  In this paper we consider neighborhood load balancing in the context of
selfish clients. We assume that a network of n processors and m tasks is given.
The processors may have different speeds and the tasks may have different
weights. Every task is controlled by a selfish user. The objective of the user
is to allocate his/her task to a processor with minimum load. We revisit the
concurrent probabilistic protocol introduced in [6], which works in sequential
rounds. In each round every task is allowed to query the load of one randomly
chosen neighboring processor. If that load is smaller the task will migrate to
that processor with a suitably chosen probability. Using techniques from
spectral graph theory we obtain upper bounds on the expected convergence time
towards approximate and exact Nash equilibria that are significantly better
than the previous results in [6]. We show results for uniform tasks on
non-uniform processors and the general case where the tasks have different
weights and the machines have speeds. To the best of our knowledge, these are
the first results for this general setting.
