Stastistics question!

I need to know how to calculate the peak (max) number of visitors to
my web site within a single hour.  I already know the total number of
visitors who
come to my web site during a typical day. Assume that
visitors-per-hour is normally distributed over 24 hours.

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Request for Question Clarification by elmarto-ga on 14 Sep 2005 03:48 PDT

Hello jb!
If the distribution is normal, then theoretically there is no upper
limit to the number of visitors you may receive in a single hour. What
you could say, using this distribution is something like "there's a
95% chance that there will be no more 1,000 visitors this hour", or
"there's a 99.5% chance that there will be no more than 1,200
visitors", etc; but a peak with 100% certainty can't be established.

If you are interested in this kind of answer, you would need to supply
the average number of visitors per hour, and also the variance of this
number.

Best regards,
elmarto

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Clarification of Question by jbambrosino-ga on 14 Sep 2005 15:40 PDT

elmarto, the 95% or 99% confidence interval is fine.  Actually I'm
really just looking to have a general formula for calculating this
information, so I can't give you any actual numbers.  Isn't there a
way to approximate a 'typical' normal distribution?

FYI this is to estimate the max amount of bandwidth I need for a web
hosting business.  So, with the formula in hand, here's what I would
envision as a telephone call with a prospective customer:

me: "how many visitors does your site get per month?"
customer: "about 200,000"
me: "what is the average number of pages viewed per site visitor?"
customer: "about 5"
me: "ok, you serve 1m pages per month, or about 33,333 per day. 
During an average day, the busiest hour will have ____ pages served,
which requires X bandwidth and thus the hosting cost is $Y."

So it's the "____" pages served that I need to calculate.

The goal here is really just to give me a way to estimate my own costs
before giving a price quote to a prospective customer.  I will only
know the total # of pageviews per month, and I'm comfortable with any
assumptions that have to be made about the curve/distribution in order
to come up with the general formula.

thanks
JB

Hello JB,

Interesting question! Just a few comments:

1. The kind of distribution assumption you make is crucial to the
answer you want. E.g., assuming it to be a Gaussian will lead to a
very different answer as opposed to a Laplace distribution. Once that
is assumed (with reasonable justification), then you can estimate the
parameters of the distribution (in case of Gaussian it is the mean and
variance) using some actual data collected. Many ways to do that, and
the most simplest/common is Maximum Likelihood Estimation. Only have
the mean of # of visits per hour will not work, as you can imagine: in
order to get an estimation of the peak value you need something which
can tell you the spread of the distribution, not just the center.

2. The max # of visits for an avg. day is actually bounded above, as
you have the total # of visits on a average day. This is useful since
even if you use a distribution with unbounded support (like Gaussian)
you may still want to consider a truncated Gaussian which removes the
tails (mainly the right tail). This typically gives you a more
reliable estimation.

3. You need to get some data (# of vists of each hour) in order to get
a sensible estimation. Actually a lot of webservices can do that, by
just simply adding a block of javascript in the webpage. For example,
http://www.webstats4u.com

Good luck!