Although ATM does not include any end$-$to$-$end error detection and control functions on the user data, it is provided with a HEC field to detect and correct header

errors. Let us consider the value of this feature. Suppose that the bit error rate of the

transmission system is B$.$ If errors are uniformly distributed, then the probability of

an error in the header is

h

h $+$ i

$*$ B

and the probability of error in the data field is

i

h $+$ i

$*$ B

where h is the number of bits in the header and i is the number of bits in the data field.

a$.$ Suppose that errors in the header are not detected and not corrected. In that case,

a header error may result in a misrouting of the cell to the wrong destination;

therefore, i bits will arrive at an incorrect destination, and i bits will not arrive at

the correct destination. What is the overall bit error rate B$1?$ Find an expression

for the multiplication effect on the bit error rate: M$1=$ B$1>$B$.$

b$.$ Now suppose that header errors are detected but not corrected. In that case, i bits

will not arrive at the correct destination. What is the overall bit error rate B$2?$

Find an expression for the multiplication effect on the bit error rate: M$2=$ B$2>$B$.$

c$.$ Now suppose that header errors are detected and corrected. What is the overall

bit error rate B$3?$ Find an expression for the multiplication effect on the bit error

rate: M$3=$ B$3>$B$.$

d$.$ Plot M$1,$ M$2,$ and M$3$ as a function of header length, for i $=48*8=384$ bits.

Comment on the results.