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A diffusion regime with non-degenerate slowdown June 18, 2009; Revised June 27, 2010 and June 15, 2011

Summary: A diffusion regime with non-degenerate slowdown
Rami Atar
June 18, 2009; Revised June 27, 2010 and June 15, 2011
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
Keywords: diffusion limits, many-server queue, heavy traffic, conventional diffusion regime,
ED and QED regimes, non-degenerate slowdown regime
1 Introduction
Diffusion asymptotic analysis of queueing systems in heavy traffic has been the subject of exten-


Source: Atar, Rami - Department of Electrical Engineering, Technion, Israel Institute of Technology


Collections: Engineering