Queueing Analysis of a Large-Scale Bike Sharing System through   Mean-Field Theory
  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 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 steps: Firstly, a multi-dimensional Markov process is set up for expressing the states of the bike sharing system, and the empirical 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 the mean-field computation. Secondly, the martingale limit is employed to investigate the limiting behavior of the empirical 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 simply, and specifically, these lead to numerical computation of 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. 
Quan-Lin Li, Chang Chen, Rui-Na Fan, Liang Xu and Jing-Yu Ma
math.PR cs.PF math.DS