Transient results for \(M/M/1/c\) queues via path counting (Q843389)

Summary: 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 and the service rate are equal, including the case \(c =\infty \). 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.