  Networks of channels conveying particles are often subject to blockages due
to the limited carrying capacity of the individual channels. If the channels
are coupled, blockage of one causes an increase in the flux entering the
remaining open channels leading to a cascade of failures. Once all channels are
blocked no additional particle can enter the system. If the blockages are of
finite duration, however, the system reaches a steady state with an exiting
flux that is reduced compared to the incoming one. We propose a stochastic
model consisting of $N_c$ channels each with a blocking threshold of $N$
particles. Particles enter the system according to a Poisson process with the
entering flux of intensity $\Lambda$ equally distributed over the open
channels. Any particle in an open channel exits at a rate $\mu$ and a blocked
channel unblocks at a rate $\mu^*$. We present a method to obtain the exiting
flux in the steady state, and other properties, for arbitrary $N_c$ and $N$ and
we present explicit solutions for $N_c=2,3$. We apply these results to compare
the efficiency of conveying a particulate stream of intensity $\Lambda$ using
different channel configurations. We compare a single "robust" channel with a
large capacity with multiple "fragile" channels with a proportionately reduced
capacity. The "robust" channel is more efficient at low intensity, while
multiple, "fragile" channels have a higher throughput at large intensity. We
also compare $N_c$ coupled channels with $N_c$ independent channels, both with
threshold $N=2$. For $N_c=2$ if $\mu^*/\mu>1/4$, the coupled channels are
always more efficient. Otherwise the independent channels are more efficient
for sufficiently large $\Lambda$.
