  We consider the problem of scheduling communication on optical WDM
(wavelength division multiplexing) networks using the light-trails technology.
We seek to design scheduling algorithms such that the given transmission
requests can be scheduled using minimum number of wavelengths (optical
channels). We provide algorithms and close lower bounds for two versions of the
problem on an $n$ processor linear array/ring network. In the {\em stationary}
version, the pattern of transmissions (given) is assumed to not change over
time. For this, a simple lower bound is $c$, the congestion or the maximum
total traffic required to pass through any link. We give an algorithm that
schedules the transmissions using $O(c+\log{n})$ wavelengths. We also show a
pattern for which $\Omega(c+\log{n}/\log\log{n})$ wavelengths are needed. In
the {\em on-line} version, the transmissions arrive and depart dynamically, and
must be scheduled without upsetting the previously scheduled transmissions. For
this case we give an on-line algorithm which has competitive ratio
$\Theta(\log{n})$. We show that this is optimal in the sense that every on-line
algorithm must have competitive ratio $\Omega(\log{n})$. We also give an
algorithm that appears to do well in simulation (for the classes of traffic we
consider), but which has competitive ratio between $\Omega(\log^2n/\log
\log{n})$ and $O(\log^2n)$. We present detailed simulations of both our
algorithms.
