##### GATE 2007 The distance between two stations M and N is

GATE 2007

The distance between two stations M and N is L kilometers. All frames are K bits
long. The propagation delay per kilometer is t seconds. Let R bits/second be the
channel capacity. Assuming that processing delay is negligible, the minimum
number of bits for the sequence number field in a frame for maximum utilization,
when the sliding window protocol is used, is:
A ) log { $(2LtR +2K) / {K }$}

B) ​ ​​ ​log { ​​ $(2LtR/K)$ ​​ ​}

C) log { $(2LtR+K)/K$}

D) log{ $(2LtR+K)/2K$}

Ans : C.

I dont how many of you know this that this problem is given in exercise of William Stallings Book Data and Network Communication. Chapter 6 Data link control. Any way that is not our concern.

Bandwidth R bps , Frame size = K bits , PT = t sec,/ kilometer, distance L kilometer. So total PT ( not RTT) = L*t sec .Processing delay =0.Assuming W is window size

Now, it says that We need to maximize utilization for sliding window protocol.

Maximum utilization if=> useful time/ total time =1 .

it is possible when ,useful time = Total time.

Useful time= W * {K/R } sec

Total time = K/R + 2* L*t sec

Now  =>  (W * {K/R } ) / ( K/R + 2* L*t ) = 1

W * {K/R } = K/R + 2* L*t

W = K + 2LTR / K

Suppose W= $2^n$where n = number of bits in sequence number

$2^n$​ ​= K + 2LTR / K

n= $log (( K + 2LTR )/K)$

ans C.

I hope it is clear.