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A multichannel microwave link is to provide telephone communicationto a remote community having 12 subscribers, each of whom uses the link 20% of the time during peak hours. How many channels are needed to make the link available during peak hours to:

a. Eighty percent of the subscribers all of the time?

b. All of the subscribers 80% of the time?

For the a) part I came up with answer 10 channels which is correct according to my TA but I am not sure about the procedure I followed (I just found out what is eighty percent of 12)

I have no clue what b part means

Aditya
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1 Answers1

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Let $c$ be the number of necessary channels and let $X \sim \text{Binomial}(12, 0.2)$ be the total number of subscribers that want to use the link at the same time during peak hours. Then we want to find the minimum $c$ such that: $$ \text{Pr}(X \leq c) \geq 0.8 $$ Using this formula: $$ \text{Pr}(X = c) = \binom{12}{c}(0.2)^c(0.8)^{12 - c} $$ we guess and check by making a table: \begin{array}{c|l} c & \text{Pr}(X \leq c) \\ \hline 0 & 0.0687195 \\ 1 & 0.274878 \\ 2 & 0.558346 \\ 3 & 0.794569 \\ 4 & 0.927445 \end{array}

So we conclude that $c = 4$ channels are more than enough.

Adriano
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  • why did u assume the number subscribers that want to usr the link at the same time during peak hour as Binomial distrubution? – Aditya Aug 20 '14 at 16:39
  • Each subscriber either wants to use the link or doesn't want to. The probability of using a link is the same for each subscriber. Each subscriber's decision as to whether or not to use the link should be independent of all the other subscribers' decisions. – Adriano Aug 20 '14 at 16:42
  • thanx buddy..!! – Aditya Aug 20 '14 at 20:45