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November, 1994 The Net Output Process of a System with Infinitely many Queues
P. A. Ferrari, L. R. G. Fontes
Ann. Appl. Probab. 4(4): 1129-1144 (November, 1994). DOI: 10.1214/aoap/1177004907


We study a system of infinitely many queues with Poisson arrivals and exponential service times. Let the net output process be the difference between the departure process and the arrival process. We impose certain ergodicity conditions on the underlying Markov chain governing the customer path. These conditions imply the existence of an invariant measure under which the average net output process is positive and proportional to the time. Starting the system with that measure, we prove that the net output process is a Poisson process plus a perturbation of order 1. This generalizes the classical theorem of Burke which asserts that the departure process is a Poisson process. An analogous result is proven for the net input process.


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P. A. Ferrari. L. R. G. Fontes. "The Net Output Process of a System with Infinitely many Queues." Ann. Appl. Probab. 4 (4) 1129 - 1144, November, 1994.


Published: November, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0812.60081
MathSciNet: MR1304777
Digital Object Identifier: 10.1214/aoap/1177004907

Primary: 60K35
Secondary: 60K25 , 90B15 , 90B22

Keywords: Burke's theorem , departure process , net output process , queuing networks , Zero range process

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.4 • No. 4 • November, 1994
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