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Limits for Cumulative Input Processes to Queues

Whitt, Ward

We establish functional central limit theorems (FCLTs) for a cumulative input process to a fluid queue from the superposition of independent on—off sources, where the on periods and off periods may have heavy-tailed probability distributions. Variants of these FCLTs hold for cumulative busy-time and idle-time processes associated with standard queueing models. The heavy-tailed on-period and off-period distributions can cause the limit process to have discontinuous sample paths (e.g., to be a non-Brownian stable process or more general Lévy process) even though the converging processes have continuous sample paths. Consequently, we exploit the Skorohod M1 topology on the function space D of right-continuous functions with left limits. The limits here combined with the previously established continuity of the reflection map in the M1 topology imply both heavy-traffic and non-heavy-traffic FCLTs for buffer-content processes in stochastic fluid networks.

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Title
Probability in the Engineering and Information Sciences
DOI
https://doi.org/10.1017/S0269964800142019

More About This Work

Academic Units
Industrial Engineering and Operations Research
Published Here
September 19, 2017