Predicting a maximum value with little data My problem is i'm trying to figure out how many servers might be required to handle a theoretical maximal load of data requests.  To do that I need to know what the maximum number of requests in a second might be.
I have a fairly limited set of data that 156,000 transactions occurred in one hour.  This is the maximum recorded transactions for a rolling 60 minute period over 4 months.  
4500 devices have access to the server and each client-server transaction takes and average of 45 seconds.  I was wanting to predict what the maximum number of client-server transactions in a single second might be with 95% certainty.  
Each server can service a maximum of 15 concurrent transactions before they start to experience lag.
It's a long time since I've done any stats so if someone could point me in the right direction then that'd be brilliant.  I figured I might need to know the max/min/average transaction time to be able to get to the variance but i'm probably just babbling now.
Thanks a million.
 A: There is wisdom and experience in the circumspect way this question is worded, for it would be foolhardy to answer it with a calculation. Any estimates based on an hour's worth of data might apply to that one hour, but the prospect that they will accurately characterize future times is very, very poor: one would have to be extremely confident that the load will continue to behave like the load observed during that hour.  It's impossible to attach a legitimate confidence to this assumption.
You need to think about the ways in which the load might vary.  In addition to unknown sources of variation, which will have to be modeled probabilistically, you can expect trends over time; cyclic patterns associated with activity that occurs periodically by the second, day, week, and year; loads that spike in response to special events; and perhaps other causes.  At a minimum, you need to characterize these trends, assess the typical variation around them, and then allow for surprises.  You may wish to model or forecast the special events that can spike the load, too.
To collect the necessary data, you need representative measures of load, perhaps at one-second intervals, taken during a set of times and over a sufficient time period to assess all these trends and cycles.  Therefore you should plan to monitor the server load with these analytical aims in mind and return to the problem of estimating the maximum number of servers once you have some relevant data.
In the meantime you also should think about how to balance the cost of maintaining additional servers against the risk of becoming overloaded: at some point there are diminishing returns and identifying that point requires you to quantify this trade-off.

Mark Twain famously lampooned this situation in his Life on the Mississippi (1883). After citing 160 years of data about how floods were altering the river's length, he wrote,

Now, if I wanted to be one of those ponderous scientific people, and
  'let on' to prove ... what will occur in the far future by what has
  occurred in late years, what an opportunity is here! ...
In the space of one hundred and seventy-six years the Lower
  Mississippi has shortened itself two hundred and forty-two miles. That
  is an average of a trifle over one mile and a third per year.
  Therefore, any calm person, who is not blind or idiotic, can see that
  in the “Old Oolitic Silurian Period,” just a million years ago next
  November, the Lower Mississippi River was upwards of one million three
  hundred thousand miles long, and stuck out over the Gulf of Mexico
  like a fishing-rod. And by the same token any person can see that
  seven hundred and forty-two years from now the Lower Mississippi will
  be only a mile and three-quarters long, and Cairo and New Orleans will
  have joined their streets together, and be plodding comfortably along
  under a single mayor and a mutual board of aldermen. There is
  something fascinating about science. One gets such wholesale returns
  of conjecture out of such a trifling investment of fact.

(Chapter 17.)  The last line beautifully sums it up.
A: Here is one approach (if I understand what you are trying for).
If the time of the transaction is short enough that each second can be considered independent of the others and the different devices are independent of each other (you will want to think through these assumptions, time of day may make all devices more likely or less likely to request a transaction), then we can model this as a binomial problem.
The average number of transactions per second is $\frac{156000}{3600} = 43.33$, and with 4500 devices that works out to a probability of about $0.00963$ of each device requesting a transaction each second.  If you want enough servers that you can handle all requests 95% of the time then the binomial tells us that for the given probability and 4500 devices then 54 servers will be enough to cover 95.18% of cases (this is just finding the value from the binomial that includes at least 95% probability).  That would mean that during just under 5% of seconds you would receive more requests that you have servers and some requests would need to wait (though if transactions take a fraction of a second, some servers could be ready again).
If things are not independent, or tranactions could last more than a second, or other complications, then you will want to look into queuing theory for tools to help you figure this out.
