A Diffusion Regime with Nondegenerate Slowdown

We study a diffusion regime---earlier considered by Gurvich, Mandelbaum, Shaikhet, and Whitt in the case of the M/M/N queue which may be regarded in a sense that we make precise, as a midpoint between two well-known heavy traffic diffusion regimes, the conventional and the quality and efficiency driven regimes. Unlike the other two, this regime, which we call the nondegenerate slowdown regime, enjoys the property that delay and service time are of the same order of magnitude, a property that is often desirable from a modeling viewpoint. Our main result is that in the case of heterogeneous exponential multiserver systems, this regime gives rise to new limit processes for the sojourn time. In particular, the joint limit law of the delay and service time processes is identified as a reflected Brownian motion and an independent process, whose marginal is a size-biased mixture of exponentials. Our results also motivate the formulation and study of new diffusion control problems based on sojourn time cost.