TY - JOUR

T1 - Interference queueing networks on grids

AU - Sankararaman, Abishek

AU - Baccelli, François

AU - Foss, Sergey

PY - 2019/10

Y1 - 2019/10

N2 - 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.

AB - 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.

KW - Coupling from the past

KW - Information theory

KW - Interacting queues

KW - Interference field

KW - Loynes' construction

KW - Mass transport theorem

KW - Monotonicity

KW - Particle systems

KW - Percolation

KW - Positive correlation

KW - Queueing theory

KW - Rate conservation principle

KW - Stability and instability

KW - Stationary distribution

KW - Wireless network

KW - EXISTENCE

KW - SYSTEM

KW - STABILITY

KW - monotonicity

KW - positive correlation

KW - interference field

KW - TRANSIENCE

KW - percolation

KW - information theory

KW - wireless network

KW - queueing theory

KW - interacting queues

KW - POWER

KW - coupling from the past

KW - stationary distribution

KW - stability and instability

KW - mass transport theorem

KW - particle systems

UR - http://www.scopus.com/inward/record.url?scp=85075126420&partnerID=8YFLogxK

U2 - 10.1214/19-AAP1470

DO - 10.1214/19-AAP1470

M3 - Article

AN - SCOPUS:85075126420

VL - 29

SP - 2929

EP - 2987

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 5

ER -