  This is an expository review paper illustrating the ``martingale method'' for
proving many-server heavy-traffic stochastic-process limits for queueing
models, supporting diffusion-process approximations. Careful treatment is given
to an elementary model -- the classical infinite-server model $M/M/\infty$, but
models with finitely many servers and customer abandonment are also treated.
The Markovian stochastic process representing the number of customers in the
system is constructed in terms of rate-1 Poisson processes in two ways: (i)
through random time changes and (ii) through random thinnings. Associated
martingale representations are obtained for these constructions by applying,
respectively: (i) optional stopping theorems where the random time changes are
the stopping times and (ii) the integration theorem associated with random
thinning of a counting process. Convergence to the diffusion process limit for
the appropriate sequence of scaled queueing processes is obtained by applying
the continuous mapping theorem. A key FCLT and a key FWLLN in this framework
are established both with and without applying martingales.
