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

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Discussion Overview

The discussion revolves around the probability of multiple phones needing access to a single network line simultaneously, with a focus on whether the situation can be modeled using a binomial or Poisson distribution. Participants explore the implications of call independence, average call duration, and the nature of the calling process.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions if the scenario can be modeled as a binomial distribution, proposing the formula P(X=k) = (10 choose k)(.2^k)(.8^(10-k)) and suggesting that E[X] = 2.
  • Another participant advocates for using a Poisson distribution, citing its suitability for time-related processes and the independence of calls.
  • There is a discussion about the rate of calls, with one participant suggesting that the average call duration should inform the rate parameter for the Poisson distribution.
  • Confusion arises regarding the interpretation of the average call time and how it relates to the distribution of requests for the line.
  • One participant proposes that the situation resembles an M/M/1 queue, indicating a need to clarify the rate at which calls are made.

Areas of Agreement / Disagreement

Participants express differing opinions on whether to model the situation with a binomial or Poisson distribution, and there is no consensus on the appropriate parameters or the nature of the calling process.

Contextual Notes

Participants highlight the need for clarity on the rate of calls and how the average call duration affects the distribution model. There is uncertainty regarding the assumptions about call timing and request distribution.

Who May Find This Useful

This discussion may be of interest to those studying probability distributions, queuing theory, or telecommunications, particularly in contexts involving limited resources and independent events.

forty
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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 :)
 
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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.
 


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


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


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


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


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

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


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

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?
 


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


Sorry what I am trying to get at is this; if you have one line and people calling in with Poisson with rate [itex]\lambda[/itex] and say they take an exponential time on the line with parameter [itex]\mu[/itex] 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.
 

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