Queues with superposition arrival processes in heavy traffic

To help provide a theoretical basis for approximating queues with superposition arrival processes, we prove limit theorems for the queue-length process in a [Sigma] GIi/G/s model, in which the arrival process is the superposition of n independent and identically distributed stationary renewal processes each with rate n-1. The traffic intensity [rho] is allowed to approach the critical value one as n increases. If n(1-[rho])2 --> c, 0 < c < [infinity], then a limit is obtained that depends on c. The two iterated limits involving [rho] and n, which do not agree, are obtained as c --> 0 and c --> [infinity].