Approximation of Queues in Bike-Sharing Systems With Finite Docks

This paper presents a closed queuing network model to address bike queues in bike-sharing systems with finite docks. The model tackles issues of bike spillover and user attrition due to fully occupied docks and bike shortages at stations. The objective is to determine throughput rates and other performance metrics for these systems. To overcome computational challenges, we propose an approximation algorithm based on the developed model. Our analysis reveals intrinsic properties of bike-sharing systems with finite docks: (i) The effective system throughput rate increases with bike fleet size and eventually converges to a ceiling value. (ii) Adding more docks at stations can unnecessarily increase or even decrease the effective throughput rate. (iii) Under certain conditions, the system can reach a self-balancing state, avoiding bike surpluses or deficiencies at each station and maximising throughput. (iv) Users can successfully return bikes with a limited number of tries, provided there is at least one station on their route with a non-zero probability of having available docks. A small-scale artificial example and a case study demonstrate the accuracy and applicability of the approximation algorithm and the properties of the systems.