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October 2019 Interference queueing networks on grids
Abishek Sankararaman, François Baccelli, Sergey Foss
Ann. Appl. Probab. 29(5): 2929-2987 (October 2019). DOI: 10.1214/19-AAP1470

Abstract

Consider a countably infinite collection of interacting queues, with a queue located at each point of the $d$-dimensional integer grid, having independent Poisson arrivals, but dependent service rates. The service discipline is of the processor sharing type, with the service rate in each queue slowed down, when the neighboring queues have a larger workload. The interactions are translation invariant in space and is neither of the Jackson Networks type, nor of the mean-field type. Coupling and percolation techniques are first used to show that this dynamics has well-defined trajectories. Coupling from the past techniques are then proposed to build its minimal stationary regime. The rate conservation principle of Palm calculus is then used to identify the stability condition of this system, where the notion of stability is appropriately defined for an infinite dimensional process. We show that the identified condition is also necessary in certain special cases and conjecture it to be true in all cases. Remarkably, the rate conservation principle also provides a closed-form expression for the mean queue size. When the stability condition holds, this minimal solution is the unique translation invariant stationary regime. In addition, there exists a range of small initial conditions for which the dynamics is attracted to the minimal regime. Nevertheless, there exists another range of larger though finite initial conditions for which the dynamics diverges, even though stability criterion holds.

Citation

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Abishek Sankararaman. François Baccelli. Sergey Foss. "Interference queueing networks on grids." Ann. Appl. Probab. 29 (5) 2929 - 2987, October 2019. https://doi.org/10.1214/19-AAP1470

Information

Received: 1 August 2018; Revised: 1 January 2019; Published: October 2019
First available in Project Euclid: 18 October 2019

zbMATH: 07155063
MathSciNet: MR4019879
Digital Object Identifier: 10.1214/19-AAP1470

Subjects:
Primary: 60D05, 60J25, 60K35, 68M20, 90B18

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 5 • October 2019
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