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, that is, in a sense that we make precise, 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, that we call the non-degenerate 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 multi-server 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. AMS subject classifications: 60K25, 60J60, 60F17, 90B22, 68M20