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Transient behavior of the Halfin-Whitt diffusion. (English) Zbl 1235.60133

The paper studies the Halfin-Whitt regime, when the number of servers and the input rate increase infinitely in a consistent way resulting in a specific form of the heavy traffic regime, where the (scaled) queue size is described by the so-called Halfin-Whitt diffusion. It is shown that this diffusion fluctuates between the Brownian motion (corresponding to the heavy-traffic regime for single-server system) and the Ornstein-Uhlenbeck process which is indeed the heavy-traffic limit for an infinite server queue. The authors solve the forward Kolmogorov equation to find the transience density of the mentioned diffusion using Laplace transforms. Then, they invert transform to find explicit expression for the density. Also, the asymptotics of the density, as time goes to infinity, is obtained in an explicit form.

MSC:

60K25 Queueing theory (aspects of probability theory)
60J60 Diffusion processes
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
34E05 Asymptotic expansions of solutions to ordinary differential equations
90B22 Queues and service in operations research
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