A Mean-Field Matrix-Analytic Method for Bike Sharing Systems under   Markovian Environment
  To reduce automobile exhaust pollution, traffic congestion and parking difficulties, bike-sharing systems are rapidly developed in many countries and more than 500 major cities in the world over the past decade. In this paper, we discuss a large-scale bike-sharing system under Markovian environment, and propose a mean-field matrix-analytic method in the study of bike-sharing systems through combining the mean-field theory with the time-inhomogeneous queues as well as the nonlinear QBD processes. Firstly, we establish an empirical measure process to express the states of this bike-sharing system. Secondly, we apply the mean-field theory to establishing a time-inhomogeneous MAP(t)/MAP(t)/1/K+2L+1 queue, and then to setting up a system of mean-field equations. Thirdly, we use the martingale limit theory to show the asymptotic independence of this bike-sharing system, and further analyze the limiting interchangeability as N goes to infinity and t goes to infinity. Based on this, we discuss and compute the fixed point in terms of a nonlinear QBD process. Finally, we analyze performance measures of this bike-sharing system, such as, the mean of stationary bike number at any station and the stationary probability of problematic stations. Furthermore, we use numerical examples to show how the performance measures depend on the key parameters of this bike-sharing system. We hope the methodology and results of this paper are applicable in the study of more general large-scale bike-sharing systems. 
Quan-Lin Li and Rui-Na Fan
math.PR cs.PF math.DS