Passengers and empty cars appear at random places and wait to meet. A match ends both waiting spells, while someone who waits too long may leave before being served. Even before accounting for pickup travel, the balance between arrivals on the two sides shapes how long each side remains idle.
Four scenes vary rider and vehicle arrival frequencies: both scarce, cars abundant, riders abundant, and both abundant. The animations show matching events and departures while recording passenger and driver waiting separately. Repeated simulations then produce two three-dimensional waiting surfaces over the arrival-rate combinations. The lesson is that improving one side's wait can worsen the other's, and that a busier market can match more quickly because participants have more potential partners nearby. These fitted surfaces describe the video's stochastic matching rules; they are not a universal formula for every real ride-hailing city.
An available taxi is useful to passengers even while it is carrying nobody. More idle vehicles spread through a service area make pickup faster, reducing the delay between requesting a ride and entering a car. Keeping those vehicles available, however, costs driver time and money.
Watch film
One ride has two different clocks. For the passenger, the trip begins with a request and includes waiting to be matched, waiting for the assigned car to arrive, and riding to the destination. For the driver, the service cycle begins while available, continues through pickup and transport, and ends only when the car is ready for another match.
Watch film
An on-demand car, a shared ride, and a fixed-route bus use vehicles very differently. An exclusive car offers direct travel but spends substantial time and distance on each passenger. Pooling shares some of that work at the cost of detours and coordination. A bus gains scale when enough people use a common corridor, but offers less flexible access.
Watch film