  We consider a system with N unit-service-rate queues in tandem, with
exogenous arrivals of rate lambda at queue 1, under a back-pressure (MaxWeight)
algorithm: service at queue n is blocked unless its queue length is greater
than that of next queue n+1. The question addressed is how steady-state queues
scale as N goes to infinity. We show that the answer depends on whether lambda
is below or above the critical value 1/4: in the former case queues remain
uniformly stochastically bounded, while otherwise they grow to infinity.
  The problem is essentially reduced to the behavior of the system with
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.
