One key design decision for ATM was whether to use fixed$-$ or variable$-$length cells. Let us consider this decision from the point of view of efficiency. We can define transmission efficiency as

$N=$ Number of information octets $/$ Number of information octets $+$ Number of overhead octets

a$.$ Consider the use of fixed$-$length packets. In this case, the overhead consists of the header octets. Define

$L=$ Data field size of the cell in octet

$H=$ Header size of the cell in octets

$x=$ Number of information octets to be transmitted as a single message

Derive an expression for $N.$ Hint: The expression will need to use the operator $*,$ where $[Y]=$ the smallest integer greater than or equal to $Y.$

b$.$ If cells have variable length, then overhead is determined by the header, plus the flags to delimit the cells or an additional length field in the header. Let $Hv=$ additional overhead octets required to enable the use of variable$-$length cells. Derive an expression for $N$ in terms of $x,H,$ and $Hv,$

c$.$ Let $L=48,H=5,$ and $Hv=2.$ Plot $N$ versus message size for fixed$-$ and variable$-$length cells. Comment on the results.