Abstract
Consider a countably infinite collection of interacting queues, with a queue located at each point of the ddimensional 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 meanfield 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.
Original language  English 

Pages (fromto)  29292987 
Number of pages  59 
Journal  Annals of Applied Probability 
Volume  29 
Issue number  5 
DOIs  
Publication status  Published  18 Oct 2019 
Keywords
 Coupling from the past
 Information theory
 Interacting queues
 Interference field
 Loynes' construction
 Mass transport theorem
 Monotonicity
 Particle systems
 Percolation
 Positive correlation
 Queueing theory
 Rate conservation principle
 Stability and instability
 Stationary distribution
 Wireless network
ASJC Scopus subject areas
 Statistics and Probability
 Statistics, Probability and Uncertainty
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Sergey Foss
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Actuarial Mathematics & Statistics  Professor
Person: Academic (Research & Teaching)