Throughput of water fountains Suppose we have a row of water fountains at a stadium.  The quantity of water fountains is fixed at N, so we can have up to N simultaneous water-drinkers.
People arrive at the water fountain area.  If there are no free fountains, they immediately get mad.  Otherwise, they quench their thirst.  For simplicity, all people take the exact same amount of time, measured in seconds, to quench their thirst.
As the facility manager, I want to guarantee that, with some high probability, everyone will find a free fountain immediately upon arrival.  In other words, I want a low probability of a customer getting mad because they have to wait any length of time.  Let's call that the "service guarantee".
I want to know how many customers per second my fountains can serve while upholding the service guarantee.
Inputs I can think of:


*

*Duration of water-drinking

*Distribution of people's arrival

*Desired probability of meeting the service guarantee


Outputs I want:


*

*How many customers per second can we serve?


I'm looking for an equation.
 A: A mostly formula-free introduction to queues.
Exponential inter-arrival times. In this kind of situation, it is typical to assume that people arrive according
to a Poisson process at a particular rate $\lambda.$ That means that inter-arrival times between customers are distributed $\mathsf{Exp}(rate = \lambda).$
The rate is specified as customers per unit time. Then the expected time
between arrivals is $1/\lambda.$
The exponential distribution is used because it has simple PDF and CDF, and
also because the exponential distribution has a no-memory property that makes
it unnecessary to take past history into account. Also, in many real-life
situations, inter-arrival times really are approximately exponentially distributed;
sometimes almost simultaneous, sometimes at random spaced relatively far
apart. 
Exponentially distributed service times. It is also typical to assume that the service time is exponentially distributed.
The rate of service is often designated as $\mu$ (not a mean). In your
case, this would imply that the average length of time a person spends at
a water fountain is $1/\mu.$ Again here the exponential assumption is often made
because of mathematical simplicity, even though in real-life situations
service times may not fit an exponential distribution extremely well.
Notation "M/M/N". In your case there are $N$ 'servers' (water fountains). With exponential
arrivals and service times, your situation would be called an 'M/M/N' queue.
The first 'M' means exponential arrivals (Markovian, or memoryless), the
second 'M' means exponential service times, and the 'N' is the numer of
servers.  
You can look at a book (or web pages) on queueing theory, stochastic processes,
or operations research for formulas that describe the behavior of an M/M/N
queue at 'steady state', given the values of $\lambda, \mu,$ and $N.$
Steady state. Roughly, 'steady state' means that the process has been going on long
enough that any anomalies in starting up the process are no longer of
consequence. In your case, you would start out with all water fountains
free, so that the first few customers wouldn't have to wait. But at steady
state a known average number of the fountains are free, and a known average
number of customers are waiting for the next free fountain. 
In your example you need to have $\lambda < N\mu$ in order for the process to achieve steady state. Otherwise, the capacity for service would not be enough to take
care of the arrivals, and the length of the waiting line ('queue') for service would
gradually grow toward infinity.
There are steady-state formulas for the number of people in the system,
the number of people waiting for service, the average time spent in the
system, the average number of free servers, and so on.
Characteristics of a Queue. Realistically, in most queueing systems it is not feasible to 'guarantee'
that no one will ever need to wait for service. That's because exponential
arrivals can be very 'clumpy', so you never know when there will be a
chance very busy period. But it may be possible to design a system so that
most people get served immediately and that those who have to wait will
not have an unduly long average wait.
Various notations. Finally, a word of caution on notation. In queueing formulas you may find
notations like $L, L_Q, W, W_Q.$ These may be averages, not random variables,
and not fixed quantities. For example, $W_Q$ is often the expected waiting time
'in the queue'; that is, the expected waiting time before being served.
Also, notations can vary among authors, $N$ (number of servers) may be $n$ or $m$; $L$ (average number in the system), may be $\bar L, N,$ or $\bar N.$ 
