Fleet-sizing and service availability for a vehicle rental system via closed queueing networks

In this paper, we address the problem of determining the optimal fleet size for a vehicle rental company and derive analytical results for its relationship to vehicle availability at each rental station in the company's network of locations. This work is motivated by the recent surge in interest for bicycle and electric car sharing systems, one example being the French program Vélib (2010). We first formulate a closed queueing network model of the system, obtained by viewing the system from the vehicle's perspective. Using this framework, we are able to derive the asymptotic behavior of vehicle availability at an arbitrary rental station with respect to fleet size. These results allow us to analyze imbalances in the system and propose some basic principles for the design of system balancing methods. We then develop a profit-maximizing optimization problem for determining optimal fleet size. The large-scale nature of real-world systems results in computational difficulties in obtaining this exact solution, and so we provide an approximate formulation that is easier to solve and which becomes exact as the fleet size becomes large. To illustrate our findings and validate our solution methods, we provide numerical results on some sample networks.