This paper describes the heavy-traffic behavior of an M/G/1
last-in-first-out preemptive resume queue. An appropriate framework for
the analysis is provided by measure-valued processes. In particular, the paper
exploits the setting of recent works by Le Gall and Le Jan.Their
finite-measure-valued exploration process corresponds to our
RES-measure (residual services measure) process, that captures all the
relevant information about the evolution of the queue, while their
height process corresponds to the queue-length process. The
heavy-traffic “diffusion” approximations for the RES-measure and
the queue-length processes are derived under the usual second moment
assumptions on the service distributions. The tightness of queue lengths
argument uses estimates for the total size and height of large
Galton–Watson trees.