Existence of Limiting Distributions in the GI/G/s Queue

A new proof is given for the Kiefer-Wolfowitz (Kiefer, J., J. Wolfowitz. 1955. On the theory of queues with many servers. Trans. Amer. Math. Soc.78 1–18.) theorem: In the GI/G/s queue the sequence of workload vectors, and thus also the sequence of waiting times, has a proper limiting distribution independent of the initial conditions if and only if p < 0.1 (assuming the interarrival times and service times have finite means). From stochastic monotonicily, convergence is easy to establish if the system starts off empty. The limit is shown lo be proper here by bounding the standard FCFS system by the corresponding s-server system with a cyclic assignment discipline, using a recent construction of Wolff (Wolff, R. W. 1977. An upper bound for multi-channel queues. J. Appl. Probab.14 884–888.). The general initial condition case is treated by appropriately bounding the original sequence above and below by sequences of countable-state Markov chains for which zero is always a regenerative state. This paper thus illustrates how recent stochastic comparison and continuity results can be applied to prove limit theorems. The bounding systems are also of special interest in regenerative simulation because they provide a way lo increase the number of regeneration points.