  Bike-sharing systems are exploding in cities around the world as more people
are adopting sustainable transportation solutions for their everyday commutes.
However, as more people use the system, riders often encounter that bikes or
docks might not be available when they arrive to a station. As a result, many
systems like CitiBike and Divvy provide riders with information about the
network via smartphone apps so that riders can find stations with available
bikes. However, not all customers have adopted the use of these smartphone apps
for their station selection. By combining customer choice modeling and finite
capacity queueing models, we explore the impact of the smartphone app
technology to increase throughput and reduce blocking in bike sharing systems.
To this end, we prove a mean-field limit and a central limit theorem for an
empirical process of the number of stations with $k$ bikes. We also prove limit
theorems for a new process called the ratio process, which characterizes the
proportion of stations whose bike availability ratio lies within a particular
partition of the interval [0,1]. For the mean field limit, we prove that the
equilibrium exists, is unique, and that the stationary distribution of the
empirical measure converges to a Dirac mass at the same equilibrium, thus
showing an interchange of limits result ($\lim_{t\rightarrow
\infty}\lim_{N\rightarrow \infty}=\lim_{N\rightarrow \infty}\lim_{t\rightarrow
\infty}$). Our limit theorems provide insight on the mean, variance, and sample
path dynamics of large scale bike-sharing systems. Our results illustrate that
if we increase the proportion of customers that use smartphone app information,
the entropy of the bike sharing network is reduced, and riders experience less
blocking in the network.
