  Motivated by interest in pedestrian traffic we study two lanes
(one-dimensional lattices) of length $L$ that intersect at a single site. Each
lane is modeled by a TASEP (Totally Asymmetric Exclusion Process). The
particles enter and leave lane $\sigma$ (where $\sigma=1,2$) with probabilities
$\alpha_\sigma$ and $\beta_\sigma$, respectively. We employ the `frozen
shuffle' update introduced in earlier work [C. Appert-Rolland et al, J. Stat.
Mech. (2011) P07009], in which the particle positions are updated in a fixed
random order. We find analytically that each lane may be in a `free flow' or in
a `jammed' state. Hence the phase diagram in the domain
$0\leq\alpha_1,\alpha_2\leq 1$ consists of four regions with boundaries
depending on $\beta_1$ and $\beta_2$. The regions meet in a single point on the
diagonal of the domain. Our analytical predictions for the phase boundaries as
well as for the currents and densities in each phase are confirmed by Monte
Carlo simulations.
