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Diffusion models for double-ended queues with renewal arrival processes. (English) Zbl 1331.60177
Summary: We study a double-ended queue where buyers and sellers arrive to conduct trades. When there is a pair of buyer and seller in the system, they immediately transact a trade and leave. Thus there cannot be a non-zero number of buyers and sellers simultaneously in the system. We assume that sellers and buyers arrive at the system according to independent renewal processes, and they would leave the system after independent exponential patience times. We establish fluid and diffusion approximations for the queue length process under a suitable asymptotic regime. The fluid limit is the solution of an ordinary differential equation, and the diffusion limit is a time-inhomogeneous asymmetric Ornstein-Uhlenbeck process (O-U process). A heavy traffic analysis is also developed, and the diffusion limit in the stronger heavy traffic regime is a time-homogeneous asymmetric O-U process. The limiting distributions of both diffusion limits are obtained. We also show the interchange of the heavy traffic and steady state limits.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 60J60 Diffusion processes 60F05 Central limit and other weak theorems 60F17 Functional limit theorems; invariance principles 60K05 Renewal theory 90B22 Queues and service in operations research
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