Expected waiting time with $m$ tellers and $n$ customers at a bank Suppose we have the following scenario:

There are $n$ customers at a bank who wish to speak with bank tellers.
  There are $m$ tellers, and each of the $n$ customers is willing to
  speak to any of the $m$ tellers. Rather than forming individual lines
  for each teller, the customers form a single queue, and the next
  person in line goes to the next available teller. The amount of time
  each customer spends with the teller is exponentially distributed with
  mean $\lambda$. Compute the expected waiting time for the person next
  in line in the queue. (You may neglect the time it takes someone to
  walk from the queue to the teller.)

Anyone know how to solve this? This isn't a homework question, but rather a question from a class that I took back in college (it is of particular relevance at the moment as I am needing to compute this sort of a quantity for a piece of software that I am writing).
 A: Because you imply there are people queued, I assume $n > m.$
Each person in line begins service when the next teller
becomes available. That happens at rate $m/\lambda,$
so the mean waiting time in the queue for the next customer
is $\lambda/m.$ The mean time until that customer finishes
being served is $\lambda/m + \lambda.$ 
Notes: (1) To avoid potential confusion, texts on queueing models typically
use $\lambda$ for the rate of arrival of new customers, and
$\mu$ for the service rate.  Means of exponential distributions
are the reciprocals of rates.
(2) The distribution of the waiting time for the minimum of
$k$ exponential waiting times (next available server) is 
exponential with a rate of $k$ times the rate for
individual servers (assuming all rates equal). 
(3) If you want to look for more on this topic, I suggest you
read about $M/M/k$ queues. The first M stands for (Markov,
memoryless, or exponential) interarrival times, the second
M for exponential service times, and the k for the number
of servers. 
