  As a favorite urban public transport mode, the bike sharing system is a
large-scale and complicated system, and there exists a key requirement that a
user and a bike should be matched sufficiently in time. Such matched behavior
makes analysis of the bike sharing systems more difficult and challenging. This
paper considers a more general large-scale bike sharing system from two
important views: (a) Bikes move in an irreducible path graph, which is related
to geographical structure of the bike sharing system; and (b) Markovian arrival
processes (MAPs) are applied to describe the non-Poisson and burst behavior of
bike-user (abbreviated as user) arrivals, while the burstiness demonstrates
that the user arrivals are time-inhomogeneous and space-heterogeneous in
practice. For such a complicated bike sharing system, this paper establishes a
multiclass closed queueing network by means of some virtual ideas, for example,
bikes are abstracted as virtual customers; stations and roads are regarded as
virtual nodes. Thus user arrivals are related to service times at station
nodes; and users riding bikes on roads are viewed as service times at road
nodes. Further, to deal with this multiclass closed queueing network, we
provide a detailed observation practically on physical behavior of the bike
sharing system in order to establish the routing matrix, which gives a
nonlinear solution to compute the relative arrival rates in terms of the
product-form solution to the steady-state probabilities of joint queue lengths
at the virtual nodes. Based on this, we can compute the steady-state
probability of problematic stations, and also deal with other interesting
performance measures of the bike sharing system. We hope that the methodology
and results of this paper can be applicable in the study of more general bike
sharing systems through multiclass closed queueing networks.
