A single disk storage device has $N$ concentric tracks and one access arm. It has been loaded with data in such a way that successive movements of the access arm (called track seeks) are independent of one another. The probability that a randomly chosen seek will take the arm to track $i$ is $p_i$. Let $X$ represent the number of tracks passed between consecutive seeks, assuming that no physical repositioning of the access arm takes place between successive seek operations. Show the following:
(a) $X$ assumes the values $0,1, \ldots, N-1$ and has the $\operatorname{pmf} p(\cdot)$ defined by
$$
p(j)=P[X=j]= \begin{cases}\sum_{i=1}^N p_i^2, & j=0 \\ 2 \sum_{i=1}^{N-j} p_i p_{i+j}, & j=1,2, \ldots, N-1 .\end{cases}
$$
(b) For the case that $p_i=1 / N$ for all $i$, it is true that
$$
\begin{gathered}
E[X]=\frac{\left(N^2-1\right)}{3 N} \approx \frac{N}{3}, \\
E\left[X^2\right]=\frac{\left(N^2-1\right)}{6} \approx \frac{N^2}{6},
\end{gathered}
$$
and
$$
\operatorname{Var}[X] \approx \frac{N^2}{18} .
$$
(c) Suppose $T$, the seek time, is a linear function of $X$; that is,
$$
T=A X+B,
$$
where $A$ and $B$ are constants. ( $A$ is then given by
$$
A=\frac{\text { maximum seek time }- \text { minimum seek time }}{N-1},
$$
and $B$ is minimum seek time.)
Then it is true that
$$
E[T]=\frac{A\left(N^2-1\right)}{3 N}+B \approx \frac{A N}{3}+B,
$$
and
$$
\operatorname{Var}[T]=A^2 \operatorname{Var}[X] \approx \frac{A^2 N^2}{18} .
$$