Phase Diagram Analysis Based on a Temporal-Spatial Queueing Model

In this paper, we propose a simple temporal-spatial queueing model to quantitatively address some typical congestion patterns that were observed around on/off-ramps. In particular, we examine three prime factors that play important roles in ramping traffic scenarios: the time τ
_in
for a vehicle to join a jam queue, the time τ
_out
for this vehicle to depart from this jam queue, and the time interval
T
for the ramping vehicle to merge into the mainline. Based on Newell's simplified car-following model, we show how τ
_in
changes with the main road flow rate
q
_main
. Meanwhile,
T
is the reciprocal of the ramping road flow rate
q
_ramp
. Thus, we analytically derive the macroscopic phase diagram plotted on the
q
_main
-versus-
q
_ramp
plane and τ
_in
-versus-
T
plane based on the proposed model. Further study shows that the new queueing model not only reserves the merits of Newell's model on the microscopic level but helps quantify the contributions of these parameters in characterizing macroscopic congestion patterns as well. Previous approaches distinguished phases merely through simulations, but our model could derive analytical boundaries for the phases. The phase transition conditions obtained by this model agree well with simulations and empirical observations. These findings help reveal the origins of some well-known phenomena during traffic congestion.