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ON THE DISCRETE-TIME G/GI/∞ QUEUE*

Published online by Cambridge University Press:  25 September 2008

Iddo Eliazar
Iddo Eliazar
Affiliation:
Department of Technology Management, Holon Institute of Technology, Holon 58102, IsraelE-mail:eliazar@post.tau.ac.il
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Abstract

The discrete-time G/GI/∞ queue model is explored. Jobs arrive to an infinite-server queuing system following an arbitrary input process X; job sizes are general independent and identically distributed random variables. The system's output process Y (of job departures) and queue process N (tracking the number of jobs present in the system) are analyzed. Various statistics of the stochastic maps X↦ Y and X↦ N are explicitly obtained, including means, variances, autocovariances, cross-covariances, and multidimensional probability generating functions. In the case of stationary inputs, we further compute the spectral densities of the stochastic maps, characterize the fixed points (in the L2 sense) of the input–output map, precisely determine when the output and queue processes display either short-ranged or long-ranged temporal dependencies, and prove a decomposition result regarding the intrinsic L2 structure of general stationary G/GI/∞ systems.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 22 , Issue 4 , October 2008 , pp. 557 - 585
Copyright
Copyright © Cambridge University Press 2008

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