## 3 Model
### Distribution Model for Taxi Arrival Time Intervals
Here, \(D\) denotes the total number of travel orders received by the ride-hailing platform at time \(t_{i}\), and \(\alpha D\) represents the number of orders assigned to taxis by the platform. In the passenger order cancellation model considered in this paper, we account for situations where passengers cancel online orders upon encountering empty cruising taxis while waiting for pick-up. In such cases, the number of occupied empty cruising taxis is the same as the number of taxis with canceled online orders. Therefore, from an overall perspective, the number of taxis changing status is effectively zero. Hence, in this model, the number of offline operating taxis, \(N_{i}\), at any given time, \(t_{i}\), is calculated as \(N_{i}=N-\alpha D\). When order overflow occurs, some drivers choose to no longer cooperate with the ride-hailing platform and instead opt to only accept orders through cruising due to the reduced cost of finding passengers offline. Let \(\theta\) represent the proportion of taxis accepting offline orders at time \(t_{i}\). Then, the number of offline operating taxis at any given time, \(t_{i}\), can be expressed as Equation (1). \[N_{i}=N\theta+(1-\theta)(N-\alpha D)=N-(1-\theta)\alpha D \tag{1}\]  
When taxis are distributed in a single lane on the road, the average spacing distance between offline cruising taxis can be expressed as \(S/N_{i}\), where \(S\) represents the total mileage of urban roads (km). Considering that taxis travel on two-way lanes, the time, \(t_{d}\), that it takes for a cruising taxi to traverse the urban roads unilaterally at any given time, \(t_{i}\), can be represented by Equation (2), where \(v\) is the average travel speed of taxis (km/h). The model in this paper, based on the assumption of taxi single-file distribution, aims to provide a theoretical analysis framework for general scenarios. Given the complexity of urban roads and buildings, which leads to variations in vehicle density across different areas, a regional division approach can be considered to classify the city into multiple grid areas with differing levels of congestion. By calculating the time taken for taxis to traverse roads in each area, the overall traversal time can be obtained, thereby enhancing the accuracy of the model. \[t_{d}=\frac{2S/N_{i}}{v}=\frac{2S}{[N-(1-\theta)\alpha D]v} \tag{2}\]  
Taking into account the vacant rate of cruising taxis, \(\eta\) (i.e., the proportion of time or distance traveled without passengers to the total operating time or total distance traveled during the taxi's operation), some of the offline operating taxis, \(N_{i}\), are already occupied with passengers. Therefore, the average time that a passenger needs to wait for the first available cruising taxi, \(T_{F}\), can be expressed as Equation (3). \[T_{F}=\frac{t_{d}}{\eta}=\frac{2S/N_{i}}{v}=\frac{2S}{[N-(1-\theta)\alpha D]v\eta} \tag{3}\]  
The arrival of taxis is a random event occurring independently at an average rate. Therefore, the arrival time of cruising taxis follows an exponential distribution with a parameter, \(\lambda\). Let \(T_{X}\) represent the arrival time of a cruising taxi. Then, the probability distribution function for the taxi arrival time intervals is given by \(f(T_{X})=\lambda e^{-\lambda T_{X}}\). Meanwhile, from Equation (3), we can derive the expression for \(\lambda\), as shown in Equation (4). Consequently, the probability distribution function for taxi arrival-time intervals during order overflow can be expressed as Equation (5). \[\lambda=\frac{1}{T_{F}}=\frac{[N-(1-\theta)\alpha D]v\eta}{2S} \tag{4}\]  
\[f(T_{X})=\frac{[N-(1-\theta)\alpha D]v\eta}{2S}e^{-\frac{[N-(1-\theta\mu D)v \eta]v\eta}{2S}TX} \tag{5}\]