  We consider the classical problem of minimizing the total weighted flow-time
for unrelated machines in the online \emph{non-clairvoyant} setting. In this
problem, a set of jobs $J$ arrive over time to be scheduled on a set of $M$
machines. Each job $j$ has processing length $p_j$, weight $w_j$, and is
processed at a rate of $\ell_{ij}$ when scheduled on machine $i$. The online
scheduler knows the values of $w_j$ and $\ell_{ij}$ upon arrival of the job,
but is not aware of the quantity $p_j$. We present the {\em first} online
algorithm that is {\em scalable} ($(1+\eps)$-speed
$O(\frac{1}{\epsilon^2})$-competitive for any constant $\eps > 0$) for the
total weighted flow-time objective. No non-trivial results were known for this
setting, except for the most basic case of identical machines. Our result
resolves a major open problem in online scheduling theory. Moreover, we also
show that no job needs more than a logarithmic number of migrations. We further
extend our result and give a scalable algorithm for the objective of minimizing
total weighted flow-time plus energy cost for the case of unrelated machines
and obtain a scalable algorithm. The key algorithmic idea is to let jobs
migrate selfishly until they converge to an equilibrium. Towards this end, we
define a game where each job's utility which is closely tied to the
instantaneous increase in the objective the job is responsible for, and each
machine declares a policy that assigns priorities to jobs based on when they
migrate to it, and the execution speeds. This has a spirit similar to
coordination mechanisms that attempt to achieve near optimum welfare in the
presence of selfish agents (jobs). To the best our knowledge, this is the first
work that demonstrates the usefulness of ideas from coordination mechanisms and
Nash equilibria for designing and analyzing online algorithms.
