Find the frame length with derivative

DODGEVIPER13 · Sep 20, 2014

Find the optimum frame length nf that maximizes transmission efficiency for a channel with random bit erros by taking the derivative and setting it to zero for the following protocols:
(a) Stop-and-Wait ARQ
(b) Go-Back-N ARQ
(c) Selective Repeat ARQ

My work has been uploaded I am a bit rusty on derivative, so I am pretty sure I made a mistake just unsure of where.

jedishrfu · Sep 20, 2014

The last factor with nf is wrong I think. I used the duv = u*dv + v*du rule with v= (nf+B)^-1 and got a different factor from yours so check it again and post your result.

DODGEVIPER13 · Sep 21, 2014

What did you use for u? I guess you did a u substitution to do this then so if I find u I can do the derivative.

jedishrfu · Sep 21, 2014

u = nf - n0 your numerator and v= (nf+B)^-1

DODGEVIPER13 · Sep 23, 2014

Handwritten derivation of d(u/V) using the quotient rule, with u = n_f - n_0 and V = (n_f + B)^-1, ending in the boxed result (n_0 + B)/(n_f + B)^2, followed by the note 'Got the same thing did I do something wrong'.

jedishrfu · Sep 23, 2014

My apologies, I must have done something wrong. Yours looks correct. How is the book answer different? That might tell you where the real error is.

jedishrfu · Sep 24, 2014

Okay I think I see your error. You differentiated the a^nf and multiplied it to the differentiated version of the second factor. Don't you have to apply the duv product rule here too?

With u=a^nf and v= the rest.

DODGEVIPER13 · Sep 25, 2014

Sorry if the post is a bit confusing and my slow responses.

Handwritten derivative working using the quotient rule: u = a^(n_f), v = (n_0 + B)/(n_f + B)^2, with du and dv expanded, followed by crossed-out steps and a simplified expression with terms like -2n_0 a^(n_f)/(n_f + B)^3.

DODGEVIPER13 · Sep 29, 2014

Is my answer improved at all?

jedishrfu · Sep 29, 2014

Yes, it looks right but you can do more by extracting out the a^nf factor and by finding a common denominator so you can combine numerator terms ie multiply the second term by (nf+B)/(nf+B).

DODGEVIPER13 · Oct 2, 2014

Posted this with my phone sorry if it is hard to read

Handwritten derivation on lined paper: top line shows A → B111 1100, followed by d(cur) = a^(nf) [ln(a)((no+B)(nf+B)/(nf+B)^3) − 2((no+B)/(nf+B)^3)], and below it = a^(nf) [ln(a)(no+B)(nf+B) − 2(no+B)] / (nf+B)^3.

DODGEVIPER13 · Oct 2, 2014

So should this be simplified further?

jedishrfu · Oct 2, 2014

I can't see anything further. Does this differ from some book answer you have? or were you expecting it to be much simpler?

Find the frame length with derivative

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