Most fairly accurate descriptions of equipment and/or process lifetimes assume that failure rates follow a three period I II III “bathtub-curve pattern” where failures/errors:

I – Decrease during the debugging or improvement time period.

II – Remain relatively constant and at their lowest levels during the normal equipment or process operating period.

III – Increase during the wearout time period.

Scientific studies of limit based natural or complex growth patterns also suggest that many processes are inherently non-linear and subject to chaotic tendencies. The logistics map3 or parabola Xt + 1 = RXt (1-Xt) where Xt + 1 the measure of the next generation is a function of the present measure Xt, R is the growth factor and t is a discrete time variable is a simple model for these processes. When the growth factor R falls within the range of 1 < R < 3 the process is stable. For R = 2, the time series iterates Xt = X1 X2 X3… converge to the constant value Xc =.5 which can be easily demonstrated (see Table 1) through the use of an Excel spreadsheet or pocket calculator.

 Table 1: Logistics Map – Xt + 1 = Rxt (1-Xt) Calculated Iterates Process Category Unstable Decreasing Stable Constant Unstable Oscillation Unstable Oscillation Unstable Chaotic R = 1 2 3 3.24 3.8 X0 0.800 0.800 0.800 0.800 0.800 X1 0.160 0.320 0.480 0.518 0.608 X2 0.134 0.435 0.749 0.809 0.906 X3 0.116 0.492 0.564 0.501 0.325 X4 0.103 0.500 0.738 0.810 0.833 X5 0.092 0.500 0.581 0.499 0.528 X6 0.084 0.500 0.730 0.810 0.947 X7 0.077 0.500 0.591 0.499 0.191 X8 0.071 0.500 0.725 0.810 0.587 X9 0.066 0.500 0.598 0.498 0.921

The critical growth factor value Rcr = 3.24 (51/2 +1 ) in Table 1 signals the start of chaotic instability in this model process and for R = 3.8 the instability is clearly evident.

### Process Variance Stability

If we assume that variance (Vt) of a process during its lifetime varies between zero and some maximum acceptable value Vm, the application of the logistics parabola model to the process results in the iterate expression Vt + 1 = RVt (Vm – Vt). In this case the process is stable1 within the growth factor R range of 1/Vm < R < 3/Vm. Also, the process attains super-stability or constancy when its variance equals one half of the maximum acceptable value (Vt = Vm/2) and when R = 2/Vm. This is illustrated for a process with a maximum-allowed variance of Vm= 9 (standard deviation =3) in Table 2.

 Table 2: Super-stable Process Variance Map – Vt + 1 = RVt (9 – Vt) Variance Category Unstable Decreasing Super-stable Constant Unstable Oscillation Unstable Oscillation Unstable Chaotic R = 0.111 0.222 0.333 0.360 0.422 V0 4.50 4.50 4.50 4.50 4.50 V1 2.25 4.50 6.74 7.29 8.55 V2 1.68 4.50 5.07 4.49 1.64 V3 1.37 4.50 6.64 7.29 5.09 V4 1.16 4.50 5.22 4.49 8.40 V5 1.01 4.50 6.57 7.29 2.13 V6 0.89 4.50 5.32 4.49 6.18 V7 0.80 4.50 6.52 7.29 7.35 V8 0.73 4.50 5.38 4.49 5.12 V9 0.67 4.50 6.48 7.29 8.39