  We begin developing the theory of the generator comparison approach of
Stein's method for continuous-time Markov processes where jumps are driven by
clocks having general distributions, as opposed to exponential distributions.
This paper handles models with a single general clock.
  Using the workload process in the $G/G/1$ queueing system as a driving
example, we develop two variants of the generator comparison approach for
models with a single general clock: the original, which we call the limiting
approach, and the recently proposed prelimit approach. The approaches are duals
of one another, yielding distinct bounds on the diffusion approximation error
of the steady-state workload.
  We also contribute to the theory of heavy-traffic approximations for the
$G/G/1$ system. Under some assumptions on the interarrival time distribution,
the prelimit approach allows us to bound the diffusion approximation error in
terms of $G/G/1$ model primitives. For example, when the interarrival time has
a nonincreasing hazard rate that is bounded from above, we show that the
diffusion approximation error of the expected workload is bounded in terms of
the first three moments of the interarrival and service-time distributions, as
well as the upper bound on the interarrival hazard rate.
