## 6 Conclusions  
Passenger transportation hubs connect inter-city large-capacity transportation options with the intra-city transportation network, which has significant impacts on the traffic management of a city. For a better understanding of the traffic characteristics of the roadway linking to a passenger transportation hub and finding bottlenecks in the system, this paper derives a fundamental diagram model from threshold queueing theory considering different designs of the pick-up zone.  
In this paper, by treating passenger boarding in the pick-up zone as a service process in a queue system, the proposed model takes human-vehicle interaction as a process of serving vehicles, where the service rate is controlled by different conditions of the queuing system: If the queue is in a non-congested condition, the service rate is high, and if the queuing system converts to the congested condition, the service is performed with a low service rate. Based on the management rules, the model supposes that vehicles receive service through a batch-by-batch mode and uses conversions between batch-based variables and vehicle-based variables to derive the fundamental diagram.  
The experiments showed that design parameters, including the number of passenger lanes at the vehicle queueing area \(n\), the number of vehicle lanes at the passenger waiting area \(m\) and the buffer of the single vehicle lane \(N\), etc., impact the capacity and the degree of capacity drop. For example, when the arrival rate of the inter-city transportation vehicle \(\lambda_{a}\) increases, \(\mu_{1}\) fluctuates within a certain range, but \(\mu_{2},\ U\) and \(D\) have a significant decrease; \(C\) and \(k_{C}\) fluctuate within a certain range, while there is a violent decrease in the value of \(\varphi_{d}\). Moreover, it can be seen that our model achieves better fitness both in comparison with the classical fundamental diagram models and with the other queueing theory-based models.  
Understanding the unique speed-density-flow characteristics of the corridor will be beneficial to the optimization of passenger flow and vehicle flow organization in hub measurement. However, the fundamental diagram reflects only the static characteristics of the system and ignores the time-variant information about the evolution of congestions caused by the pick-up zone. For a deep implementation of the results, there is still a need for dynamic volume-delay functions that can connect the fundamental diagram and the actual control of the vehicle flow and passenger flow. Moreover, the assumption of M/M/1 queueing cannot fit all scenarios in reality; a more general model based on G/G/1 queueing (e.g., PH/PH/1 [33]) would be the focus of future work. Thus, in future work, we will explore the construction of dynamic volume-delay functions based on the FD studied in this paper. Meanwhile, we will study more scalable and applicable FD models based on the PH/PH/1 queuing model.  
Data curation, H.Z.; funding acquisition, H.Z.; methodology, H.Z., Y.Y., G.G. and K.Y.; supervision, J.C.; writing--original draft, H.Z., Y.Y., G.G. and K.Y.; writing--review and editing, J.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by (1) The National Key Research and Development Program of China, grant number 2018 YFB1601200. (2) China Postdoctoral Science Foundation, grant number 2021M700186 and (3) China Postdoctoral Science Foundation, grant number 2021T140003.  
Not applicable.  
Thanks to [PERSON] for his support and help in data processing.  
The authors declare no conflict of interest.