How to calculate probability of users generating distributed events reaching n events per 15 minutes?

We have games & apps that connect to services such as Facebook and Twitter to fetch information. These services have various rate-limit caps that you cannot exceed - typically based on a 15 minute window of time. If we exceed this rate - the service blocks for a while. This makes my users sad. For a concrete example - you can only fetch a users tweets about 300 times per 15 minute window.

I would like to estimate how many users it might take before I could reasonably expect to hit this quota of 300 events in any given 15 minute window. This is so I can look ahead from our usage trends and maybe cache this data or pool it or whatever.

Assumptions:

• A user can be expected to use the app for 5 minutes then quit
• There is a 1/5 any given user will access this twitter feed during a session (based on actual usage)
• I would restrict this usage to daylight hours (Not yet concerned about lower levels of - users at night - most of our gamers are in the US mainland and not night owls)
• I assume usage is evenly spread across this time.

I see it is connected to questions such as this: ( How to calculate the probability of two events happening within a certain time period using exponential distribution ) but I can't quite connect the dots :)

Thanks for any input!

• Just for clarification: The model is that $20\%$ of the $N$ users will cause $1$ "query" event within a duration of $5$ minutes. $N$ is to be determined from the number of installations divided by the time window you see fit, sliced into $5$-minute slices. The relevant Poisson-parameter is $\lambda = \frac15 + \frac4{25} + \frac{16}{125}$ (the probability of access in at least one of the three $5$-minute slices we are watching.) – AlexR Jan 28 '15 at 20:42
• Sorry for the assumption formatting. I had lots of newlines that were removed when I posted - I'll try to reformat it now :) Any given user has a %20 chance to cause a query within the 5 minutes they run. I think that's equivalent to what you wrote. So am I correct that 1/λ is the number of users where I can expect it to happen? I'm guessing from the term "mean" on the Wikipedia page of Exponential Distributions here ;) – S Wilkinson Jan 29 '15 at 15:50