Discrete flow networks: bottleneck analysis and fluid approximations
Mathematics of Operations Research
Asymptotic analysis of closed queueing networks with bottlenecks
Proceedings of the IFIP WG 7.3 International Conference on Performance of Distributed Systems and Integrated Communication Networks
Asymptotic Expansions for the Congestion Period for the M/M/∞ Queue
Queueing Systems: Theory and Applications
HEAVY TRAFFIC APPROXIMATIONS FOR A SYSTEM OF INFINITE SERVERS WITH LOAD BALANCING
Probability in the Engineering and Informational Sciences
Queueing Systems: Theory and Applications
On the fluid limit of the M/G/∞ queue
Queueing Systems: Theory and Applications
Approximation of initial loading of infinite-server systems
Automation and Remote Control
Multiserver Loss Systems with Subscribers
Mathematics of Operations Research
Dynamics of New Product Introduction in Closed Rental Systems
Operations Research
Two-parameter heavy-traffic limits for infinite-server queues
Queueing Systems: Theory and Applications
Large-time asymptotics for the Gt/Mt/st+GIt many-server fluid queue with abandonment
Queueing Systems: Theory and Applications
Queues with Many Servers and Impatient Customers
Mathematics of Operations Research
The Impact of Dependent Service Times on Large-Scale Service Systems
Manufacturing & Service Operations Management
Two-parameter heavy-traffic limits for infinite-server queues with dependent service times
Queueing Systems: Theory and Applications
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This paper studies the heavy-traffic behavior of a closed system consisting of two service stations. The first station is an infinite server and the second is a single server whose service rate depends on the size of the queue at the station. We consider the regime when both the number of customers, n, and the service rate at the single-server station go to infinity while the service rate at the infinite-server station is held fixed. We show that, as n\rightarrow\infty, the process of the number of customers at the infinite-server station normalized by n converges in probability to a deterministic function satisfying a Volterra integral equation. The deviations of the normalized queue from its deterministic limit multiplied by \sqrt{n} converge in distribution to the solution of a stochastic Volterra equation. The proof uses a new approach to studying infinite-server queues in heavy traffic whose main novelty is to express the number of customers at the infinite server as a time-space integral with respect to a time-changed sequential empirical process. This gives a new insight into the structure of the limit processes and makes the end results easy to interpret. Also the approach allows us to give a version of the classical heavy-traffic limit theorem for the G/GI/\infty queue which, in particular, reconciles the limits obtained earlier by Iglehart and Borovkov.