Large number of queues in tandem: Scaling properties under back-pressure algorithm

We consider a system with N unit-service-rate queues in tandem, with exogenous arrivals of rate λ at queue 1, under a back-pressure (MaxWeight) algorithm: service at queue n is blocked unless its queue length is greater than that of the next queue n + 1. The question addressed is how steady-state queues scale as N →∞. We show that the answer depends on whether λ is below or above the critical value 1/4: in the former case the queues remain uniformly stochastically bounded, while otherwise they grow to infinity. The problem is essentially reduced to the behavior of the system with an infinite number of queues in tandem, which is studied using tools from interacting particle systems theory. In particular, the criticality of load 1/4 is closely related to the fact that this is the maximum possible flux (flow rate) of a stationary totally asymmetric simple exclusion process.