U. Glaeser

3. Move horizontally to the intersection with the linear segment.
4. Move down to the horizontal axis and determine the point x ∗ .
Because the linear segment of the seek time characteristic determines the values of x ∗ and T ∗ (as well as a
and v max), it follows that the nonlinear segment must terminate exactly in the (x ∗ , T ∗ ) point. A numerical
method for computing x ∗ and T ∗ can be based on any two points (x 1, T 1) and (x 2, T 2) taken on the linear
segment. From the linear segment function T = T 1 + (x − x 1)(T 2 − T 1)/(x 2 − x 1), it follows:
–
-------------------------- , T ∗
2 T1x 2 – T2x 1
= --------------------------
–
x ∗ T1x 2 T2x 1
=
T2 – T1 The presented model is suitable for a qualitative description of the seek time behavior and for providing
insight into disk characteristics, but its flexibility is limited because it has only two parameters. In addition
to limited flexibility, this model is not appropriate for those disk units that do not satisfy the assumptions
of constant acceleration/deceleration, and for units where the seek time for short distances is significantly
affected by the head settling time. The accuracy of modeling can be improved using numerical models
that fit the measured characteristic and use more than two adjustable parameters.
Numerical Computation of the Average Seek Time
A simple numerical model can be based on three points: (x 1, T 1), (x ∗ , T ∗ ), and (x 2, T 2), where x 1 < x ∗ < x 2.
Here, (x 1, T 1) is the point from the initial nonlinear segment. Contrary to the approach presented in the
section on “A Fixed Maximum Velocity Model of Seek Time,” the middle point (x ∗ , T ∗ ) is not computed
from the linear part of the measured characteristic, but directly selected from the seek time graph as the
beginning of the linear segment. The point (x 2, T 2) denotes the end of the linear segment (the maximum
distance heads can travel). The corresponding model is
r
tx1 T( x)
∗ x
T
x ∗
⎛---- ⎞
⎝ ⎠
r
tx r T
, t
∗
x ∗
( ) r ----------- , x x ∗
= = ≤
T ∗ T2 T ∗
–
x2 x ∗ ----------------- x x
–
∗
+ ( – ) Ax B, x x ∗
⎧
⎪
⎪
= ⎨
⎪
⎪
= + ≥
⎩
From T 1 = and T ∗ = t(x ∗ ) r it follows that T ∗ /T 1 = (x ∗ /x 1) r . Therefore, the parameters are
r
A
T ∗ /T 1
log(
)
r
= --------------------------, t = T1/x1= log(
)
x ∗ /x 1
T2 T ∗
–
x ∗ ----------------- , B T
–
∗ Ax ∗
= = –
x 2
Using this model it is possible to numerically compute the mean seek time. Suppose that a file occupies
N cylinders. The probability that the seek distance x is less than or equal to a given value z is
This probability distribution function yields the following probability density function:
© 2002 by CRC Press LLC
x 2
x 1
T ∗ / x ∗
( ) r
Px( z)
P[ x≤ z]
2Nz z 2
( – )/N 2
= =
px( z)
dP/dz 2( N– z)/N
2
= =