Transfer Time Model

Transfer time ( $t_{transfer}$), the time to actually read/write the data, has four components. The first is the time to read or write the data by moving the mover over the tip array with a constant velocity (read/write time, or $t_{RW}$). Second, if a request spans more than one bit column, we must add the time that it takes for the mover to reverse direction (e.g., from moving down to moving up); we call this time the turnaround time ($t_{TA}$). To minimize wasted space, a sector can begin at any bit within a column, and continue at the next column. The MEMS device controller is responsible for tracking the starting positions of sectors. Thus, a single read/write request may require one or more bit column movements in the $X$ direction. This third component is called an x-move ($t_{XM}$). The last component is the time that it takes to switch between sets of active tips, or tip switch time ($t_{TS}$).

Thus, the transfer time is a sum of four terms: $n_{TA}$, the number of turnarounds, multiplied by $t_{TA}$, the turnaround time; $n_{TS}$, the number of tip switches times $t_{TS}$, the tip switching time; $n_{XM}$, the number of moves in $X\!$, times $t_{XM}$, the time to move one bit in $X\!$, and $t_{RW}$, the time it takes to actually read the data. Equation 3 shows this combination.

$$t_{transfer} = n_{TA} \times t_{TA} + n_{TS} \times t_{TS} +$$

$$n_{XM} \times t_{XM} + t_{RW}$$ (3)

As described in Section 3.2,the data to be read or written is divided among all active tips, which work in parallel. Each tip reads $r_f$ bits, which is equal to the number of bits per request (8 $r$) divided by the number of active tips $T_{active}$, as shown in Equation 4.

$$r_f = \frac{8r}{T_{active}}$$ (4)

The read/write time, $t_{RW}$, depends on the number of bits $r_f$ that each active tip has to read, the velocity of the device ($v_0$), and the bit width $d_b$ (to translate the number of bits into distance). To read or write each bit, it will take $d_b/v_0$ time for each active tip to move over it. This relationship is given in Equation 5.

$$t_{RW} = \frac{r_f \times d_b}{v_0}$$ (5)

The number of turnarounds, $X$-moves, and tip changes depend on the number of requested bits that each active tip accesses, the starting position, and the the number of bits in a bit column ( $n_{bitsinY}$). To calculate this quantity we divide the movement range in Y by the bit-width. Using the layout described in Section 3.2, the number of turnarounds is the same as the number of tip switches, because a tip switch is necessary at every turnaround. The $X$-move component also depends on the number of tips in $Y$ per cluster that determines the cluster dimension (i.e., the number of bits in one cluster column).

Specifically, the average number of turnarounds (or tip switches) per request is the ratio of the number of bits each tip must access to service the average size request ($r_f$) and the number of bits in each column ( $n_{bitsinY}$), as shown in Equation 6.

$$n_{TA} = \frac{r_f}{n_{bits in Y}}$$ (6)

The average number of $X$-moves is the ratio of the average requested number of bits per active tip ($r_f$) divided by the product of the number of bits in $Y$ times the number of tips in $Y$ per cluster ($T_y$), as shown in Equation 7.

$$n_{XM} = \frac{r_f}{n_{bits in Y} \times T_y}.$$ (7)

The number of bits in $Y$ is equal to the movement range in $Y$ divided by the bit-width.

Substituting Equations 4-6 in Equation 3 gives us an expression for the transfer time in milliseconds shown in Equation 8.

$$t_{transfer}(\delta_y, T_{active}, T_y, r) =$$

$$\frac{8 r d_b}{T_{active}} (\frac{1}{\delta_y} (t_{TA} + \frac{t_{XM}}{T_y}) + \frac{1}{v_0})$$ (8)