  We present a study of exclusion process on a peculiar topology of network
with two intersected lanes, competing for the particles in a reservoir with
finite capacity. To provide a theoretical ground for our findings, we exploit
mean-field approximation along with domain-wall theory. The stationary
properties of the system including phase transitions, density profiles and
position of the domain-wall are derived analytically. Under the similar
dynamical rules, the particles of both the lanes interact only at the
intersected site. The symmetry of system is maintained till number of particles
do not exceed total number of sites. However, beyond this the symmetry breaking
phenomenon occurs resulting in the appearance of asymmetric phases and
continues to persist even for infnite number of particles. The complexity of
phase diagram shows a non-monotonic behaviour with increasing number of
particles in the system. A bulk induced shock appears in a symmetric phase,
whereas, a boundary induced shock is observed in symmetric as well as
asymmetric phase. Monitoring the location of shock with increasing entry of
particles, we explain the possible phase transitions. The theoretical results
are supported by extensive Monte Carlo simulations and explained using simple
physical arguments.
