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U. Glaeser

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In addition, to assure perfect continuity of this model, for x = x ∗ the nonlinear and linear model must<br />

have the same first derivatives:<br />

dT<br />

----dxx<br />

x ∗<br />

=<br />

Inserting this value of A in the connection relation, we have the expression for B:<br />

The linear function can now be written as<br />

A cr x ∗<br />

( – 1)<br />

r−1<br />

= =<br />

.<br />

B t c x ∗<br />

( – 1)<br />

r<br />

cr x ∗<br />

( – 1)<br />

r−1<br />

x ∗ = + –<br />

.<br />

Ax + B<br />

cr x x ∗<br />

( – )<br />

x ∗<br />

( – 1)<br />

1−r ----------------------- t c x ∗<br />

( – 1)<br />

r<br />

= + + .<br />

Therefore, using formulas (8.1) and (8.2) our exponential model is now<br />

T( x)<br />

=<br />

⎧<br />

⎪<br />

⎨<br />

⎪<br />

⎩<br />

t c( x– 1)<br />

r + ,<br />

1 x x ∗<br />

≤ ≤<br />

cr x x ∗<br />

( – )<br />

x ∗<br />

( – 1)<br />

1−r ----------------------- t c x ∗<br />

( – 1)<br />

r<br />

+ + , x x ∗<br />

≥<br />

The model has four independent parameters t, c, r, x ∗ that can be determined using a calibration<br />

procedure. The objective of calibration is to make the model as close as possible to the measured values<br />

(x 1,T 1),…, (x n,T n). Optimum values of parameters can be computed from the measured values<br />

(x 1,T 1),…, (x n,T n) by minimizing one of the following criterion functions:<br />

E1 t,c,r,x ∗<br />

( )<br />

E2 t,c,r,x ∗<br />

( )<br />

E 1 is a traditional mean square error, E 2 is the mean absolute error and E 3 is used to minimize the<br />

maximum error (“minimax”). These criterion functions yield consistent or similar results and in this<br />

paper we primarily used E 1. For all the above criteria the most suitable minimization method is the<br />

Nelder-Mead simplex algorithm [8]. The resulting calibrated (optimum) values of parameters t,c,r,x ∗ are<br />

those that yield the minimum value of the selected criterion function.<br />

Experiments with modern disk units show that the four-parameter exponential model regularly<br />

achieves high accuracy. Typical average relative errors are between 2% and 3%. The quality of this model<br />

is illustrated in Fig. 8.4 for the Quantum Atlas III disk that has 8057 cylinders, and capacity of 9.1 GB.<br />

The optimum parameters of the model are t = 1.55 ms, c = 0.32 ms, r = 0.387, and x ∗ = 1686 cylinders.<br />

Note that the optimum value of the exponent r is not 1 /2 as expected from constant acceleration/deceleration<br />

models, and frequently used in disk performance literature.<br />

© 2002 by CRC Press LLC<br />

E 3 t,c,r,x ∗<br />

n<br />

1<br />

-- ( T( xi) – Ti) n<br />

2<br />

= ∑<br />

i=1<br />

n<br />

1<br />

= -- |T( xi) – T<br />

n∑<br />

i|<br />

i=1<br />

( ) = max<br />

1≤i≤n<br />

|T( xi) – Ti| (8.1)<br />

(8.2)

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