# Ten phones are linked by only one line to the network. Each phone

1. Apr 9, 2009

### forty

Ten phones are linked by only one line to the network. Each phone needs to log on average 12mins an hour to the network. calls made by different phones are independent of each other. They can't simultaneously call.

What is the probability that k phones (k=0,1,2...10) simultaneously need the line? what is the most likely number of phones requiring the line at one time?

Is this a binomial distribution?

if so, does P(X=k) = (10 choose k)(.2^k)(.8^(10-k)) ??

and also would E[X] = 2?

OR is this a Poisson (or other :S) distribution?

Any help as always is greatly appreciated :)

2. Apr 9, 2009

### Focus

Re: Binomial?

Personally I would use Poisson, esspecialy as it involves time you may want to use a Poisson process. The nice thing about Poisson is independent Poisson processes do not jump at the same time and you can add Poisson processes very easily.

It looks like a M/M/1 queue.

3. Apr 9, 2009

### forty

Re: Binomial?

So if i was to use a Poisson distribution would my parameter be the average? which is 2?

4. Apr 9, 2009

### Focus

Re: Binomial?

Is there not a rate of calls? The rate should be the rate of people calling in.

5. Apr 9, 2009

### forty

Re: Binomial?

So by rate you mean 12/60 (12 minutes an hour)?

6. Apr 9, 2009

### Focus

Re: Binomial?

No I mean the rate of calls being made or the line requested.

7. Apr 9, 2009

### forty

Re: Binomial?

I don't really know what you mean by this. On average the likely hood of any one line being used is (12/60)? Hmmm now I'm really lost >.<

8. Apr 10, 2009

### Focus

Re: Binomial?

Sorry I think I confused myself here. The callers need 12mins each per hour right? How is their requests distributed? Do they have Poisson arrival then get served for 12 mins or can they call in for 2 mins then hang up and phone again after a while and spend 10 mins?

9. Apr 11, 2009

### forty

Re: Binomial?

So... I think what you are getting at is that its a Binomial distribution with the probability being a poisson distribution? if not I'm completely lost :(

10. Apr 11, 2009

### Focus

Re: Binomial?

Sorry what I am trying to get at is this; if you have one line and people calling in with Poisson with rate $\lambda$ and say they take an exponential time on the line with parameter $\mu$ then you have a M/M/1 queue. Your question looks very similar to this but it depends on the rate they request the line at. The exponential parameter looks like 5 as they need on average 12 mins on the network, but I don't understand what rate they are calling with.