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U. Glaeser

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Hence, the average seek time for the N-cylinder file can be computed as follows:<br />

Tseek<br />

Using the presented seek time model for N ≤ x ∗ , we have<br />

Similarly, for N ≥ x ∗ , we get<br />

Tseek<br />

=<br />

=<br />

Tseek<br />

N<br />

2 N<br />

= T( z)px(<br />

z)dz<br />

= ------ T( z)<br />

( N– z)dz<br />

∫<br />

0<br />

N 2<br />

2<br />

N 2<br />

x<br />

------ T( z)<br />

( N– z)dz<br />

0<br />

∗<br />

∫<br />

0<br />

Therefore, the average seek time as a function of file size is given by the following function:<br />

Tseek( N)<br />

Advantages of the presented exponential model are: (1) parameters can be quickly computed from<br />

three selected points of the characteristic, and (2) parameter r enables modeling of disk characteristics<br />

that are different from the square root model. From T(1) = t it follows that t is interpreted as the single<br />

cylinder seek time. The limitation of this model is that by determining t and r from points (x 1, T 1) and<br />

(x ∗ , T ∗ ) it is not possible to have the exact value of t and optimum modeling of the curvature. To improve<br />

this model we can introduce one more parameter as follows:<br />

This new model has a nonlinear part (for x ≤ x ∗ ) and a linear part (for x ≥ x ∗ ). Since T(1) = t and T(2) =<br />

t + c the parameter t is the single cylinder seek time and the parameter c is the difference c = T(2) − T(1).<br />

The parameters of this model (t,c,r,A,B,x ∗ ) are not independent. First, for x = x ∗ the nonlinear part must<br />

be connected to the linear part:<br />

© 2002 by CRC Press LLC<br />

N 2<br />

2t<br />

------ z r N<br />

2tN<br />

∫ ( N– z)dz<br />

r<br />

= = -------------------------------<br />

( r + 1)<br />

( r + 2)<br />

+<br />

∫<br />

N<br />

x ∗<br />

∫<br />

T( z)<br />

( N– z)dz<br />

2<br />

N 2 ------ tz r x<br />

( N– z)dz<br />

0<br />

∗<br />

∫ ( Az + B)<br />

( N– z)dz<br />

x ∗<br />

N<br />

+ ∫<br />

2t x ∗<br />

( ) r+1<br />

-------------------- N x--------r<br />

+ 1<br />

∗<br />

⎛ – ---------- ⎞ N x<br />

⎝ r + 2⎠<br />

∗<br />

( – ) N AN 3B Ax ∗<br />

( + + ) x ∗ 3B 2Ax ∗<br />

[ – ( + ) ]<br />

=<br />

+ ----------------------------------------------------------------------------------------------------------------<br />

N 2<br />

2tN r<br />

------------------------------- , N x<br />

( r + 1)<br />

( r + 2)<br />

∗<br />

≤<br />

2t x ∗<br />

( ) r+1<br />

N 2 -------------------- N x--------r<br />

+ 1<br />

∗<br />

⎛ – ---------- ⎞ N x<br />

⎝ r + 2⎠<br />

∗<br />

( – ) N AN 3B Ax ∗<br />

( + + ) x ∗ 3B 2Ax ∗<br />

[ – ( + ) ]<br />

3N 2<br />

----------------------------------------------------------------------------------------------------------------, N x ∗<br />

⎧<br />

⎪<br />

⎪<br />

= ⎨<br />

⎪<br />

⎪<br />

+<br />

≥<br />

⎩<br />

T( x)<br />

=<br />

⎧<br />

⎪<br />

⎨<br />

⎪<br />

⎩<br />

0,<br />

t c( x– 1)<br />

r + ,<br />

Ax + B,<br />

0<br />

x = 0<br />

3N 2<br />

1 x x ∗<br />

≤ ≤<br />

x x ∗<br />

≥<br />

t c x ∗<br />

( – 1)<br />

r<br />

+ Ax ∗ = + B.

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