Queueing Analysis of a Large-Scale Bike Sharing System through Mean-Field Theory

Abstract

The bike sharing systems are fast increasing as a public transport mode in urban short trips, and have been developed in many major cities around the world. A major challenge in the study of bike sharing systems is that some large-scale and complex queueing networks have to be applied through multi-dimensional Markov processes, while their discussion always suffers a common difficulty: State space explosion. For this reason, this paper provides a mean-field computational method to study such a large-scale bike sharing system. Our mean-field computation is established in the following three steps: Firstly, a multi-dimensional Markov process is set up for expressing the states of the bike sharing system, and the empirical measure process of the multi-dimensional Markov process is given to partly overcome the difficulty of state space explosion. Based on this, the mean-field equations are derived by means of a virtual time-inhomogeneous M(t)/M(t)/1/K queue whose arrival and service rates are determined by using some mean-field computation. Secondly, the martingale limit is employed to investigate the limiting behavior of the empirical measure process, the fixed point is proved to be unique so that it can be computed by means of a nonlinear birth-death process, the asymptotic independence of this system is discussed, and specifically, these lead to numerical computation for the steady-state probability of the problematic (empty or full) stations. Finally, some numerical examples are given for valuable observation on how the steady-state probability of the problematic stations depends on some crucial parameters of the bike sharing system.