Distance travelled on average between successive seeks

Assuming that next I/O request resides in any cylinder with equal probability, what is the distance travelled by head, on average, between successive seeks on disk with n cylinders.

D = random variable denoting the distance travelled between successive seeks

\begin{array}{rcl} E[D] &=& \sum_{d=0}^{n-1} d . pr(D=d) \\ &=& 0 . \frac{n}{n^2} + \sum_{d=1}^{n-1} d . \frac{2(n-d)}{n^2} \\ &=& \frac{n^2-1}{3n} \\ &\approx& \frac{n}{3} \end{array}

So, if all cylinders have equal probability of access, the average seek distance is 1/3 the maximum seek

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