Consider a countably infinite collection of coupled queuesrepresenting a large wireless network with a queue  at each point of the d-dimensional integer grid . These queues have independent Poisson arrivals, but are coupled through their service rateswhich is the signal to interference ratio of wireless network theory. More precisely, the service discipline is translation invariant and of the processor sharing type,with the service rate in each queue slowed down, when the neighboring queues have a larger workload. The dynamics is infinite dimensional Markov, with each queue having a non compact state space . It is neither reversible nor asymptotically product form, as in the mean-field setting . Coupling and percolation techniques are first used to show that this dynamics has well defined trajectories. Coupling from the past techniques of the Loynes' type are then proposed to build its minimal stationary regime. This regime is the one obtained when starting from the all empty initial condition in the distant past. The rate conservation principle of Palm calculus is then used to identify the stability condition of this system, namely the condition on the interference sequence and arrival rates guaranteeing the finiteness of this minimal regime . Remarkably, the rate conservation principle also provides a closed form expression for its mean queue size. When the stability condition holds, this minimal solution is the unique stationary regime, provided it has finite second moments, and this is the case if the arrival rate is small enough . In addition, there exists a range of small initial conditions for which the dynamics is attracted to the minimal regime. Surprisingly however , there exists another range of larger though finite initial conditions for which the dynamics diverges, even though stability criterion holds.
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.
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