  This paper presents an interactive bathtub model for describing the traffic
dynamics of ride-sourcing vehicles including non-shared taxis and ride-pooling
cars. A city with a network of undifferentiated streets and solely served by
ride-sourcing services is assumed to facilitate the modeling, isolate the
congestion contribution, and accordingly develop control strategies. The
proposed model is parsimonious with only input information of the total lane
length of the network, the in-flux of demand, and the travel distance
distributions. The output of the model, however, captures not only the traffic
dynamics of vehicles but also the dynamic states of passengers in ride-pooling
services in terms of the total number, the remaining travel distances, and the
queue of unmatched requests at any system time. Useful system metrics can be
exploited for use of the authorities to monitor, predict, and control the
traffic, as well as for the TNCs to determine the fleet sizes, dispatch
vehicles, and measure the service productivity. For illustration, we propose a
robust control rule to manage the traffic efficiently and avoid gridlock, and
also present time-varying ride-pooling sizes to eliminate the queue of
unmatched requests. Numerical examples demonstrate the effectiveness of the
proposed model and the control and operation strategies.
