# Modeling concurrent internet users

I'm feeling generous in the new year and want to open my Wifi connection to the public. I want to estimate the effect that $N$ additional users on my router would have on download times. In other words, how does my waiting time for an average piece of data degrade as the number of users increases?

To keep things simple I've formalized the problem in the following way:

N:         the number of users.
B:         the bandwidth I have (e.g. 1024 KB/s).
D(x):    the distribution of file sizes (for simplicity sake, I'm going to assume that each piece of data is a file $x$ KB large).
I(y):     the distribution of interval between requests in seconds
WT:     denote the waiting time in seconds for a file.

Now I know that in the real-world a user makes parallel requests, but for this particular version of the problem a single user will only issue requests in sequence.

My question is: what is E[WT]?

For example, if $N = 1, E[WT]$ is simply $E[D(x)]/b$.

But for $N = 2$, I get all confused because the users could have overlapping requests that could affect each others waiting time, and subsequently, $I(y)$.

How do I model this problem? What branch of mathematics is suitable for it?

• Queueing theory might be helpful. – Jonathan Christensen Jan 17 '13 at 23:24
• Looked at queuing theory but there's a lot of material there and I'm not sure where to begin. – user1988778 May 2 '13 at 18:48