We consider Ramp Metering (RM) at the microscopic level subject to vehicle
following safety constraints for a freeway with arbitrary number of on- and
off-ramps. The arrival times of vehicles to the on-ramps, as well as their
destinations are modeled by exogenous stochastic processes. Once a vehicle is
released from an on-ramp, it accelerates towards the free flow speed if it is
not obstructed by another vehicle; once it gets close to another vehicle, it
adopts a safe gap vehicle following behavior. The vehicle exits the freeway
once it reaches its destination off-ramp. We design traffic-responsive RM
policies that maximize the throughput. For a given routing matrix, the
throughput of a RM policy is characterized by the set of on-ramp arrival rates
for which the expected queue size at all the on-ramps remain bounded. The
proposed RM policies work in synchronous cycles during which an on-ramp does
not release more vehicles than its queue size at the beginning of the cycle.
Moreover, all the policies operate under vehicle following safety constraints,
where new vehicles are released only if there is sufficient gap between
vehicles on the mainline at the moment of release. We provide three mechanisms
under which each on-ramp: (i) pauses release for a time interval at the end of
a cycle, or (ii) adjusts the release rate during a cycle, or (iii) adopts a
conservative safe gap criterion for release during a cycle. All the proposed
policies are reactive, meaning that they only require real-time traffic
measurements without the need for demand prediction. The throughput of these
policies is characterized by studying stochastic stability of the induced
Markov chains, and is proven to be maximized when the merging speed at all the
on-ramps equals the free flow speed. Simulations are provided to illustrate the
performance of our policies and compare with a well-known RM policy from the
literature.