  Filtration, flow in narrow channels and traffic flow are examples of
processes subject to blocking when the channel conveying the particles becomes
too crowded. If the blockage is temporary, which means that after a finite time
the channel is flushed and reopened, one expects to observe a maximum
throughput for a finite intensity of entering particles. We investigate this
phenomenon by introducing a queueing theory inspired, circular Markov model.
Particles enter a channel with intensity $\lambda$ and exit at a rate $\mu$. If
$N$ particles are present at the same time in the channel, the system becomes
blocked and no more particles can enter until the blockage is cleared after an
exponentially distributed time with rate $\mu^*$. We obtain an exact expression
for the steady state throughput (including the exiting blocked particles) for
all values of $N$. For $N=2$ we show that the throughput assumes a maximum
value for finite $\lambda$ if $\mu^*/\mu < 1/4$. The time-dependent throughput
either monotonically approaches the steady state value, or reaches a maximum
value at finite time. We demonstrate that, in the steady state, this model can
be mapped to a previously introduced non-Markovian model with fixed transit and
blockage times.
  We also examine an irreversible, non-Markovian blockage process with constant
transit time exposed to an entering flux of fixed intensity for a finite time
and we show that the first and second moments of the number of exiting
particles are maximized for a finite intensity.
