# Question on calculating throughput

`Here is the necessary information:

A file contains:

• A header of size of h bits
• Data component of size d bits
• There is a probability b that a bit will be dropped (ruining the file)

The connection can support a maximum of c bits/second

I need to:

• Calculate the probability a file will be ruined. Already did this: (1-b)^(h+d)
• Compute the throughput over the link. (h remains the same size, so what is the optimal value for d?)

I'm having problems coming up with a function to model the number of files being sent through. If I could do that, I think it would be simple to maximize it.

Any help is appreciated.

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I have removed [calculus] and [logic] since they most definitely did not fit the question. I am not even sure about [probability] and [statistics], but I cannot decide in this case. I am also not too sure what tag would fit here instead of these four. I will leave it to more competent folks, then... – Asaf Karagila Oct 10 '11 at 5:35

Your throughput is about $c (1-b)^{h+d} \dfrac{d}{h+d}$, which you want to maximise by changing $d$.

If you take the derivative with respect to $d$ and set this to 0, I think you may find yourself solving $d^2+hd +h/\log_e(1-b)=0$

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Since $d$ is an integer, I don't like taking the derivative with respect to $d$ as @Henry has suggested. So, as an alternative, taking the throughput to be $c(1-b)^{h+d} \frac{d}{h+d}$, we look at what happens if we increase $d$ by $1$. The throughput is now $c(1-b)^{h+d+1} \frac{d+1}{h+d+1}$ and we can examine the ratio $$\frac{c(1-b)^{h+d}\frac{d}{h+d}}{c(1-b)^{h+d+1} \frac{d+1}{h+d+1}} = \frac{d(h+d+1)}{(1-b)(d+1)(h+d)}$$ to find the smallest value of $d$ for which the ratio has value $1$ or more. Once again, the answer is the solution to a quadratic equation (and only slightly different from Henry's result).

The maximum throughput of a different model of the transmission process, in which the transmission errors are detected at the receiving end, a request for a repeat of the file transmission is sent to the transmitter, and file transmissions are repeated until the file is successfully received (an ARQ system), is discussed here.

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