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Heavy-traffic limits for infinite-server queues in series with . . .

By Guodong Pang and Ward Whitt


We study a stochastic network with two service stations in series, each equipped with infinitely many servers, together with a probabilistic and time-dependent splitting mechanism after service completions at the first station. External arrivals enter the system at the first station according to a general arrival process with time-varying arrival rate, assumed to satisfy a functional central limit theorem (FCLT). The service-time distributions are allowed to be non-exponential. At each station, the service times are identically distributed but allowed to be weakly dependent. We establish heavy-traffic limits (first a FWLLN and then a FCLT refinement) for the two-parameter stochastic processes {(Qe 1(t, y), Qe 2(t, y)) : t ≥ 0, 0 ≤ y ≤ t}, where Qe l (t, y), l = 1, 2, represents the number of customers in the lth service station at time t with elapsed service times less than or equal to y. The FCLT limit is a continuous two-parameter Gaussian processes (random field). We give explicit formulas for the time-dependent means and variances of the resulting Gaussian approximation when the arrival limit process is a Brownian motion

Year: 2011
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