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Suppose that the stop-and-wait protocol is used on a link with a bit rate of $64$ $\text{kilobits}$ per second and $20$ $\text{milliseconds}$ propagation delay. Assume that the transmission time for the acknowledgment and the processing time at nodes are negligible. Then the minimum frame size in bytes to achieve a link utilization of at least $50$ $\text{%}$ is_________________.

edited
@arjun sir

as bandwidth is given and propagation delay is mentioned we can find total capacity of channel by RTT *BANDWIDTH i.e 64 kbps * 40 ms which gives 2560 as total capacity in bits now  2560/8 i.e 320 B for 100 % utilisation so for 50% it should be 160 B!

Why this approach is wrong? What am I missing?

when link utilization of at least 50%

then TT =2PT

That is not right. @Raj

Efficiency = etta(n)

Link utilization = etta * Bandwidth

Was the guy who made the question was high on weed, or is there some concept behind the 160 in the official answer. Anyone know?
haha

Link Utilization $=\dfrac{\text{Amount of data sent}}{\text{Max. amount of data that could be sent}}$

Let $x$ be the frame size in bits.

In stop-and-wait protocol, once a frame is sent, next frame won't be sent until ACK is received.

Time for this,

RTT $=\text{Propagation delay for frame + Transmission time for frame}$
$\qquad\text{+ Propagation delay for ACK + Transmission time for ACK}$

$= 20\ \text{ms} +\dfrac{ x }{64\ \text{ms}}+20\ \text{ms} + 0$ (as given in question)

$=\left(40 +\dfrac{x}{64}\right)\ \text{ms}$.

Amount of data sent during RTT $= x$

Max. amount of data that could be sent $= \left(40 +\dfrac{x}{64}\right)\times 64=2560+x\ \text{bits}$.

So, link utilization,$0.5 = \dfrac{x}{(2560 + x)}$

$x= 2560\ \text{bits} = 320\ \text{bytes}$.

Alternative Approach ,

Link utilization or efficiency of stop and wait protocol is ,

efficiency $=\dfrac{T_x}{(T_x + 2T_p)}=\dfrac{1}{\left(1+2\left(\dfrac{T_p}{T_x}\right)\right)}=\dfrac{1}{(1 +2a)},$

where , Transmission time $=T_x=\dfrac{\text{packet size}}{\text{bandwidth}}=\dfrac{L}{B}$

Propagation time$=T_p=\dfrac{\text{distance}}{\text{speed}}=\dfrac{d}{v},$ and
$a=\dfrac{\text{Propagation time}}{\text{Transmission time}}=\dfrac{T_p}{T_x},$

Now for $50\%$ efficiency ,

efficiency $=\dfrac{1}{(1+2a)}$

$50\%=\dfrac{1}{(1+2a)}$

$\dfrac{1}{2}=\dfrac{1}{(1 +2a)}$

$2=(1+2a)$

$2-1=2a$

$1=2\left(\dfrac{T_p}{T_x}\right)$

$T_x =2\times T_p$

$\dfrac{L}{B}=2\times 20\ \text{ms}$

$L=2\times 20\ \text{ms} \times B=2\times 20\times 10^{-3}\times 64\text{ k bits}$

$=2\times 20\times 10^{-3}\times 64\times 10^3\ \text{bits}$

$L=40\times 64\ \text{bits} = 40\times \dfrac{64}{8}\ \text{bytes} = 40\times 8\ \text{bytes} = 320\text{ bytes (answer)}$

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was answer changed or given bonus or

Don’t think answer debate option was there during GATE 2015. May be this question was a motivation for adding that :)

That’s a relief :)
for a stop and wait ARQ we send only a single packet at once.

\begin{align*} \eta &= \frac{TT}{TT+2\times PT} \\ \frac{50}{100}&= \frac{f/64k}{f/64k+2\times 20ms}\\ \frac{f}{64}&=40\\ f &= 2560bits\\ &= 2560/8\\ &= 320Bytes \end{align*}

160 was in key and I guess ACE had given some formula like that. But AFAIK that is wrong..
Why they have given wrong answer?  @arjun sir

let x be Bandwidth delay product = $2*T_{propagation}*BW$

$\Rightarrow$ Bandwidth delay product = $2*20*10^{-3}*64*10^{3}$ = $2560 bits$

Now,Let y be the length of packet.

$\frac{y}{x} >= \frac{1}{2}$

$\Rightarrow$ $y >= \frac{2560}{8*2}$Bytes

Therefore length of packet should be atleast $160 bytes$.

We end up with a wrong answer if include Transmission delay in RTT! But Forouzan has not included Transmission delay in his examples. Moreover if you don't include Transmission delay in RTT you will come up with 16 as an answer for the question in the following link which is wrong.  (Same year question). This is ambiguity.

https://gateoverflow.in/8056/gate2015-2_8

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edited by

Aghori as per your given definition and the way  @reena_kandari has derived LU=1/(1+2a), we can say that it is link efficiency too. As both giving same result. Than how are they different? plz clear

@reena_kandari  so for this question, if we follow Arjun Sir approach than correct ans is 320 B since Tt for sending frame isn't negligible and we have to include this in RTT? please correct me.

This is the right way to get the answer of this question..:)

What is the difference between link utilization and throughput?

## ½ = tf/(tf + 2tp)

on evaluation above we get

tf +2tp = 2tf

tf = 2tp   , and we know that tf = length / bandwidth,

length/bandwidth = 2tp

length = 2tp * Bandwidth

length = 2 * 20 ms * 64kilobits persec

length = 2560 bits

length = 2560/8 = 320 bytes.