  In order to obtain Markov heavy-traffic approximations for infinite-server
queues with general non-exponential service-time distributions and general
arrival processes, possibly with time-varying arrival rates, we establish
heavy-traffic limits for two-parameter stochastic processes. We consider the
random variables $Q^e(t,y)$ and $Q^r(t,y)$ representing the number of customers
in the system at time $t$ that have elapsed service times less than or equal to
time $y$, or residual service times strictly greater than $y$. We also consider
$W^r(t,y)$ representing the total amount of work in service time remaining to
be done at time $t+y$ for customers in the system at time $t$. The
two-parameter stochastic-process limits in the space $D([0,\infty),D)$ of
$D$-valued functions in $D$ draw on, and extend, previous heavy-traffic limits
by Glynn and Whitt (1991), where the case of discrete service-time
distributions was treated, and Krichagina and Puhalskii (1997), where it was
shown that the variability of service times is captured by the Kiefer process
with second argument set equal to the service-time c.d.f.
