Statistical Multiplexing
[This is a variation of problem 8 in the book Kurose Ross]
Suppose 40 users are sharing an outgoing link that statistically multiplexes the packets they are sending. Each user transmits only 10% of the time.
a)Find the probability that exactly n of these users are transmitting simultaneously.
b)Find the probability that n or more users are transmitting simultaneously. Compute the value for n=11. Give your result with an accuracy of 4 decimal digits after the period.
c)Solve the general case: Suppose that a link has a capacity R and that each user transmits at a rate that is a fraction f of the link capacity. Given N users that share this link, what is the probability that more than 1/f users are transmitting simultaneously? (In the example given above, f=0.1 and R=1Mbps, and N=40.) Plot your result over an interesting range.
Hint: Use the binomial distribution to model part a). You may use the central limit theorem to approximate parts b) & c), although this is not required; in particular if you use such software as Mathematica, which allows exact computation. If you approximate, recall that the mean μ of a binomial distribution with trials is nnp=μand its standard deviation σ is given as )1(pnp-=σ.
Consider doing part c) before part b).