Transient results for M/M/1/c queues via path counting

We find combinatorially the probability of having n customers in an M/M/1/c queueing system at an arbitrary time t when the arrival rate λ and the service rate μ are equal, including the case c = ∞. Our method uses pathcounting methods and finds a bijection between the paths of the type needed for the queueing model and paths of another type which are easy to count. The bijection involves some interesting geometric methods.