Annals of Applied Probability

Interference queueing networks on grids

Abishek Sankararaman, François Baccelli, and Sergey Foss

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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.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 5 (2019), 2929-2987.

Dates
Received: August 2018
Revised: January 2019
First available in Project Euclid: 18 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1571385626

Digital Object Identifier
doi:10.1214/19-AAP1470

Mathematical Reviews number (MathSciNet)
MR4019879

Zentralblatt MATH identifier
07155063

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J25: Continuous-time Markov processes on general state spaces 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 90B18: Communication networks [See also 68M10, 94A05] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

Keywords
Rate conservation principle mass transport theorem monotonicity positive correlation percolation stationary distribution coupling from the past Loynes’ construction particle systems interacting queues queueing theory stability and instability information theory interference field wireless network

Citation

Sankararaman, Abishek; Baccelli, François; Foss, Sergey. Interference queueing networks on grids. Ann. Appl. Probab. 29 (2019), no. 5, 2929--2987. doi:10.1214/19-AAP1470. https://projecteuclid.org/euclid.aoap/1571385626


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