Methods for the Reliability of Repairable Systems, Wiley: New York, 2000.

12. Tobias, P. A. and Trindade, D. C. Applied Reliability, Second Edition, Chapman and Hall: New York, 1995.

Appendix: Proof of Theorem 1

The proof is by induction. When i = 1, we have, using (6.3.2) from Abramowitz and Stegun (1964),

Γ(1)ψ(1) = 1 × (−γ) = −γ = A_{1}.

Now suppose that for k ≥ 2, A_{k} = Γ(k)ψ(k). Then,

A_{k+1} = kA_{k} + Γ(k)

= kΓ(k)ψ(k) + Γ(k) = Γ(k) (kψ(k) + 1) = Γ(k)kψ(k + 1) = Γ(k + 1)ψ(k + 1).

The second last line follows from (6.3.5) in Abramowitz and Stegun (1964).

Figure 1: Duane plot for data in Table 1. 8 Figure 2: Expectations of points on a Duane plot for β = 1.1 and n = 20.

9 Figure 3: Expectations of points on a Duane plot for β = 1.5 and n = 20.

10

March 15, 2001

Douglas Montgomery Editor, Quality and Reliability Engineering International Department of Industrial Engineering Arizona State University Tempe, AZ 85287-5906

Dear Doug,

Enclosed is a manuscript entitled “Properties of the Duane Plot for Repairable Systems” which I am submitting for possible publication in Quality and Reliability Engineering International. I look forward to hearing from you.

Sincerely,

Steve Rigdon