Elementary Methods for Some Waiting Time Problems

Let W_n be the waiting time of the nth arrival in a single server queuing system, and T = min{n ∣ W_n = 0}. The distribution of T is obtained for the system M/G/1, and by applying a duality principle, Smith's theorem for GI/M/1 is deduced. The waiting time distribution in the system E_k/D/1 is also obtained.