For stable FIFO GI/G/s queues, s ⩾ 2, we show that finite (k + 1)st moment of service time, S, is not in general necessary for finite kth moment of steady-state customer delay, D, thus weakening some classical conditions of Kiefer and Wolfowitz. Further, we demonstrate that the conditions required for E[Dk] < ∞ are closely related to the magnitude of traffic intensity ρ (defined to be the ratio of the expected service time to the expected interarrival time). In particular, if ρ is less than the integer part of s/2, then E[D] < ∞ if E[S3/2] < ∞, and E[Dk] < ∞ if E[Sk] < ∞, k ⩾ 2. On the other hand, if s – 1 < ρ < s, then E[Dk] < ∞ if and only if E[Sk+1] < ∞, k ⩾ 1. Our method of proof involves three key elements; a novel recursion for delay which reduces the problem to that of a reflected random walk with dependent increments, a new theorem for proving the existence of finite moments of the steady-state distribution of reflected random walks with stationary increments, and use of the classic Kiefer and Wolfowitz conditions.
